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Chapter 2

Fluid mechanics & Orlicz spaces

Abstract

We extend some classical tools from fluid mechanics – Korn's inequality, the Bogovskiĭ operator and the pressure recovery – to the setting of Orlicz spaces. As a special case the known $L^{p}$ -theory is included as well as the case of Orlicz spaces generated by a nice Young function (i.e., under $Δ_{2}$ and $\nabla_{2}$ condition). In the general case there is some loss of integrability, for instance in the limit cases $L \log L \to L^{1}$ and $L^{\infty} \to Exp (L)$ . The results are shown to be optimal in the sense of Orlicz spaces.

Keywords

Orlicz spaces; Divergence equation; Bogovskiĭ operator; Negative norm theorem; Pressure recovery; Korn's inequality

A crucial tool in the mathematical approach to the behaviour of Newtonian fluids is Korn's inequality: given a bounded open domain $G \subset R^{d}$ , $d ⩾ 2$ , with Lipschitz boundary ∂G we have

$\int_{G} | \nabla v |^{2} d x ⩽ 2 \int_{G} | ε (v) |^{2} d x$

(2.0.1)

for all $v \in W_{0}^{1, 2} (G)$ . For smooth functions with compact support (2.0.1) can be shown by integration by parts. The general case is treated by approximation. A first proof was given by Korn in [104]. We note that variants of Korn's inequality in $L^{2}$ have been established by Courant and Hilbert [53], Friedrichs [84], Èidus [70] and Mihlin [114]. Many problems in the mathematical theory of generalized Newtonian fluids and mechanics of solids lead to the following question (compare for example the monographs of Málek, Nečas, Rokyta and Růžička [111], of Duvaut and Lions [66] and of Zeidler [143]): is it possible to bound a suitable energy depending on $\nabla v$ by the corresponding functional of $ε (v)$ , that is

$\int_{G} | \nabla v |^{p} d x ⩽ c (p, G) \int_{G} | ε (v) |^{p} d x$

(2.0.2)

for functions $v \in W_{0}^{1, p} (G)$ ? As shown by Gobert [91,92], Nečas [119], Mosolov and Mjasnikov [116], Temam [138] and later by Fuchs [74] this is true for all $1 < p < \infty$ (we remark that the inequality fails in the case $p = 1$ , see [120] and [52]).

A first step in the generalization of (2.0.2) is mentioned in [4]: Acerbi and Mingione prove a variant for the Young function

$A (t) = {(1 + t^{2})}^{\frac{p - 2}{2}} t^{2} .$

More precisely, they show that

${‖ \nabla v ‖}_{L^{A} (G)} ⩽ c (φ, G) {‖ ε (v) ‖}_{L^{A} (G)}$

(2.0.3)

for all functions $v \in W_{0}^{1, A} (G)$ . Although they only consider a special case they provide tools for much more general situations. Note that they only obtain inequalities in the Luxembourg-norm which is not appropriate in many situations (for example in regularity theory, see [35]). A general theorem is proved in [64], namely that

$\int_{G} A (| \nabla v - {(\nabla v)}_{G} |) d x ⩽ c (A, G) \int_{G} A (| ε (v) - {(ε (v))}_{G} |) d x$

(2.0.4)

for all $v \in W^{1, A} (G)$ , where A is a Young function satisfying the $Δ_{2}$ - and $\nabla_{2}$ -condition. Furthermore, Fuchs [75] obtains (2.0.4) for functions with zero traces and the same class of Young functions by a different approach. It is shown in [32] that the $Δ_{2}$ - and $\nabla_{2}$ -condition are also necessary for the inequality (2.0.4). We remark that the constitutive law

$S = \frac{A^{'} (| ε (v) |)}{| ε (v) |} ε (v)$

for a Young function A is a quite general model to describe the motion of generalized Newtonian fluids (see, i.e., [35], [25] and [59]).

In order to characterize the behaviour of Prandtl–Eyring fluids (see Chapter 4) Eyring [69] suggested the constitutive law

$S = D W (ε (u)), W (ε) = h (| ε |) = | ε | \log (1 + | ε |) .$

(2.0.5)

This leads in a natural way to the question about Korn's inequality in the space $L^{h} (G)$ . Since we have

$\tilde{h} (t) \approx t (\exp (t) - 1),$

the $\nabla_{2}$ -condition fails in this case, hence the results mentioned above do not apply. So the following question remains: given some integrability of the symmetric gradient – in the sense of Orlicz spaces – what is the best integrability for the full gradient we can expect?

A second fundamental question in fluid mechanics is the recovery of the pressure. It is common (and very useful) to study pressure-free formulations of (generalized) Navier–Stokes equations. So one starts by finding a velocity field which solves the corresponding system in the sense of distributions on divergence-free test-functions. Afterwards there is the question about the existence of the pressure function in order to have a weak solution in the sense of distributions.

Let us be a bit more precise and consider the equation

$\int_{G} H : \nabla φ d x = 0 for all φ \in C_{0, div}^{\infty} (G),$

(2.0.6)

where H is an integrable function (in case of a stationary generalized Navier–Stokes equation we have $H = S (ε (v)) + \nabla Δ^{- 1} f - v \otimes v$ ). Secondly, the pressure π is reconstructed in the sense that

$\int_{G} H : \nabla φ d x = \int_{Ω} π div φ d x for all φ \in C_{0}^{\infty} (G) .$

(2.0.7)

The existence of a pressure in the sense of distributions is a consequence of the classical theorem by De Rahm (see [131] for an appropriate version). It is also well-known that – if $1 < p < \infty$ – then $H \in L^{p} (G)$ implies $π \in L^{p} (G)$ . This result breaks down in the limit cases. Again motivated by the Prandtl–Eyring model the following question remains: given a function H solving (2.0.6) – located in some Orlicz space – what is the optimal integrability of the pressure in (2.0.7)?

In classical $L^{p}$ -spaces, both questions raised above can be answered by Nečas' negative norm theorem [119]. The negative Sobolev norm of the distributional gradient of a function $u \in L^{1} (G)$ can be defined as

${‖ \nabla u ‖}_{W^{- 1, p} (G)} = \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u div φ d x}{{‖ \nabla φ ‖}_{L^{p^{'}} (G)}} d x,$

(2.0.8)

where $1 ⩽ p ⩽ \infty$ . In (2.0.8), and in similar occurrences throughout this chapter, we tacitly assume that the supremum is extended over all functions v which do not vanish identically. We remark that the quantity on the right-hand side of (2.0.8) agrees with the norm of ∇u, when regarded as an element of the dual of $W_{0}^{1, p^{'}} (G)$ . Nečas showed that, if G is regular enough – a bounded Lipschitz domain, say – and $1 < p < \infty$ , then the $L^{p} (G)$ norm of a function is equivalent to the $W^{- 1, p} (G)$ norm of its gradient. Namely, there exist positive constants $C_{1} = C_{1} (G, p)$ and $C_{2} = C_{2} (d)$ , such that

$C_{1} {‖ u - u_{G} ‖}_{L^{p} (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1, p} (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L^{p} (G)}$

(2.0.9)

for every $u \in L^{1} (G)$ .

Using the formula

$Δ u = div V (u), V_{i j} (u) = 2 ε^{D} (u) - (\frac{1}{2} - \frac{1}{d}) (div u) I,$

where $ε^{D} = ε - \frac{1}{d} tr ε I$ , the proof of Korn's inequality based on (2.0.9) is elementary. Moreover, a combination of De Rahm's Theorem and (2.0.9) shows that if $H \in L^{p} (G)$ satisfies (2.0.6) then there is $π \in L^{p} (G)$ such that (2.0.7) holds.

In order to understand how Korn's inequality and the pressure recovery work in Orlicz spaces, we have to understand Orlicz versions of (2.0.9). Let A be a Young function, and let G be a bounded domain in $R^{d}$ . We define the negative Orlicz–Sobolev norm associated with A of the distributional gradient of a function $u \in L^{1} (G)$ as

${‖ \nabla u ‖}_{W^{- 1, A} (G)} = \sup_{φ \in C_{0}^{\infty} (G, R^{n})} \frac{\int_{G} u div φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} .$

(2.0.10)

The alternative notation $W^{- 1} L^{A} (G)$ will also occasionally be employed to denote the negative Orlicz–Sobolev norm $W^{- 1, A} (G)$ associated with the Orlicz space $L^{A} (G)$ . As (2.0.9) is known to break down in the limit cases, an Orlicz-version with the same Young function on both sides cannot hold in general. In fact, our Orlicz–Sobolev space version of the negative norm theorem involves pairs of Young functions A and B which obey the following balance conditions:

$t \int_{0}^{t} \frac{B (s)}{s^{2}} d s ⩽ A (c t) for t ⩾ 0,$

(2.0.11)

and

$t \int_{0}^{t} \frac{\tilde{A} (s)}{s^{2}} d s ⩽ \tilde{B} (c t) for t ⩾ 0,$

(2.0.12)

for some positive constant c. Note that the same conditions come into play in the study of singular integral operators in Orlicz spaces [47].

If either (2.0.11) or (2.0.12) holds, then A dominates B globally [48, Proposition 3.5]. In a sense, the assumptions (2.0.11) and (2.0.12) provide us with a quantitative information about how much weaker the norm ${‖ \cdot ‖}_{L^{B} (G)}$ is than ${‖ \cdot ‖}_{L^{A} (G)}$ . Under these assumptions a version of the negative norm theorem can be restored in Orlicz–Sobolev spaces.

Theorem 2.0.5

Let A and B be Young functions, fulfilling (2.0.11) and (2.0.12). Assume that G is a bounded domain with the cone property in $R^{d}$ , $d ⩾ 2$ . Then there exist constants $C_{1} = C_{1} (G, c)$ and $C_{2} = C_{2} (d)$ such that

$C_{1} {‖ u - u_{G} ‖}_{L^{B} (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1, A} (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L^{A} (G)}$

(2.0.13)

for every $u \in L^{1} (G)$ . Here, c denotes the constant appearing in (2.0.11) and (2.0.12).

Remark 2.0.6

Inequality (2.0.13) continues to hold even if conditions (2.0.11) and (2.0.12) are just fulfilled for $t ⩾ t_{0}$ for some $t_{0} > 0$ , but with constants $C_{1}$ and $C_{2}$ depending also on A, B, $t_{0}$ and $| G |$ . Indeed, the Young functions A and B can be replaced, if necessary, with Young functions which are equivalent near infinity and fulfil (2.0.11) and (2.0.12) for every $t > 0$ . Due to (1.2.11), such replacement leaves the quantities ${‖ \cdot ‖}_{L^{A} (G)}$ , ${‖ \cdot ‖}_{L^{B} (G)}$ and ${‖ \nabla \cdot ‖}_{W^{- 1, A} (G)}$ unchanged, up to multiplicative constants depending on A, B, $t_{0}$ and $| G |$ .

The situations when (2.0.11), or (2.0.12), holds with $B = A$ can be precisely characterized. Membership of A to $Δ_{2}$ is a necessary and sufficient condition for (2.0.12) to hold with $B = A$ [103, Theorem 1.2.1]. Therefore, under this condition, assumption (2.0.12) can be dropped in Theorem 2.0.5. On the other hand, $A \in \nabla_{2}$ if and only if $\tilde{A} \in Δ_{2}$ . Hence membership of A to $\nabla_{2}$ is a necessary and sufficient condition for (2.0.11) to hold with $B = A$ . Thus, under this condition, assumption (2.0.11) can be dropped in Theorem 2.0.5. Particularly, if $A \in Δ_{2} \cap \nabla_{2}$ , then both conditions (2.0.11) and (2.0.12) are fulfilled with $B = A$ . Hence, we have the following corollary which also follows from the results of [64].

Corollary 2.0.1

Assume that G is a bounded domain with the cone property in $R^{d}$ , $d ⩾ 2$ . Let A be a Young function in $Δ_{2} \cap \nabla_{2}$ . Then there are two constants $C = C (G, A)$ and $C_{2} = C_{2} (d)$ such that

$C_{1} {‖ u - u_{G} ‖}_{L^{A} (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1, A} (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L^{A} (G)}$

(2.0.14)

for every $u \in L^{1} (G)$ .

A typical situation where condition (2.0.11) does not hold with $B = A$ is when A grows linearly, or “almost linearly”, near infinity. In this case, $A \notin \nabla_{2}$ . In fact, as already mentioned, the standard negative norm theorem expressed by (2.0.9) breaks down in the borderline case $p = 1$ . On the other hand, condition (2.0.12) fails, with $B = A$ , if, for example, A has a very fast – faster than any power – growth. In this case, $A \notin Δ_{2}$ . Loosely speaking, the norm ${‖ \cdot ‖}_{L^{A} (G)}$ is now “close” to ${‖ \cdot ‖}_{L^{\infty} (G)}$ , and, as a matter of fact, equation (2.0.9) is not true with $p = \infty$ .

These, however, are not the only situations when (2.0.11), or (2.0.12), fail with $B = A$ . For instance, there are functions A which neither satisfy the $Δ_{2}$ condition, nor the $\nabla_{2}$ condition. Therefore, neither (2.0.12) nor (2.0.11) can hold with $B = A$ . In those cases $A (t)$ “oscillates” between two different powers $t^{p}$ and $t^{q}$ , with $1 < p < q < \infty$ . Functions of this kind are referred to as $(p, q)$ -growth in the literature. Partial differential equations, and associated variational problems, whose nonlinearity is governed by this growth, have been extensively studied. In the framework of non-Newtonian fluids, they have been analysed in [22].

All the circumstances described above can be handled via Theorem 2.0.5. A few examples involving customary families of Young functions are presented hereafter.

Example 1

Assume that $A (t)$ is a Young function equivalent to $t^{p} \log^{α} (1 + t)$ near infinity, where either $p > 1$ and $α \in R$ , or $p = 1$ and $α ⩾ 1$ . Hence, if $| G | < \infty$ , then

$L^{A} (G) = L^{p} \log^{α} L (G) .$

Assume that G is a bounded domain with the cone property in $R^{d}$ . If $p > 1$ , then $A \in Δ_{2} \cap \nabla_{2}$ , and hence Corollary 2.0.1 tells us that

$C_{1} {‖ u - u_{G} ‖}_{L^{p} \log^{α} L (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1} L^{p} \log^{α} L (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L^{p} \log^{α} L (G)}$

(2.0.15)

for every $u \in L^{1} (G)$ . However, if $p = 1$ , then $A \in Δ_{2}$ , but $A \notin \nabla_{2}$ . An application of Theorem 2.0.5 now implies

$C_{1} {‖ u - u_{G} ‖}_{L \log^{α - 1} L (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1} L \log^{α} L (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L \log^{α} L (G)}$

(2.0.16)

for every $u \in L^{1} (G)$ . In particular,

$C_{1} {‖ u - u_{G} ‖}_{L^{1} (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1} L \log L (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L \log L (G)}$

(2.0.17)

for every $u \in L^{1} (G)$ .

Example 2

Let $β > 0$ , and let $A (t)$ be a Young function equivalent to $\exp (t^{β})$ near infinity. Then

$L^{A} (G) = \exp L^{β} (G)$

if $| G | < \infty$ . One has that $A \in \nabla_{2}$ , but $A \notin Δ_{2}$ . Theorem 2.0.5 ensures that, if G is a bounded domain with the cone property in $R^{d}$ , then

$C_{1} {‖ u - u_{G} ‖}_{\exp L^{\frac{β}{β + 1}} (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1} \exp L^{β} (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{\exp L^{β} (G)}$

(2.0.18)

for every $u \in L^{1} (G)$ . Moreover,

$C_{1} {‖ u - u_{G} ‖}_{\exp L (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1} L^{\infty} (G)} ⩽ C_{2} {‖ u - u_{G} ‖}_{L^{\infty} (G)}$

(2.0.19)

for every $u \in L^{1} (G)$ .

Our approach is based on a study of the Bogovskiĭ operator [24] in Orlicz spaces in Theorem 2.1.7 which is already interesting itself. The Bogovskiĭ operator is a solution operator to the divergence equation with respect to zero boundary conditions. The continuity of the Bogovskiĭ operator implies the negative norm Theorem from which we can deduce both, the pressure recovery and Korn's inequality.

In Theorem 2.2.10 we give the precise statement of the pressure recovery in Orlicz spaces. In fact, $H \in L^{A} (G)$ implies $π \in L^{B} (G)$ where A and B are linked through (2.0.11) and (2.0.12). Moreover, the following inequality holds

$\int_{G} B (| π |) d x ⩽ \int_{G} A (C | H - H_{G} |) d x .$

Theorem 2.3.12 contains a version of Korn's inequality in general Orlicz spaces which says

$\int_{G} B (| \nabla u - {(\nabla u)}_{G} |) d x ⩽ \int_{G} A (C | ε (u) - {(ε (u))}_{G} |) d x .$

The final question which remains is the sharpness of the mentioned results. In Section 2.3 we are going to show that the balance conditions (2.0.11) and (2.0.12) are also necessary for a Korn's inequality. This implies that also the results about negative norms in Theorem 2.0.5 and the Bogovskiĭ operator in Theorem 2.1.6 are optimal.

2.1 Bogovskiĭ operator

Our proof of Theorem 2.0.5 relies upon an analysis of the divergence equation

${\begin{matrix} div u & = f in G, \\ u & = 0 on \partial G, \end{matrix}$

(2.1.20)

in Orlicz spaces which we analyse in the following. Subsequently, we set

$\begin{matrix} C_{0, ⊥}^{\infty} (G) & = {u \in C_{0}^{\infty} (G) : u_{G} = 0}, \\ L_{⊥}^{A} (G) & = {u \in L^{A} (G) : u_{G} = 0}, \end{matrix}$

where $u_{G} = ⨍_{G} u d x$ denotes the mean value of the function u.

Theorem 2.1.6

Assume that G is a bounded domain with the cone property in $R^{d}$ , $d ⩾ 2$ . Let A and B be Young functions fulfilling (2.0.11 and (2.0.12). Then there exists a bounded linear operator

${Bog}_{G} : L_{⊥}^{A} (G) \to W_{0}^{1, B} (G)$

(2.1.21)

such that

${Bog}_{G} : C_{0, ⊥}^{\infty} (G) \to C_{0}^{\infty} (G)$

(2.1.22)

and

$div ({Bog}_{G} f) = f in G$

(2.1.23)

for every $f \in L_{⊥}^{A} (G)$ . In particular, there exists a constant $C = C (G, c)$ such that

${‖ \nabla ({Bog}_{G} f) ‖}_{L^{B} (G)} ⩽ C {‖ f ‖}_{L^{A} (G)}$

(2.1.24)

and

$\int_{G} B (| \nabla ({Bog}_{G} f) |) d x ⩽ \int_{G} A (C | f |) d x$

(2.1.25)

for every $f \in L_{⊥}^{A} (G)$ . Here, c denotes the constant appearing in (2.0.11) and (2.0.12).

Although it will not be used for our main purposes, we state in Theorem 2.1.7 below a result parallel to Theorem 2.1.6, dealing with a version of problem (2.1.20) in the case when the right-hand side of the equation is in divergence form. Namely,

${\begin{matrix} div u & = div g & in G, \\ u & = 0 & on \partial G, \end{matrix}$

(2.1.26)

where $g : G \to R^{d}$ is a given function satisfying the compatibility condition (in a weak sense) that its normal component on ∂G vanishes. As a precise formulation of this condition we consider the space $H^{A} (G)$ of those vector-valued functions $u : G \to R^{n}$ for which the norm

${‖ u ‖}_{H^{A} (G)} = {‖ u ‖}_{L^{A} (G, R^{n})} + {‖ div u ‖}_{L^{A} (G)}$

(2.1.27)

is finite. We denote by $H_{0}^{A} (G)$ its subspace of those functions $u \in H^{A} (G)$ whose normal component on ∂G vanishes, in the sense that

$\int_{G} φ div u d x = - \int_{G} u \cdot \nabla φ d x$

(2.1.28)

for every $φ \in C^{\infty} (\overline{G})$ . It is easy to see that both $H^{A} (G)$ and $H_{0}^{A} (G)$ are Banach spaces.

Theorem 2.1.7

Assume that G is a bounded Lipschitz domain in $R^{d}$ , $d ⩾ 2$ . Let A and B be Young functions fulfilling (2.0.11) and (2.0.12). Then there exists a bounded linear operator

$E_{G} : H_{0}^{A} (G) \to W_{0}^{1, B} (G)$

(2.1.29)

such that

$div (E_{G} g) = div g in G$

(2.1.30)

for every $g \in H_{0}^{A} (G)$ . In particular, there exists a constant $C = C (G, c)$ such that

${‖ \nabla (E_{G} g) ‖}_{L^{B} (G)} ⩽ C {‖ div g ‖}_{L^{A} (G)}$

(2.1.31)

and

${‖ E_{G} g ‖}_{L^{B} (G)} ⩽ C {‖ g ‖}_{L^{A} (G)}$

(2.1.32)

for every $g \in H_{0}^{A} (G)$ . Here, c denotes the constant appearing in (2.0.11) and (2.0.12).

The proofs of Theorems 2.1.6 and 2.1.7 make use of a rearrangement estimate, which extends those of [18, Theorem 16.12] and [15], for a class of singular integral operators of the form

$T f (x) = \lim_{ε \to 0^{+}} \int_{{y : | y - x | > ε}} K (x, y) f (y) d y for x \in R^{d},$

(2.1.33)

for an integrable function $f : R^{d} \to R$ . Here $K (x, y) = N (x, x - y)$ , where the kernel $N : R^{d} \times R^{d} \to R$ fulfills the following properties:

$N (x, λ z) = λ^{- d} N (x, z) for x, z \in R^{d};$

(2.1.34)

$\int_{S^{d - 1}} N (x, z) d H^{d - 1} (z) = 0 for x \in R^{d};$

(2.1.35)

For every $σ \in [1, \infty)$ , there exists a constant $C_{1}$ such that

${(\int_{S^{d - 1}} | N (x, z) |^{σ} d H^{d - 1} (y))}^{\frac{1}{σ}} ⩽ C_{1} {(1 + | x |)}^{d} for x \in R^{d},$

(2.1.36)

where $S^{s - 1}$ denotes the unit sphere, centered at 0 in $R^{d}$ , and $H^{s - 1}$ stands for the $(s - 1)$ -dimensional Hausdorff measure.

There exists a constant $C_{2}$ such that

$| K (x, y) | ⩽ C_{2} \frac{{(1 + | x |)}^{d}}{| x - y |^{s}} for x, y \in R^{d}, x \neq y,$

(2.1.37)

and, if $2 | x - z | < | x - y |$ , then

$| K (x, y) - K (z, y) | ⩽ C_{2} {(1 + | y |)}^{d} \frac{| x - z |}{| x - y |^{d + 1}},$

(2.1.38)

$| K (y, x) - K (y, z) | ⩽ C_{2} {(1 + | y |)}^{d} \frac{| x - z |}{| x - y |^{d + 1}} .$

(2.1.39)

Theorem 2.1.8

Let G be a bounded open set in $R^{d}$ , and let K be a kernel satisfying (2.1.34)–(2.1.39). If $f \in L^{1} (R^{d})$ and $f = 0$ in $R^{d} ∖ G$ , then the singular integral operator T given by (2.1.33) is well defined for a.e. $x \in R^{d}$ , and there exists a constant $C = C (C_{1}, C_{2}, d, diam (G))$ for which

${(T f)}^{⁎} (s) ⩽ C (\frac{1}{s} \int_{0}^{s} f^{⁎} (r) d r + \int_{s}^{| G |} f^{⁎} (r) \frac{d r}{r}) for s \in (0, | G |) .$

(2.1.40)

As a consequence of Theorem 2.1.8, the boundedness of singular integral operators given by (2.1.33) between Orlicz spaces associated with Young functions A and B fulfilling (2.0.11) and (2.0.12) can be established.

Theorem 2.1.9

Let G, K and T be as in Theorem 2.1.8. Assume that A and B are Young functions satisfying (2.0.11) and (2.0.12). Then there exists a constant $C = C (C_{1}, C_{2}, d, diam (G), c)$ such that

${‖ T f ‖}_{L^{B} (G)} ⩽ C {‖ f ‖}_{L^{A} (G)},$

(2.1.41)

and

$\int_{G} B (| T f |) d x ⩽ \int_{G} A (C | f |) d x$

(2.1.42)

for every $f \in L^{A} (G)$ . Here, c denotes the constant appearing in (7.0.1) and (5.3.12).

Proof

According to Lemma 1.2.1, if A and B are Young functions satisfying (2.0.11), then there exists a constant $C = C (c)$ such that

${‖ \frac{1}{s} \int_{0}^{s} φ (r) d r ‖}_{L^{B} (0, \infty)} ⩽ C {‖ φ ‖}_{L^{A} (0, \infty)}$

(2.1.43)

for every $φ \in L^{A} (0, \infty)$ . Moreover, if A and B fulfil (2.0.12), then there exists a constant $C = C (c)$ such that

${‖ \int_{s}^{\infty} φ (r) \frac{d r}{r} ‖}_{L^{B} (0, \infty)} ⩽ C {‖ φ ‖}_{L^{A} (0, \infty)}$

(2.1.44)

for every $φ \in L^{A} (0, \infty)$ . Combining (2.1.40), (2.1.43) and (2.1.44), and making use of property (1.2.12) implies inequality (2.1.41).

As far as (2.1.42) is concerned, observe that, inequalities (2.0.11) and (2.0.12) continue to hold, with the same constant c, if A and B are replaced with kA and kB, where k is any positive constant. Thus, inequality (2.1.41) continues to hold, with the same constant C, after this replacement, whatever k is, namely

${‖ T f ‖}_{L^{k B} (G)} ⩽ C {‖ f ‖}_{L^{k A} (G)}$

(2.1.45)

for every $f \in L^{A} (G)$ . Now, given any such f, choose $k = \frac{1}{\int_{G} A (| f |) d x}$ . The very definition of Luxemburg norm tells us that ${‖ f ‖}_{L^{k A} (G)} ⩽ 1$ . Hence, by (2.1.45), ${‖ T f ‖}_{L^{k B} (G)} ⩽ C$ . The definition of Luxemburg norm again implies that $\int_{G} k B (\frac{| T f |}{C}) d x ⩽ 1$ , namely (2.1.42). □

Proof of Theorem 2.1.8

Let $R > 0$ be such that $G \subset B_{R} (0)$ , the ball centered at 0, with radius R. Fix a smooth function $η : [0, \infty) \to [0, \infty)$ for which $η = 1$ in $[0, 3 R]$ and $η = 0$ in $[4 R, \infty)$ . Define

$\begin{matrix} \hat{N} (x, z) = η (| x |) N (x, z) for x, z \in R^{d}, \\ \hat{K} (x, y) = \hat{N} (x, x - y) for x, y \in R^{d} . \end{matrix}$

By properties (2.1.34)–(2.1.39), one has that:

$\hat{N} (x, λ y) = λ^{- d} \hat{N} (x, z) for x, z \in R^{d};$

(2.1.46)

$\int_{S^{d - 1}} \hat{N} (x, z) d H^{d - 1} (z) = 0 for x \in R^{d};$

(2.1.47)

for every $σ \in [1, \infty)$ , there exists a constant ${\hat{C}}_{1} = {\hat{C}}_{1} (C_{1}, σ, R, d)$ such that

${(\int_{S^{d - 1}} | \hat{N} (x, z) |^{σ} d H^{d - 1} (z))}^{\frac{1}{σ}} ⩽ {\hat{C}}_{1} for x \in R^{d},$

(2.1.48)

where $C_{1}$ is the constant appearing in (2.1.36); there exists a constant ${\hat{C}}_{2} = {\hat{C}}_{2} (C_{2}, R, d)$ for which

$| \hat{K} (x, y) | ⩽ \frac{{\hat{C}}_{2}}{| x - y |^{d}} for x, y \in R^{d}, x \neq y,$

(2.1.49)

and, if $x \in R^{d}$ , $y \in G$ and $2 | x - z | < | x - y |$ , then

$| \hat{K} (x, y) - \hat{K} (z, y) | ⩽ {\hat{C}}_{2} \frac{| x - z |}{| x - y |^{d + 1}},$

(2.1.50)

$| \hat{K} (y, x) - \hat{K} (y, z) | ⩽ {\hat{C}}_{2} \frac{| x - z |}{| x - y |^{d + 1}},$

(2.1.51)

where $C_{2}$ is the constant appearing in (2.1.37)–(2.1.39). Define

$\begin{matrix} {\hat{T}}_{ε} f (x) & = \int_{{y : | y - x | > ε}} \hat{K} (x, y) f (y) d y, \\ {\hat{T}}_{S} f (x) & = \sup_{ε > 0} | {\hat{T}}_{ε} (f) (x) | . \end{matrix}$

Inequality (2.1.40) will follow if we prove that

${({\hat{T}}_{S} f)}^{⁎} (s) ⩽ C (\frac{1}{s} \int_{0}^{s} f^{⁎} (r) d r + \int_{s}^{| G |} f^{⁎} (r) \frac{d r}{r}) for s \in (0, \infty)$

(2.1.52)

for some constant $C = C (C_{1}, C_{2}, d, R)$ , and for every $f \in L^{1} (R^{d})$ for which $f = 0$ in $R^{d} ∖ B_{R} (0)$ . A proof of inequality (2.1.52) can be accomplished along the same lines as that of Theorem 1 of [15], which in turn relies upon similar techniques as in [51]. For completeness, we give the details of the proof hereafter.

The key step in the derivation of (2.1.52) consists in showing that, for every $γ \in (0, 1)$ , there exists a constant $C = C (C_{1}, C_{2}, γ, d, R)$ such that

${({\hat{T}}_{S} f)}^{⁎} (s) ⩽ C {(M f)}^{⁎} (γ s) + {({\hat{T}}_{S} f)}^{⁎} (2 s) for s \in (0, \infty)$

(2.1.53)

for every $f \in L^{1} (R^{d})$ with $f = 0$ in $R^{d} ∖ B_{R} (0)$ . Fix $s > 0$ , and define

$E = {x \in R^{d} : {\hat{T}}_{S} f (x) > {({\hat{T}}_{S} f)}^{⁎} (2 s)} .$

Then, there exists an open set $U \supset E$ for which $| U | ⩽ 3 s$ . By Whitney's covering theorem, there exist a family of disjoint cubes ${Q_{k}}$ such that $U = \cup_{k = 1}^{\infty} Q_{k}$ , $\sum_{k = 1}^{\infty} | Q_{k} | = | U | ⩽ 3 s$ , and

$diam (Q_{k}) ⩽ dist (Q_{k}, R^{d} ∖ U) ⩽ 4 diam (Q_{k}) for k \in N .$

The operator ${\hat{T}}_{S}$ is of weak type $(1, 1)$ , namely, there exists a constant $C^{'}$ such that

$| {x \in R^{d} : {\hat{T}}_{S} f (x) > λ} | ⩽ \frac{C^{'}}{λ} {‖ f ‖}_{L^{1} (R^{d})}$

(2.1.54)

for $f \in L^{1} (R^{d})$ . The proof of (2.1.54) follows from classical arguments: By (2.1.51) we have for all $r > 0$ , $y \in R^{d}$ and all $x \in B_{r} (z)$ that

$\int_{| y - x | ⩾ 2 r} | \hat{K} (y, x) - \hat{K} (y, z) | d y ⩽ C .$

So $\hat{K}$ satisfies condition (10) in [134, p. 33]. By [134, Cor. 1, p. 33] we obtain

$| {x \in R^{d} : {\hat{T}}_{ε} f (x) > λ} | ⩽ \frac{C}{λ} {‖ f ‖}_{L^{1} (R^{d})},$

(2.1.55)

where C does not depend on ε. Now (2.1.55) implies (2.1.54) by taking the supremum in ε.

We now show that there exists a constant $\overline{C}$ such that

$| {x \in Q_{k} : {\hat{T}}_{S} f (x) > \overline{C} M f (x) + {({\hat{T}}_{S} f)}^{⁎} (2 s)} | ⩽ \frac{1 - γ}{3} | Q_{k} | for k \in N .$

(2.1.56)

Fix any $k \in N$ , choose $x_{k} \in R^{d} ∖ U$ such that $dist (x_{k}, Q_{k}) ⩽ 4 diam (Q_{k})$ , and denote by Q the cube, centered at $x_{k}$ , with $diam (Q) = 20 diam (Q_{k})$ . Define

$g = f χ_{Q}, h = f χ_{R^{d} ∖ Q},$

so that $f = g + h$ . If we prove that there exist constants ${\overline{C}}_{1}$ and ${\overline{C}}_{2}$ such that

${\hat{T}}_{S} h (x) ⩽ {\overline{C}}_{1} M f (x) + {({\hat{T}}_{S} f)}^{⁎} (2 s) for x \in Q_{k},$

(2.1.57)

and

$| {x \in Q_{k} : {\hat{T}}_{S} g (x) > {\overline{C}}_{2} M f (x)} | ⩽ \frac{1 - γ}{3} | Q_{k} |,$

(2.1.58)

then (2.1.56) follows with $\overline{C} = {\overline{C}}_{1} + {\overline{C}}_{2}$ . Consider (2.1.58) first. Let ${\overline{C}}_{2}$ be a constant for which $\frac{C^{'} | Q |}{{\overline{C}}_{2}} ⩽ \frac{1 - γ}{3} | Q_{k} |$ . Let $λ = \frac{{\overline{C}}_{2}}{| Q |} \int_{Q} | g | d x$ . Since ${\overline{C}}_{2} M f (x) ⩾ λ$ for $x \in Q_{k}$ , an application of (2.1.54) with this choice of λ tells us that

$\begin{matrix} | {x \in Q_{k} : {\hat{T}}_{S} g (x) > {\overline{C}}_{2} M f (x)} | & ⩽ | {{\hat{T}}_{S} g (x) > λ} | \\ ⩽ \frac{C^{'}}{λ} \int_{Q} | g | d x ⩽ \frac{C^{'} | Q |}{{\overline{C}}_{2}} ⩽ \frac{1 - γ}{3} | Q_{k} |, \end{matrix}$

namely (2.1.58). In order to establish (2.1.57), it suffices to prove that, for every $ε > 0$ ,

$| {\hat{T}}_{ε} h (x) | ⩽ {\overline{C}}_{1} M f (x) + {\hat{T}}_{S} f (x_{k}) for x \in Q_{k} .$

(2.1.59)

Indeed, since $x_{k} \notin U$ , we have that ${\hat{T}}_{S} f (x_{k}) ⩽ {({\hat{T}}_{S} f)}^{⁎} (2 s)$ , and hence (2.1.59) implies (2.1.57). We may thus focus on (2.1.59). Fix $ε > 0$ , and set $r = \max {ε, dist (x_{k}, R^{d} ∖ Q)}$ . Observe that $r > 10 diam (Q_{k})$ . Given any $x \in Q_{k}$ , define $V = B_{ε} (x) △ B_{ε} (x_{k})$ . One has that

$\begin{matrix} | {\hat{T}}_{ε} h (x) | & = | \int_{{y : | y - x | > ε}} \hat{K} (x, y) h (y) d y | \\ ⩽ | \int_{{y : | y - x_{k} | > ε}} \hat{K} (x, y) h (y) d y | + \int_{V} | \hat{K} (x, y) h (y) | d y . \end{matrix}$

(2.1.60)

Observe that, if $y \in supp h$ , then $| x - y | > \frac{r}{2}$ and hence $\frac{1}{| x - y |^{d}} < \frac{2^{d}}{r^{d}}$ . Thus, due to (2.1.49),

$| \hat{K} (x, y) | ⩽ \frac{{\hat{C}}_{2}}{r^{d}} .$

Moreover, $V \subset B_{3 r} (x)$ . Therefore, there exists a constant $\hat{C}$ such that

$\int_{V} | \hat{K} (x, y) h (y) | d y ⩽ \hat{C} \underset{B_{3 r} (x)}{⨍} | h (y) | d y ⩽ \hat{C} M h (x) ⩽ \hat{C} M f (x) .$

(2.1.61)

On the other hand,

$\begin{matrix} | \int_{{y : | y - x_{k} | > ε}} \hat{K} (x, y) h (y) d y | ⩽ | \int_{{y : | y - x_{k} | > r}} \hat{K} (x, y) h (y) d y | \\ ⩽ | \int_{{y : | y - x_{k} | > r}} \hat{K} (x_{k}, y) f (y) d y | + \int_{{y : | y - x_{k} | > r}} | \hat{K} (x_{k}, y) - \hat{K} (x, y) | | f (y) | d y \\ ⩽ {\hat{T}}_{S} (x_{k}) + \int_{{y : | y - x_{k} | > r}} | \hat{K} (x_{k}, y) - \hat{K} (x, y) | | f (y) | d y, \end{matrix}$

(2.1.62)

where the first inequality holds since $h (y) = 0$ in ${y : | y - x_{k} | ⩽ r}$ if $r = dist (x_{k}, R^{d} ∖ Q)$ , and trivially holds (with equality) if $r = ε$ . Since $2 | x - x_{k} | ⩽ | x - y |$ in the last integral in (2.1.62), and f vanishes in $R^{d} ∖ B_{R} (0)$ , by (2.1.50)

$| \hat{K} (x_{k}, y) - \hat{K} (x, y) | ⩽ {\hat{C}}_{2} \frac{| x_{k} - x |}{| x - y |^{d + 1}} ⩽ {\hat{C}}_{2} \frac{diam (Q_{k})}{| x - y |^{d + 1}} .$

Hence,

$\begin{matrix} \int_{{y : | y - x_{k} | > r}} | \hat{K} (x_{k}, y) - \hat{K} (x, y) | | f (y) | d y \\ ⩽ \int_{{y : | y - x | > diam (Q_{k})}} | f (y) | \frac{diam (Q_{k})}{| x - y |^{d + 1}} d y ⩽ \tilde{C} M f (x) \end{matrix}$

(2.1.63)

for some constant $\tilde{C}$ . Note that, in the first inequality, we made use of the inclusion ${y : | y - x_{k} | > r} \subset {y : | y - x | > diam (Q_{k})}$ , which holds since $| x - x_{k} | < 5 diam (Q_{k})$ , and $10 diam (Q_{k}) < r$ .

Combining inequalities (2.1.60)–(2.1.63) implies (2.1.59). Inequality (2.1.56) is fully established. Via summation in $k \in Q_{k}$ , we obtain from (2.1.56) that

$| {x \in R^{d} : {\hat{T}}_{S} f (x) > \hat{C} M f (x) + {({\hat{T}}_{S} f)}^{⁎} (2 s)} | ⩽ (1 - γ) s .$

(2.1.64)

Combining (2.1.64) with the inequality

$| {x \in R^{d} : M f (x) > {(M f)}^{⁎} (γ s)} | ⩽ γ s$

(2.1.65)

tells us that

$\begin{matrix} | {x \in R^{d} : {\hat{T}}_{S} f (x) > \hat{C} {(M f)}^{⁎} (γ s) + {({\hat{T}}_{S} f)}^{⁎} (2 s)} | \\ ⩽ | {{\hat{T}}_{S} f (x) > \hat{C} M f (x) + {({\hat{T}}_{S} f)}^{⁎} (2 s)} | + | {M f (x) > {(M f)}^{⁎} (γ s)} | ⩽ s . \end{matrix}$

Hence (2.1.53) follows, by the very definition of decreasing rearrangement.

Starting from inequality (2.1.53) we apply the iteration argument from [15, Lemma 3.2] and obtain for $γ = 1 / 2$

${({\hat{T}}_{S} f)}^{⁎} (s) ⩽ C \sum_{k = 0}^{\infty} {(M f)}^{⁎} (2^{k - 1} s) + \lim_{s \to \infty} {({\hat{T}}_{S} f)}^{⁎} (s) .$

Therefore, we have for all f satisfying $\lim_{s \to \infty} {({\hat{T}}_{S} f)}^{⁎} (s) = 0$ that

${({\hat{T}}_{S} f)}^{⁎} (s) ⩽ C \sum_{k = 2}^{\infty} {(M f)}^{⁎} (2^{k - 1} s) + 2 C {(M f)}^{⁎} (\frac{s}{2}) .$

Since

${(M f)}^{⁎} (2^{k - 1} s) ⩽ \int_{{2^{k - 2} s ⩽ σ ⩽ 2^{k - 1} s}} {(M f)}^{⁎} (σ) \frac{d σ}{σ}$

we conclude

${({\hat{T}}_{S} f)}^{⁎} (s) ⩽ C \int_{s}^{\infty} {(M f)}^{⁎} (σ) \frac{d σ}{σ} + 2 C {(M f)}^{⁎} (\frac{s}{2}) .$

(2.1.66)

Now we fix $s > 0$ and assume that $\int_{s}^{\infty} {(M f)}^{⁎} (σ) \frac{d σ}{σ}$ is finite. Since each f has compact support, ${\hat{T}}_{S} f (x)$ converges to zero for all k as $| x | \to \infty$ (recall (2.1.49) and the definition of ${\hat{T}}_{S}$ ). Therefore, we have that $\lim_{s \to \infty} {({\hat{T}}_{S} f)}^{⁎} (s) = 0$ for every k. Hence (2.1.66) implies

$\begin{matrix} {({\hat{T}}_{S} f)}^{⁎} (s) & ⩽ C \int_{s}^{\infty} {(M f)}^{⁎} (σ) \frac{d σ}{σ} + 2 C {(M f)}^{⁎} (\frac{s}{2}) \\ ⩽ C (\int_{s}^{\infty} f^{⁎} (r) \frac{d r}{r} + f^{⁎} (\frac{s}{2})) \\ ⩽ C (\int_{s}^{\infty} f^{⁎} (r) \frac{d r}{r} + \frac{2}{s} \int_{\frac{s}{2}}^{s} f^{⁎} (r) d r) \\ ⩽ C (\int_{s}^{\infty} f^{⁎} (r) \frac{d r}{r} + \frac{1}{s} \int_{0}^{s} f^{⁎} (r) d r) \end{matrix}$

which shows (2.1.40). □

Lemma 2.1.1

Let G be a bounded domain with the cone property in $R^{d}$ , with $n ⩾ 2$ . Then there exist $N \in N$ and a finite family ${G_{i}}_{i = 0, \dots N}$ of domains which are starshaped with respect to balls, for which $G = \cup_{i = 0}^{N} G_{i}$ . Moreover, given $f \in L_{⊥}^{A} (G)$ , there exist $f_{i} \in L_{⊥}^{A} (G)$ , $i = 0, \dots N$ , such that $f_{i} = 0$ in $G ∖ G_{i}$ ,

$f = \sum_{i = 0}^{N} f_{i}$

and

${‖ f_{i} ‖}_{L^{A} (G)} ⩽ C {‖ f ‖}_{L^{A} (G)} for i = 0, \dots, N,$

(2.1.67)

for some constant $C = C (G)$ .

Proof, sketched

Any bounded open set with the cone property can be decomposed into a finite union of Lipschitz domains [5, Lemma 4.22]. On the other hand, any Lipschitz domain can be decomposed into a finite union of open sets which are starshaped with respect to balls [85, Lemma 3.4, Chapter 3]. This proves the existence of the domains ${G_{i}}_{i = 0, \dots N}$ as in the statement. The same argument as in the proof of [85, Lemma 3.2, Chapter 3] then enables us to construct the desired family of functions $f_{i}$ on G, $i = 1, \dots, N$ , according to the following iteration scheme. We set $D_{i} = \cup_{j = i + 1}^{N} G_{j}$ , $g_{0} = f$ , and, for $i = 1, \dots, N - 1$ ,

$g_{i} (x) = {\begin{matrix} (1 - χ_{G_{i} \cap D_{i}} (x)) g_{i - 1} (x) - \frac{χ_{G_{i} \cap D_{i}} (x)}{| G_{i} \cap D_{i} |} \int_{D_{i} ∖ G_{i}} g_{i - 1} (y) d y & if x \in D_{i}, \\ 0 & otherwise, \end{matrix}$

(2.1.68)

and

$f_{i} (x) = {\begin{matrix} g_{i - 1} (x) - \frac{χ_{G_{i} \cap D_{i}} (x)}{| G_{i} \cap D_{i} |} \int_{G_{i}} g_{i - 1} (y) d y & if x \in G_{i}, \\ 0 & otherwise . \end{matrix}$

(2.1.69)

Observe that, since G is connected, we can always relabel the sets $G_{i} \cap D_{i}$ in such a way that $| G_{i} \cap D_{i} | > 0$ for $i = 1, \dots, N - 1$ . Finally, we define

$f_{N} = g_{N - 1} .$

(2.1.70)

The family ${f_{i}}$ satisfies the required properties. The only nontrivial property is (2.1.67). To verify it, fix i, and observe that, by (2.1.69), the second inequality in (1.2.10), inequality (1.2.3), and inequality (1.2.7)

$\begin{matrix} {‖ f_{i} ‖}_{L^{A} (G)} & ⩽ {‖ g_{i - 1} ‖}_{L^{A} (G)} (1 + \frac{2}{| G_{i} \cap D_{i} |} {‖ 1 ‖}_{L^{A} (G_{i} \cap D_{i})} {‖ 1 ‖}_{L^{\tilde{A}} (G_{i})}) \\ = {‖ g_{i - 1} ‖}_{L^{A} (G)} (1 + \frac{2}{| G_{i} \cap D_{i} | A^{- 1} (1 / | G_{i} \cap D_{i} |)} \frac{1}{{\tilde{A}}^{- 1} (1 / | G_{i} |)}) \\ ⩽ {‖ g_{i - 1} ‖}_{L^{A} (G)} (1 + 4 \frac{{\tilde{A}}^{- 1} (1 / | G_{i} \cap D_{i} |)}{{\tilde{A}}^{- 1} (1 / | G_{i} |)}) \\ ⩽ {‖ g_{i - 1} ‖}_{L^{A} (G)} (1 + 4 \frac{| G_{i} |}{| G_{i} \cap D_{i} |}) . \end{matrix}$

(2.1.71)

On the other hand, by (2.1.68) and a chain similar to (2.1.71), one has that

$\begin{matrix} {‖ g_{i - 1} ‖}_{L^{A} (G)} & ⩽ {‖ g_{i - 2} ‖}_{L^{A} (G)} (1 + \frac{2}{| G_{i - 1} \cap D_{i - 1} |} {‖ 1 ‖}_{L^{A} (G_{i - 1} \cap D_{i - 1})} {‖ 1 ‖}_{L^{\tilde{A}} (D_{i - 1})}) \\ ⩽ {‖ g_{i - 2} ‖}_{L^{A} (G)} (1 + 4 \frac{{\tilde{A}}^{- 1} (| 1 / G_{i - 1} \cap D_{i - 1} |)}{{\tilde{A}}^{- 1} (| 1 / D_{i - 1} |)}) \\ ⩽ {‖ g_{i - 2} ‖}_{L^{A} (G)} (1 + 4 \frac{| D_{i - 1} |}{| G_{i - 1} \cap D_{i - 1} |}) . \end{matrix}$

(2.1.72)

From (2.1.71), and an iteration of (2.1.72), one infers that

${‖ f_{i} ‖}_{L^{A} (G)} ⩽ (1 + 4 \frac{| G_{i} |}{| G_{i} \cap D_{i} |}) \prod_{j = 1}^{i - 1} (1 + 4 \frac{| D_{j} |}{| G_{j} \cap D_{j} |}) {‖ f ‖}_{L^{A} (G)},$

and (2.1.67) follows. □

Proof of Theorem 2.1.6

By Lemma 2.1.1, it suffices to prove the statement in the case when G is a domain starshaped with respect to a ball $B$ , which, without loss of generality, can be assumed to be centered at the origin and with radius 1. In this case, we are going to show that the (gradient of the) Bogovskiĭ operator ${Bog}_{G}$ , defined at a function $f \in L_{⊥}^{A} (G)$ is

${Bog}_{G} f (x) = \int_{G} f (y) (\frac{x - y}{| x - y |^{d}} \int_{| x - y |}^{\infty} ω (y + ζ \frac{x - y}{| x - y |}) ζ^{d - 1} d ζ) d y$

(2.1.73)

for $x \in G$ . Here ω is any (nonnegative) function in $C_{0}^{\infty} (B)$ with $\int_{B} ω d x = 1$ , agrees with a singular integral operator, whose kernel fulfills (2.1.34)–(2.1.39), plus two operators enjoying stronger boundedness properties. If $f \in C_{0}^{\infty} (G)$ it is easy to see that the same is true for ${Bog}_{G} f$ using the representations

$\begin{matrix} {Bog}_{G} f (x) \\ = \int_{G} f (y) (x - y) (\frac{x - y}{| x - y |^{d}} \int_{1}^{\infty} ω (y + ζ (x - y)) ζ^{d - 1} d ζ) d y \\ = \int_{G} f (y) (x - y) (\frac{x - y}{| x - y |^{d}} \int_{0}^{\infty} ω (y + ζ \frac{(x - y)}{| x - y |}) {(| x - y | ζ)}^{d - 1} d ζ) d y . \end{matrix}$

Setting $u = {Bog}_{G} f$ , we have

$u (x) = \int_{G} f (y) N (x, y) d y for x \in G,$

where

$N (x, y) = \frac{x - y}{| x - y |^{d}} \int_{| x - y |}^{\infty} ω (y + ζ \frac{x - y}{| x - y |}) ζ^{d - 1} d ζ for x, y \in G .$

By standard arguments we obtain (see [85, Proof of Lemma III.3.1] for more details)

$\partial_{j} u_{i} (x) = \int_{G} f (y) \partial_{j} N_{i} (x, y) d y + f (x) \int_{G} \frac{{(x - y)}_{i} {(x - y)}_{j}}{| x - y |^{2}} ω (y) d y .$

Computing $\partial_{j} N$ we see that

$\partial_{j} N = K_{i j} (x, y) + G_{i j} (x, y)$

with

$\begin{matrix} K_{i j} (x, y) & = \frac{δ_{i j}}{| x - y |^{d}} \int_{0}^{\infty} ω (x + ζ \frac{x - y}{| x - y |}) ζ^{d - 1} d ζ \\ + \frac{x_{i} - y_{i}}{| x - y |^{d + 1}} \int_{0}^{\infty} \partial_{j} ω (x + ζ \frac{x - y}{| x - y |}) ζ^{d} d ζ, \end{matrix}$

(2.1.74)

$G_{i j} (x, y) = \frac{x_{i} - y_{i}}{| x - y |^{d + 1}} \int_{0}^{\infty} \partial_{j} ω (x + ζ \frac{x - y}{| x - y |}) \sum_{k = 0}^{d - 1} (\begin{matrix} d \\ k \end{matrix}) | x - y |^{k} ζ^{d - k} d ζ .$

(2.1.75)

Now we want to justify the formula for a non-smooth function f. We claim that $u \in W_{0}^{1, 1} (G)$ , and

$\frac{\partial u_{i}}{\partial x_{j}} = H_{i j} f for a.e. x \in G,$

(2.1.76)

where $H_{i j}$ is the linear operator defined at f as

$\begin{matrix} (H_{i j} f) (x) & = \int_{G} K_{i j} (x, y) f (y) d y + \int_{G} G_{i j} (x, y) f (y) d y \\ + f (x) \int_{G} \frac{{(x - y)}_{i} {(x - y)}_{j}}{| x - y |^{2}} ω (y) d y for x \in G, \end{matrix}$

(2.1.77)

for $i, j = 1, \dots d$ . Here, $K_{i j}$ is the kernel of a singular integral operator satisfying the same assumptions as the kernel K in Theorem 2.1.8. Moreover, the following holds

$| G_{i j} (x, y) | ⩽ \frac{c}{| x - y |^{d - 1}} for x, y \in R^{d}, x \neq y .$

(2.1.78)

Computing the divergence based on (2.1.76) and (2.1.77) we see that

$\begin{matrix} div u & = \int_{G} d f (y) \int_{1}^{\infty} ω (y + r (x - y)) r^{d - 1} d r d y \\ + \int_{G} f (y) \sum_{i = 1}^{d} \int_{1}^{\infty} (x_{i} - y_{i}) \partial_{i} ω (y + r (x - y)) r^{d} d r d y \\ + \sum_{i = 1}^{d} f (x) \int_{G} \frac{{(x_{i} - y_{i})}^{2}}{| x - y |^{2}} ω (y) d y \\ = \int_{G} d f (y) \int_{1}^{\infty} ω (y + r (x - y)) r^{d - 1} d r d y \\ + \int_{G} f (y) \int_{1}^{\infty} \frac{d}{d r} ω (y + r (x - y)) r^{d} d r d y + f (x) \\ = - ω (x) \int_{G} f (y) d y + f (x) \end{matrix}$

using (2.1.74), (2.1.75) and $\int_{G} ω d y = 1$ . As $\int_{G} f d y = 0$ we obtain $div u = f$ .

Now we pass to general functions $f \in L_{⊥}^{A} (G)$ . Recall that, if $f \in C_{0, ⊥}^{\infty} (G)$ , then $u \in C_{0}^{\infty} (G)$ , and moreover the equations (2.1.76) and (2.1.23) hold for every $x \in G$ . Due to (2.0.11), $L_{⊥}^{A} (G) \to L Log L_{⊥} (G)$ , since $B (t)$ grows at least linearly near infinity, and hence $A (t)$ dominates the function $t \log (1 + t)$ near infinity. Since the space $C_{0, ⊥}^{\infty} (G)$ is dense in $L \log L_{⊥} (G)$ , there exists a sequence of functions ${f_{k}} \subset C_{0, ⊥}^{\infty} (G)$ such that $f_{k} \to f$ in $L \log L (G)$ . Hence

${Bog}_{G} : L \log L (G) \to L^{1} (G)$

(in fact, ${Bog}_{G}$ is also bounded into $L \log L (G)$ ). Furthermore,

$H_{i j} : L \log L (G) \to L^{1} (G),$

as a consequence of (2.1.78) and of a special case of Theorem 2.1.9, with $L^{A} (G) = L \log L (G)$ and $L^{B} (G) = L^{1} (G)$ . Thus, $Bog f_{k} \to {Bog}_{G} f$ in $L^{1} (G)$ and $H_{i j} f_{k} \to H_{i j} f$ in $L^{1} (G)$ . This implies that $u \in W_{0}^{1, 1} (G)$ , and (2.1.76) and (2.1.23) hold.

By Theorem 2.1.9, the singular integral operator defined by the first term on the right-hand side of (2.1.77) is bounded from $L^{A} (G)$ into $L^{B} (G)$ . By inequality (2.1.78), the operator defined by the second term on the right-hand-side of (2.1.77) has (at least) the same boundedness properties as a Riesz potential operator with kernel $\frac{1}{| x - y |^{d - 1}}$ . Such an operator is bounded in $L^{1} (G)$ and in $L^{\infty} (G)$ , with norms depending only on $| G |$ and on d. An interpolation theorem by Calderon [19, Theorem 2.12, Chap. 3] then ensures that it is also bounded from $L^{A} (G)$ into $L^{A} (G)$ , and hence from $L^{A} (G)$ into $L^{B} (G)$ , with norm depending on d and $| G |$ . Finally, the operator given by the last term on the right-hand-side of (2.1.77) is pointwise bounded (in absolute value) by $| f (x) |$ . Thus, it is bounded from $L^{A} (G)$ into $L^{A} (G)$ , and hence from $L^{A} (G)$ into $L^{B} (G)$ . Equations (2.1.21) and (2.1.24) are thus established.

Inequality (2.1.25) follows from (2.1.24) via a scaling argument analogous to that which leads to (2.1.42) from (2.1.41) – see the Proof of Theorem 2.1.9. □

Proof of Theorem 2.1.7

By an argument as in the proofs of [85, Lemmas 3.4 and 3.5, and Theorem 3.3], it suffices to show that, if G and G are bounded Lipschitz domains, for which the domain $G_{0} = G \cap D$ is star-shaped with respect to a ball $B ⋐ G_{0}$ , and f has the form

$f = ζ div g + θ \int_{G} φ div g d y,$

for some functions $ζ \in C_{0}^{\infty} (G)$ , $θ \in C_{0}^{\infty} (G_{0})$ and $φ \in C^{\infty} (\overline{G})$ , and fulfills

$\int_{G_{0}} f (x) d x = 0,$

then there exists a function $w \in W_{0}^{1, B} (G)$ such that

$div w = f in G_{0},$

(2.1.79)

${‖ \nabla w ‖}_{L^{B} (G_{0})} ⩽ C {‖ div g ‖}_{L^{A} (G)},$

(2.1.80)

and

${‖ w ‖}_{L^{B} (G_{0})} ⩽ C {‖ g ‖}_{L^{A} (G)},$

(2.1.81)

for some constant $C = C (φ, θ, ζ, c, B, G, G)$ , where c is the constant appearing in (2.0.11) and (2.0.12).

Since $f \in L_{⊥}^{A} (G_{0})$ , an inspection of the proof of Theorem 2.1.6 then reveals that the function w, given by

$w (x) = \int_{G_{0}} f (y) N (x, y) d y for x \in G_{0},$

(2.1.82)

where

$N (x, y) = \frac{x - y}{| x - y |^{d}} \int_{| x - y |}^{\infty} ω (y + ζ \frac{x - y}{| x - y |}) ζ^{d - 1} d ζ for x, y \in G,$

(2.1.83)

and ω is any (nonnegative) function in $C_{0}^{\infty} (B)$ with $\int_{B} ω d x = 1$ , satisfies (2.1.79), and

${‖ \nabla w ‖}_{L^{B} (G_{0})} ⩽ C {‖ f ‖}_{L^{A} (G_{0})}$

(2.1.84)

for some constant C. Since

${‖ f ‖}_{L^{A} (G_{0})} ⩽ C {‖ div g ‖}_{L^{A} (G)}$

(2.1.85)

for some constant C, inequality (2.1.80) follows. It remains to prove (2.1.81). To this purpose, assume, for the time being, that $div g \in C_{0}^{\infty} (G)$ . Then, by [85, Equation 3.35],

$\begin{matrix} w_{i} (x) & = - \int_{G_{0}} N_{i} (x, y) g (y) \cdot \nabla ζ (y) d y \\ - \int_{G_{0}} K_{i j} (x, x - y) ζ (y) g_{j} (y) d y \\ - \int_{G_{0}} \sum_{j = 1}^{n} G_{i j} (x, y) ζ (y) g_{j} (y) d y \\ - ζ (x) \sum_{j = 1}^{d} g_{j} (x) \int_{G_{0}} \frac{(x_{i} - y_{i}) (x_{j} - y_{j})}{| x - y |^{2}} ω (y) d y \\ - (\int_{G} g \cdot \nabla φ d y) \int_{G_{0}} N_{i} (x, y) θ (y) d y for a.e. x \in G_{0}, \end{matrix}$

(2.1.86)

where the kernels $K_{i j}$ and $G_{i j}$ satisfy the same assumptions as the kernels in (2.1.77), and $| N (x, y) | ⩽ C | x - y |^{1 - d}$ for some constant C. Note that condition (2.1.28) has been used in writing the last term on the right-hand side of equation (2.1.86).

We now drop the assumption that $div g \in C_{0}^{\infty} (G)$ . Condition (2.0.11) entails that $L^{A} (G) \to L \log L (G)$ , and hence $H_{0}^{A} (G) \to H_{0}^{L \log L} (G)$ , where the latter space denotes $H_{0}^{A} (G)$ with $A (t)$ equivalent to $t \log (1 + t)$ near infinity. Thus, $g \in H_{0}^{L \log L} (G)$ , and hence it can be approximated by a sequence of functions ${g_{k}} \subset C_{0}^{\infty} (G)$ in such a way that $g_{k} \to g$ in $H^{L \log L} (G)$ , cf. Theorem A.1.46. The first and third term on the right-hand side of (2.1.86) are integral operators applied to g whose kernel is bounded by a multiple of $| x - y |^{1 - d}$ . The fourth term is just bounded by a constant multiple of $| g |$ . The last term is a constant multiple of an integral of g against a bounded vector-valued function ∇φ. Hence, all these operators are bounded from $L^{A} (G)$ into $L^{A} (G_{0})$ . The second term is a singular integral operator enjoying the same properties as the singular integral operator K in Theorems 2.1.8 and 2.1.9. Thus, since all operators appearing on the right-hand side of (2.1.86) are bounded from $L \log L (G)$ into $L^{1} (G_{0})$ , the right-hand side of (2.1.86), evaluated with g replaced by $g_{k}$ , converges in $L^{1} (G_{0})$ to the right-hand-side of (2.1.86). On the other hand, equation (2.1.82) and the properties of the Bogovskiĭ operator tell us that the left-hand-side of (2.1.86), with $w_{i}$ corresponding to $g_{k}$ , converges in $L^{1} (G_{0})$ to the left-hand-side of (2.1.86). Altogether, we conclude that (2.1.86) actually holds even if g is just in $H_{0}^{A} (G)$ .

The properties of the operators on the right-hand-side of (2.1.86) mentioned above ensure that they are bounded from $L^{A} (G)$ into $L^{B} (G_{0})$ . Inequality (2.1.81) thus follows from (2.1.86). □

2.2 Negative norms & the pressure

We need a last preliminary result in preparation of the proof of Theorem 2.0.5.

Proposition 2.2.1

Let G be an open subset in $R^{d}$ such that $| G | < \infty$ , and let A be a Young function. Assume that $u \in L^{A} (G)$ . Then we have

$\sup_{v \in L^{\tilde{A}} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u φ d x}{{‖ φ ‖}_{L^{\tilde{A}} (G)}},$

(2.2.87)

and

$\sup_{v \in L_{⊥}^{\tilde{A}} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} u φ d x}{{‖ φ ‖}_{L^{\tilde{A}} (G)}} .$

(2.2.88)

Note that equation (2.2.87) is well known under the assumption that $A \in \nabla_{2}$ near infinity, namely $\tilde{A} \in Δ_{2}$ near infinity. This is because $C_{0}^{\infty} (G)$ is dense in $L^{\tilde{A}} (G)$ in this case. Equation (2.2.88) also easily follows from this property when $A \in \nabla_{2}$ near infinity. The novelty of Proposition 2.2.1 is in the arbitrariness of A.

Proof of Proposition 2.2.1

Consider first (2.2.87). It clearly suffices to show that

$\sup_{v \in L^{\tilde{A}} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{v \in L^{\infty} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}},$

(2.2.89)

and

$\sup_{v \in L^{\infty} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u φ d x}{{‖ φ ‖}_{L^{\tilde{A}} (G)}} .$

(2.2.90)

Given any $v \in L^{\tilde{A}} (G)$ , define, for $k \in N$ , the function $v_{k} : G \to R$ as

$v_{k} = sign (v) \min {| v |, k} .$

(2.2.91)

Clearly, $v_{k} \in L^{\infty} (G)$ , and $0 ⩽ | v_{k} | ↗ | v |$ a.e. in G as $k \to \infty$ . Hence,

$\int_{G} | u v_{k} | d x ↗ \int_{G} | u v | d x as k \to \infty,$

by the monotone convergence theorem for integrals, and, by the Fatou property of the Luxemburg norm,

${‖ v_{k} ‖}_{L^{\tilde{A}} (G)} ↗ {‖ v ‖}_{L^{\tilde{A}} (G)} as k \to \infty .$

Thus, since

$\sup_{v \in L^{\tilde{A}} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{v \in L^{\tilde{A}} (G)} \frac{\int_{G} | u v | d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}},$

equation (2.2.89) follows.

As far as (2.2.90) is concerned, consider an increasing sequence of compact sets $E_{k}$ such that $dist (E_{k}, R^{d} ∖ G) ⩾ \frac{2}{k}$ , $E_{k} \subset E_{k + 1} \subset G$ for $k \in N$ , and $\cup_{k} E_{k} = G$ . Moreover, let ${ϱ_{k}}$ be a family of (nonnegative) smooth mollifiers in $R^{d}$ , such that $supp ϱ_{k} \subset B_{\frac{1}{k}} (0)$ and $\int_{R^{d}} ϱ_{k} d x = 1$ for $k \in N$ . Given $v \in L^{\infty} (G)$ , define $w_{k} : R^{d} \to R$ as

$w_{k} = {\begin{matrix} v & in E_{k}, \\ 0 & elsewhere, \end{matrix}$

and $φ_{k} : R^{d} \to R$ as

$φ_{k} (x) = \int_{R^{d}} w_{k} (y) ϱ_{k} (x - y) d y for x \in R^{d} .$

(2.2.92)

Classical properties of mollifiers ensure that

$\begin{matrix} φ_{k} \in C_{0}^{\infty} (G), φ_{k} \to v a.e. in G as k \to \infty, \\ {‖ φ_{k} ‖}_{L^{\infty} (G)} ⩽ {‖ v ‖}_{L^{\infty} (G)} for k \in N . \end{matrix}$

Thus, if $u \in L^{A} (G)$ , then

$\int_{G} u φ_{k} d x \to \int_{G} u v d x as k \to \infty,$

(2.2.93)

by the dominated convergence theorem for integrals. Moreover,

${‖ φ_{k} ‖}_{L^{\tilde{A}} (G)} \to {‖ v ‖}_{L^{\tilde{A}} (G)} as k \to \infty .$

(2.2.94)

Indeed, by dominated convergence and the definition of Luxemburg norm,

$\int_{G} \tilde{A} (\frac{| φ_{k} |}{{‖ v ‖}_{L^{\tilde{A}} (G)}}) d x \to \int_{G} \tilde{A} (\frac{| v |}{{‖ v ‖}_{L^{\tilde{A}} (G)}}) d x ⩽ 1 as k \to \infty .$

In particular, for every $ε > 0$ , there exists $k_{ε}$ such that

$\int_{G} \tilde{A} (\frac{| φ_{k} |}{{‖ v ‖}_{L^{\tilde{A}} (G)}}) d x < 1 + ε if k > k_{ε} .$

Hence, by the arbitrariness of ε and the definition of Luxemburg norm,

$\underset{k \to \infty}{\lim \inf} {‖ φ_{k} ‖}_{L^{\tilde{A}} (G)} ⩾ {‖ v ‖}_{L^{\tilde{A}} (G)} .$

(2.2.95)

We also have that

$\underset{k \to \infty}{\lim \sup} {‖ φ_{k} ‖}_{L^{\tilde{A}} (G)} ⩽ {‖ v ‖}_{L^{\tilde{A}} (G)} .$

(2.2.96)

Indeed, assume that (2.2.96) fails. Then, there exists $σ > 0$ and a subsequence of ${φ_{k}}$ , still denoted by ${φ_{k}}$ , such that

$1 < \int_{G} \tilde{A} (\frac{| φ_{k} |}{{‖ v ‖}_{L^{\tilde{A}} (G)} + σ}) d x \to \int_{G} \tilde{A} (\frac{| v |}{{‖ v ‖}_{L^{\tilde{A}} (G)} + σ}) d x ⩽ 1,$

which is a contradiction. Equation (2.2.94) follows from (2.2.95) and (2.2.96). Coupling (2.2.93) with (2.2.94) implies (2.2.90). The proof of (2.2.87) is complete.

The proof of (2.2.88) follows along the same lines, and, in particular, via the equations

$\sup_{v \in L_{⊥}^{\tilde{A}} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{v \in L_{⊥}^{\infty} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}},$

(2.2.97)

and

$\sup_{v \in L_{⊥}^{\infty} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{A}} (G)}} = \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} u φ d x}{{‖ φ ‖}_{L^{\tilde{A}} (G)}} .$

(2.2.98)

For any $v \in L_{⊥}^{\tilde{A}} (G)$ , we define the sequence of functions ${{\overline{v}}_{k}} \subset L_{⊥}^{\infty} (G)$ by

${\overline{v}}_{k} = v_{k} - {(v_{k})}_{G} .$

Here we have $k \in N$ and $v_{k}$ is given by (2.2.91). We can prove equation (2.2.97) via a slight variant of the argument used for (2.2.89). Here, one has to use the fact that ${(v_{k})}_{G} \to 0$ as $k \to \infty$ .

Equation (2.2.98) can be established similarly to (2.2.90). Let $v \in L_{⊥}^{\infty} (G)$ be given. We have to replace the sequence ${φ_{k}}$ defined by (2.2.92) with the sequence ${{\overline{φ}}_{k}} \subset C_{0, ⊥}^{\infty} (G)$ defined by

${\overline{φ}}_{k} = φ_{k} - {(φ_{k})}_{G} ψ for k \in N .$

Here ψ is any function in $C_{0}^{\infty} (G)$ such that $\int_{G} ψ d x = 1$ . For every $ε > 0$ , there exists $k_{ε} \in N$ such that ${‖ {\overline{φ}}_{k} ‖}_{L^{\infty} (G)} ⩽ {‖ v ‖}_{L^{\infty} (G)} + ε$ , provided that $k > k_{ε}$ . □

Proof of Theorem 2.0.5

Let $u \in L^{1} (G)$ . Then

$\begin{matrix} {‖ u - u_{G} ‖}_{L^{B} (G)} & = \sup_{v \in L^{\tilde{B}} (G)} \frac{\int_{G} (u - u_{G}) v d x}{{‖ v ‖}_{L^{\tilde{B}} (G)}} = \sup_{v \in L^{\tilde{B}} (G)} \frac{\int_{G} (u - u_{G}) (v - v_{G}) d x}{{‖ v ‖}_{L^{\tilde{B}} (G)}} \\ = \sup_{v \in L^{\tilde{B}} (G)} \frac{\int_{G} u (v - v_{G}) d x}{{‖ v ‖}_{L^{\tilde{B}} (G)}} ⩽ 3 \sup_{v \in L^{\tilde{B}} (G)} \frac{\int_{G} u (v - v_{G}) d x}{{‖ v - v_{G} ‖}_{L^{\tilde{B}} (G)}} \\ = 3 \sup_{v \in L_{⊥}^{\tilde{B}} (G)} \frac{\int_{G} u v d x}{{‖ v ‖}_{L^{\tilde{B}} (G)}} = 3 \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} u φ d x}{{‖ φ ‖}_{L^{\tilde{B}} (G)}} . \end{matrix}$

(2.2.99)

Note that the inequality in (2.2.99) holds since, by the first inequality in (1.2.3),

$\begin{matrix} {‖ v - v_{G} ‖}_{L^{\tilde{B}} (G)} & ⩽ {‖ v ‖}_{L^{\tilde{B}} (G)} + {‖ v_{G} ‖}_{L^{\tilde{B}} (G)} ⩽ {‖ v ‖}_{L^{\tilde{B}} (G)} + | v_{G} | {‖ 1 ‖}_{L^{\tilde{B}} (G)} \\ ⩽ {‖ v ‖}_{L^{\tilde{B}} (G)} + \frac{2}{| G |} {‖ v ‖}_{L^{\tilde{B}} (G)} {‖ 1 ‖}_{L^{B} (G)} {‖ 1 ‖}_{L^{\tilde{B}} (G)} \\ = {‖ v ‖}_{L^{\tilde{B}} (G)} + \frac{2}{| G |} {‖ v ‖}_{L^{\tilde{B}} (G)} \frac{1}{B^{- 1} (| G |)} \frac{1}{{\tilde{B}}^{- 1} (| G |)} ⩽ 3 {‖ v ‖}_{L^{\tilde{B}} (G)} . \end{matrix}$

The last equality in (2.2.99) follows from (2.2.88). By Theorem 2.1.6, applied with A and B replaced with $\tilde{B}$ and $\tilde{A}$ , respectively, there exists a constant $C = C (G, c)$ such that

$\begin{matrix} \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} u φ d x}{{‖ φ ‖}_{L^{\tilde{B}} (G)}} = \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} u div ({Bog}_{G} φ) d x}{{‖ φ ‖}_{L^{\tilde{B}} (G)}} \\ ⩽ C \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} u div ({Bog}_{G} φ) d x}{{‖ \nabla {Bog}_{G} φ ‖}_{L^{\tilde{A}} (G)}} \\ ⩽ C \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u div φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} d x = C {‖ u ‖}_{W^{- 1, A} (G)} . \end{matrix}$

(2.2.100)

The first inequality in (2.0.13) follows from (2.2.99) and (2.2.100). The second inequality is trivial, since

$\begin{matrix} {‖ u ‖}_{W^{- 1, A} (G)} \\ = \sup_{φ \in C_{0}^{\infty} (G)} \int_{G} \frac{u div φ}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} d x = \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} (u - u_{G}) div φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} \\ ⩽ C \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} (u - u_{G}) div φ d x}{{‖ div φ ‖}_{L^{\tilde{A}} (G)}} d x ⩽ C \sup_{φ \in C_{0, ⊥}^{\infty} (G)} \frac{\int_{G} (u - u_{G}) φ d x}{{‖ φ ‖}_{L^{\tilde{A}} (G)}} d x \\ ⩽ 2 C {‖ u - u_{G} ‖}_{L^{A} (G)}, \end{matrix}$

for some constant $C = C (d)$ . □

In the remaining part of this section, we focus on the second question raised in the introduction of this chapter, namely the reconstruction of the pressure π in a correct Orlicz space. In case of fluids governed by a general constitutive law, the function H belongs to some Orlicz space $L^{A} (G)$ . If $A \in Δ_{2} \cap \nabla_{2}$ , then $π \in L^{A} (G)$ as well. However, in general, one can only expect that π belongs to some larger Orlicz space $L^{B} (G)$ . The balance between the Young functions A and B is determined by conditions (2.0.11) and (2.0.12), as stated in the following result.

Theorem 2.2.10

Let A and B be Young functions fulfilling (2.0.11) and (2.0.12). Let G be a bounded domain with the cone property in $R^{d}$ , $d ⩾ 2$ . Assume that $H \in L^{A} (G)$ satisfies

$\int_{G} H : \nabla φ d x = 0$

for every $φ \in C_{0, div}^{\infty} (G)$ . Then there exists a unique function $π \in L_{⊥}^{B} (G)$ such that

$\int_{G} H : \nabla φ d x = \int_{G} π div φ d x$

(2.2.101)

for every $φ \in C_{0}^{\infty} (G)$ . Moreover, there exists a constant $C = C (G, c)$ such that

${‖ π ‖}_{L^{B} (G)} ⩽ C {‖ H - H_{G} ‖}_{L^{A} (G)},$

(2.2.102)

and

$\int_{G} B (| π |) d x ⩽ \int_{G} A (C | H - H_{G} |) d x .$

(2.2.103)

Here, c denotes the constant appearing in (2.0.11) and (2.0.12).

In particular, Theorem 2.2.10 reproduces, within a unified framework, various results appearing in the literature. For instance, when the power law model is in force, the function $A (t)$ is just a power $t^{q}$ for some $q > 1$ . So $L^{A} (G)$ agrees with the Lebesgue space $L^{q} (G)$ , and Theorem 2.2.10 recovers the fact that π belongs to the same Lebesgue space $L^{q} (G)$ .

As far as the simplified system (without convective term) for the Eyring–Prandtl model (see Chapter 4) is concerned, under appropriate assumptions on the function f one has that $H = S (ε (v)) + \nabla Δ^{- 1} f \in \exp L (G)$ . Hence, via Theorem 2.2.10, we obtain the existence of a pressure $π \in \exp L^{\frac{1}{2}} (G)$ . More generally, if $H \in \exp L^{β} (G)$ for some $β > 0$ , one has that $π \in \exp L^{β / (β + 1)} (G)$ . The complete system for the Eyring–Prandtl model, in the 2-dimensional case, admits a weak solution v such that $v \otimes v \in L \log L^{2} (G)$ and hence $H = S (ε (v)) + \nabla Δ^{- 1} f - v \otimes v \in L \log L^{2} (G)$ , see Chapter 4. Again, one cannot expect that the pressure π belongs to the same space. In fact, Theorem 2.2.10 implies the existence of a pressure $π \in L \log L (G)$ . This reproduces a result from [33]. In general, if $H \in L \log L^{α} (G)$ for some $α ⩾ 1$ , then we obtain that $π \in L \log L^{α - 1} (G)$ .

Proof of Theorem 2.2.10

By De Rahms Theorem, in the version of [131], there exists a distribution Ξ such that

$\int_{G} H : \nabla φ d x = Ξ (div φ)$

(2.2.104)

for every $φ \in C_{0}^{\infty} (G)$ . Replacing φ with ${Bog}_{G} (φ - φ_{G})$ in (2.2.104), where $φ \in C_{0}^{\infty} (G)$ , implies

$\int_{G} H : \nabla {Bog}_{G} (φ - φ_{G}) d x = Ξ (φ - φ_{G})$

for every $φ \in C_{0}^{\infty} (G)$ . We claim that the linear functional $C_{0}^{\infty} (G) ∋ φ \mapsto Ξ (φ - φ_{G})$ is bounded on $C_{0}^{\infty} (G)$ equipped with the $L^{\infty} (G)$ norm. Indeed, by (2.0.11), one has that $L^{A} (G) \to L \log L (G)$ . Moreover, by a special case of Theorem 2.1.6, $\nabla {Bog}_{G} : L_{⊥}^{\infty} (G) \to \exp L (G)$ . Thus, since $L \log L (G)$ and $\exp L (G)$ are Orlicz spaces generated by Young functions which are conjugate of each other,

$\begin{matrix} | \int_{G} H : \nabla {Bog}_{G} (φ - φ_{G}) d x | \\ ⩽ C {‖ H ‖}_{L \log L (G)} {‖ \nabla {Bog}_{G} {(φ - φ)}_{G}) ‖}_{\exp L (G)} \\ ⩽ C^{'} {‖ H ‖}_{L^{A} (G)} {‖ φ - φ_{G} ‖}_{L^{\infty} (G)} \\ ⩽ C^{″} {‖ H ‖}_{L^{A} (G)} {‖ φ ‖}_{L^{\infty} (G)}, \end{matrix}$

(2.2.105)

for every $φ \in C_{0}^{\infty} (G)$ , where $C = C (| G |, d)$ and $C^{'} = C^{'} (G, c)$ . Hence, the relevant functional can be continued to a bounded linear functional on $φ \in C_{0} (G)$ , with the same norm.

Now, as a consequence of Riesz's representation Theorem, there exists a Radon measure μ such that

$Ξ (φ - φ_{G}) = \int_{G} φ d μ$

for every $φ \in C_{0} (G)$ . Fix any open set $E \subset G$ . By Theorem 2.1.6 again, there exists a constant C such that

$\begin{matrix} μ (E) & = \sup_{φ \in C_{0}^{0} (E), {‖ φ ‖}_{\infty} = 1} Ξ (φ - φ_{G}) \\ = \sup_{φ \in C_{0}^{0} (E), {‖ φ ‖}_{\infty} = 1} \int_{G} H : \nabla {Bog}_{G} (φ - φ_{G}) d x \\ ⩽ \sup_{φ \in C_{0}^{0} (E), {‖ φ ‖}_{\infty} = 1} {‖ H ‖}_{L \log L (E)} {‖ \nabla {Bog}_{G} (φ - φ_{G}) ‖}_{\exp L (G)} \\ ⩽ C \sup_{φ \in C_{0}^{0} (E), {‖ φ ‖}_{\infty} = 1} {‖ H ‖}_{L \log L (E)} {‖ φ - {(φ)}_{G} ‖}_{L^{\infty} (G)} ⩽ C {‖ H ‖}_{L \log L (E)} . \end{matrix}$

(2.2.106)

One can verify that the norm ${‖ \cdot ‖}_{L \log L (E)}$ is absolutely continuous in the following sense. For every $ε > 0$ there exists $δ > 0$ such that ${‖ H ‖}_{L \log L (E)} < ε$ if $| E | < δ$ . Since any Lebesgue measurable set can be approximated from outside by open sets, inequality (2.2.106) implies that the measure μ is absolutely continuous with respect to the Lebesgue measure. Hence, μ has a density with respect to the Lebesgue measure. So Ξ can be represented by a function $π \in L^{1} (G)$ fulfilling (2.2.101) holds. The function π is uniquely determined if we assume that $π_{G} = 0$ . By this assumption, Theorem 2.0.5, and equation (2.2.101) we have that

$\begin{matrix} {‖ π ‖}_{L^{B} (G)} & ⩽ C {‖ \nabla π ‖}_{W^{- 1, A} (G)} = C \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} π div φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} \\ = C \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} H : \nabla φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} = C \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} (H - H_{G}) : \nabla φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} \\ ⩽ 2 C {‖ H - H_{G} ‖}_{L^{A} (G)}, \end{matrix}$

where $C = C (G, c)$ . This proves inequality (2.2.102). Inequality (2.2.103) follows from (2.2.102), by replacing A and B with kA and kB, respectively, with $k = \frac{1}{\int_{G} A (| H - H_{G} |) d x}$ , via an argument analogous to that of the proof of (2.1.42). □

2.3 Sharp conditions for Korn-type inequalities

In order to formulate the main result of this section we need to introduce the Banach function space

$\begin{matrix} E^{A} (G) & : = {u \in L^{A} (G) : ε (u) \in L^{A} (G)}, \\ {‖ u ‖}_{E^{A} (G)} & : = {‖ u ‖}_{L^{A} (G)} + {‖ ε (u) ‖}_{L^{A} (G)}, \end{matrix}$

and its subspace

$E_{0}^{A} (G) : = {u \in E^{A} (G) : the continuation of u by zero belongs to E^{A} (R^{d})} .$

If A is of power growth (belongs to $Δ_{2} \cap \nabla_{2}$ ) then there is no need for this definition. In this case the space $E^{A} (G)$ coincides with the standard Lebesgue spaces (Orlicz spaces). However, this is not true in our general context.

Theorem 2.3.11

Let G be any open bounded set in $R^{d}$ . Let A and B be Young functions such that

$t \int_{t_{0}}^{t} \frac{B (s)}{s^{2}} d s ⩽ A (c t) for t ⩾ t_{0},$

(2.3.107)

and

$t \int_{t_{0}}^{t} \frac{\tilde{A} (s)}{s^{2}} d s ⩽ \tilde{B} (c t) for t ⩾ t_{0},$

(2.3.108)

for some constants $c > 0$ and $t_{0} ⩾ 0$ . Then $E_{0}^{A} (G) \subset W_{0}^{1, B} (G)$ , and

${‖ \nabla u ‖}_{L^{B} (G)} ⩽ C {‖ ε (u) ‖}_{L^{A} (G)}$

(2.3.109)

for some constant $C = C (t_{0}, G, c, A, B)$ and for every $u \in E_{0}^{A} (G)$ . Moreover,

$\int_{G} B (C | \nabla u |) d x ⩽ C_{1} + \int_{G} A (C | ε (u) |) d x$

for every $u \in E_{0}^{A} (G)$ with $C_{1} = C_{1} (t_{0}, G, c, A, B)$ . If $t_{0} = 0$ then $C_{1} = 0$ and $C = C (G, c)$ .

Theorem 2.3.12

Let G be an open bounded Lipschitz domain in $R^{d}$ . Assume that A and B are Young functions fulfilling conditions (2.3.107) and (2.3.108). Then $E^{A} (G) \subset W^{1, B} (G)$ , and

${‖ \nabla u - {(\nabla u)}_{G} ‖}_{L^{B} (G)} ⩽ C {‖ ε (u) - {(ε (u))}_{G} ‖}_{L^{A} (G)}$

(2.3.110)

for some constant $C = C (t_{0}, G, c; A, B)$ and for every $u \in E^{A} (G)$ . Moreover,

$\int_{G} B (C | \nabla u - {(\nabla u)}_{G} |) d x ⩽ C_{1} + \int_{G} A (C | ε (u) - {(ε (u))}_{G} |) d x$

(2.3.111)

for every $u \in E^{A} (G)$ with $C_{1} = C_{1} (t_{0}, G, c, A, B)$ . If $t_{0} = 0$ then $C_{1} = 0$ and $C = C (G, c)$ .

Instead of subtracting the mean value it is also useful to subtract an element from the kernel of the differential operator ε given by

$R = {w : R^{d} \to R^{d} : w (x) = b + Q x : b \in R^{d}, Q \in R^{d \times d}, Q = - Q^{T}} .$

Corollary 2.3.1

Let G be an open bounded Lipschitz domain in $R^{d}$ . Assume that A and B are Young functions fulfilling conditions (2.3.107) and (2.3.108). Then $E^{A} (G) \subset W^{1, B} (G)$ , and there is $w \in R$ such that

${‖ \nabla u - \nabla w ‖}_{L^{B} (G)} ⩽ C {‖ ε (u) ‖}_{L^{A} (G)}$

(2.3.112)

for some constant $C = C (t_{0}, G, c; A, B)$ and for every $u \in E^{A} (G)$ . Moreover,

$\int_{G} B (C | \nabla u - \nabla w |) d x ⩽ C_{1} + \int_{G} A (C | ε (u) |) d x$

(2.3.113)

for every $u \in E^{A} (G)$ with $C_{1} = C_{1} (t_{0}, G, c, A, B)$ . If $t_{0} = 0$ then $C_{1} = 0$ and $C = C (G, c)$ .

Theorem 2.3.13

Let G be an open bounded Lipschitz domain in $R^{d}$ and let A be any Young function. Then there is $w \in R$ such that

${‖ u - w ‖}_{L^{A} (G)} ⩽ C {‖ ε (u) ‖}_{L^{A} (G)}$

(2.3.114)

for some constant $C = C (G, A)$ and for every $u \in E^{A} (G)$ . Moreover,

$\int_{G} A (C | u - w |) d x ⩽ \int_{G} A (C | ε (u) |) d x$

(2.3.115)

for every $u \in E^{A} (G)$ .

Proof of Theorem 2.3.12

Let us introduce negative norms for single partial derivatives as follows. Given $u \in L^{1} (G)$ , we set

${‖ \frac{\partial u}{\partial x_{k}} ‖}_{W^{- 1, A} (G)} = \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u \frac{\partial φ}{\partial x_{k}} d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} for k = 1, \dots, d .$

Obviously, the following holds

${‖ \frac{\partial u}{\partial x_{k}} ‖}_{W^{- 1, A} (G)} ⩽ {‖ \nabla u ‖}_{W^{- 1, A} (G)} for k = 1, \dots d .$

(2.3.116)

On the other hand,

$\begin{matrix} {‖ \nabla u ‖}_{W^{- 1, A} (G)} & = \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u div φ d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} = \sup_{φ \in C_{0}^{\infty} (G)} \sum_{k = 1}^{d} \frac{\int_{G} u \frac{\partial φ_{k}}{\partial x_{k}} d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} \\ ⩽ \sup_{φ \in C_{0}^{\infty} (G)} \sum_{k = 1}^{d} \frac{\int_{G} u \frac{\partial φ_{k}}{\partial x_{k}} d x}{{‖ \nabla φ_{k} ‖}_{L^{\tilde{A}} (G)}} ⩽ \sum_{k = 1}^{d} \sup_{φ \in C_{0}^{\infty} (G)} \frac{\int_{G} u \frac{\partial φ}{\partial x_{k}} d x}{{‖ \nabla φ ‖}_{L^{\tilde{A}} (G)}} \\ = \sum_{k = 1}^{d} {‖ \frac{\partial u}{\partial x_{k}} ‖}_{W^{- 1, A} (G)}, \end{matrix}$

(2.3.117)

where $φ_{k}$ denotes the k-th component of φ. Next, notice the identity

$\frac{\partial^{2} v_{i}}{\partial x_{k} \partial x_{j}} = \frac{\partial {(ε (v))}_{i j}}{\partial x_{k}} + \frac{\partial {(ε (v))}_{i k}}{\partial x_{j}} - \frac{\partial {(ε (v))}_{j k}}{\partial x_{i}}$

(2.3.118)

for every weakly differentiable function $v : G \to R^{d}$ . Thus, the following chain holds for every $u \in W^{1, 1} (G) \cap E^{A} (G)$

$\begin{matrix} {‖ \nabla u - {(\nabla u)}_{G} ‖}_{L^{B} (G)} ⩽ C \sum_{i, j = 1}^{d} {‖ \frac{\partial u_{i}}{\partial x_{j}} - {(\frac{\partial u_{i}}{\partial x_{j}})}_{G} ‖}_{L^{B} (G)} \\ ⩽ C \sum_{i, j = 1}^{n} {‖ \nabla \frac{\partial u_{i}}{\partial x_{j}} ‖}_{W^{- 1, A} (G)} ⩽ C \sum_{i, j, k = 1}^{d} {‖ \frac{\partial^{2} u_{i}}{\partial x_{k} \partial x_{j}} ‖}_{W^{- 1, A} (G)} \\ ⩽ C \sum_{i, j, k = 1}^{d} ({‖ \frac{\partial {(ε (u))}_{i j}}{\partial x_{k}} ‖}_{W^{- 1, A} (G)} + {‖ \frac{\partial {(ε (u))}_{i k}}{\partial x_{j}} ‖}_{W^{- 1, A} (G)} + {‖ \frac{\partial {(ε (u))}_{j k}}{\partial x_{i}} ‖}_{W^{- 1, A} (G)}) \\ ⩽ C \sum_{i, j = 1}^{d} {‖ \nabla {(ε (u))}_{i j} ‖}_{W^{- 1, A} (G)} ⩽ C \sum_{i, j = 1}^{n} {‖ {(ε (u))}_{i j} - {({(ε (u))}_{i j})}_{G} ‖}_{L^{A} (G)} \\ ⩽ C {‖ ε (u) - {(ε (u))}_{G} ‖}_{L^{A} (G)} . \end{matrix}$

(2.3.119)

Note that the second inequality holds by Theorem 2.0.5 (see Remark 2.0.6 for the case $t_{0} > 0$ ), the third by (2.3.117), the fourth by (2.3.118), the fifth by (2.3.116), and the sixth by Theorem 2.0.5 again.

Let us turn to the modular version. Suppose first that $t_{0} = 0$ in (2.3.107) and (2.3.108). An inspection of the proof of Theorem (2.3.119) and of the statement of Theorem 2.0.5 tells us that the constant C in (2.3.119) depends only on G and on the constant c appearing in conditions (2.3.107) and (2.3.108). These conditions continue to hold if the functions A and B are replaced by the functions $A_{M}$ and $B_{M}$ given by $A_{M} (t) = A (t) / M$ and $B_{M} (t) = B (t) / M$ for some positive constant M. Given a function $u \in E_{0}^{A} (G)$ , set

$M = \int_{Ω} A (C | ε (u) - {(ε (u))}_{G} |) d x .$

If $M = \infty$ , then inequality (2.3.111) holds trivially. We may thus assume that

${‖ \nabla u - {(\nabla u)}_{G} ‖}_{L^{A_{M}} (G)} ⩽ 1 .$

Hence, by inequality (2.3.110) applied with A and B replaced by $A_{M}$ and $B_{M}$ , we obtain

$\int_{Ω} B (| \nabla u - {(\nabla u)}_{G} |) d x ⩽ \int_{Ω} A (C | ε (u) - {(ε (u))}_{G} |) d x .$

(2.3.120)

This is (2.3.111) with $C_{1} = 0$ .

Assume next that (2.3.107) and (2.3.108) just hold for some $t_{0} > 0$ . The functions A and B can be replaced by new Young functions $\overline{A}$ and $\overline{B}$ , equivalent to A and B near infinity, and such that (2.3.107) and (2.3.108) hold for the new functions with $t_{0} = 0$ . The same argument as above implies (2.3.120) with A and B replaced by $\overline{A}$ and $\overline{B}$ ., i.e., we have

$\int_{Ω} \overline{B} (| \nabla u - {(\nabla u)}_{G} |) d x ⩽ \int_{Ω} \overline{A} (C | ε (u) - {(ε (u))}_{G} |) d x$

(2.3.121)

for some constant C. Since $\overline{A}$ and $\overline{B}$ are equivalent to A and B near infinity, there exist constants $t_{0} > 0$ and $c > 0$ such that

$\overline{A} (t) ⩽ A (c t) if t ⩾ t_{0}, B (t) ⩽ \overline{B} (c t) if t ⩾ t_{0} .$

(2.3.122)

From (2.3.121) and (2.3.122) we obtain

$\begin{matrix} \int_{G} B (| \nabla u - {(\nabla u)}_{G} |) d x \\ = \int_{{| \nabla u - {(\nabla u)}_{G} | < t_{0}}} B (| \nabla u - {(\nabla u)}_{G} |) d x \\ + \int_{{| \nabla u - {(\nabla u)}_{G} | ⩾ t_{0}}} B (| \nabla u - {(\nabla u)}_{G} |) d x \\ ⩽ B (t_{0}) | G | + \int_{G} \overline{B} (c | \nabla u - {(\nabla u)}_{G} |) d x \\ ⩽ B (t_{0}) | G | + \int_{G} \overline{A} (C c | ε (u) - {(ε (u))}_{G} |) d x \\ ⩽ B (t_{0}) | G | + \int_{{C c | ε (u) - {(ε (u))}_{G} | < t_{0}}} \overline{A} (C c | ε (u) - {(ε (u))}_{G} |) d x \\ + \int_{{C c | ε (u) - {(ε (u))}_{G} | ⩾ t_{0}}} \overline{A} (C c | ε (u) - {(ε (u))}_{G} |) d x \\ ⩽ (B (t_{0}) + A (c t_{0})) | G | + \int_{G} A (C c^{2} | ε (u) - {(ε (u))}_{G} |) d x, \end{matrix}$

namely (2.3.111) with $C_{1} = (B (t_{0}) + A (c t_{0})) | G |$ . □

Proof of Theorem 2.3.11

If $u \in W_{0}^{1, 1} (G)$ , then ${(\nabla u)}_{G} = {(ε (u))}_{G} = 0$ , and Theorem 2.3.11 implies the claim. □

Proof of Corollary 2.3.1

Let us choose $w \in R$ through the condition $\nabla w = {(\nabla u)}_{G} - {(ε (u))}_{G}$ . Then the following holds by (2.3.110)

${‖ \nabla u - \nabla w ‖}_{L^{B} (G)} ⩽ {‖ \nabla u - {(\nabla u)}_{G} ‖}_{L^{B} (G)} + {‖ {(ε (u))}_{G} ‖}_{L^{B} (G)} ⩽ c {‖ ε (u) ‖}_{L^{A} (G)} .$

The integral version follows form the norm version as in the proof of (2.3.111). □

Proof of Theorem 2.3.13

Step 1: Star-shaped domains.

First, we assume that G is star-shaped with respect to some ball $B \subset G$ . Then, according to formula (2.39) in [126] each $v \in C^{\infty} (G)$ can be represented as

$u (x) = P_{G} u (x) + L_{ε} (ε (u)) (x),$

(2.3.123)

where $P_{G} : L^{1} (G) \to R$ is a suitable linear projection operator into the space of rigid motions (compare (2.33)–(2.39) in [126]) defined even for $L^{1} (G)$ functions. Furthermore, the operator $L_{ε}$ is a weakly singular integral operator (compare (2.37) in [126]) given by

$\begin{matrix} L_{ε} (φ) & : = S_{ε} (ψ) + T_{ε} (ψ), ψ \in C^{\infty} (G), \\ S_{ε}^{i} (φ) (x) & : = \int_{G} \frac{G^{i} (x, e)}{| x - z |^{d - 1}} : ψ (z) d z, i = 1, . . ., d, \\ T_{ε}^{i} (φ) (x) & : = \int_{G} θ^{i} (x, z) : ψ (z) d z, i = 1, . . ., d, \end{matrix}$

(2.3.124)

for $x \in G$ . Here, $G^{i} (x, e)$ are smooth functions ( $e : = (x - z) / | x - z |$ ) and $θ^{i} (x, z)$ are bounded continuous functions (both with values in $R^{d \times d}$ ); see [126] after (2.38).

Since the kernel of $S_{ε}$ is essentially homogeneous of degree $1 - d$ the theory of Riesz potentials applies (see, e.g. [134] or [90]). Hence we have $S_{ε} : L^{1} (G) \to L^{1} (G)$ . Of course, the same is true for $T_{ε}$ because the $θ^{i} (x, z)$ are bounded. So, we obtain

$L_{ε} : L^{1} (G) \to L^{1} (G)$

(2.3.125)

continuously. We claim that equation (2.3.123) continues to hold even if $u \in E^{A} (G)$ . From [138, Proposition 1.3, Chapter 1], we deduce that $C^{\infty} (G)$ is dense in $E^{1} (G)$ . Thus, there exists a sequence ${u_{m}} \subset C^{\infty} (G)$ such that

$u_{m} \to u in E^{1} (G) .$

We already know that formula (2.3.123) holds with u replaced by $u_{m}$ . Thus we can pass to the limit (passing to a subsequence if necessary) in the representation formula (2.3.123) applied to $u_{m}$ . This implies that it continues to hold also for u. Note that this is a consequence of (2.3.125).

Now, we turn to the case of a Lipschitz domain. In order to do so, we consider a general linear projection operator $Π : L^{1} (G) \to Σ$ such that

${‖ Π u ‖}_{L^{1} (G)} ⩽ C {‖ u ‖}_{L^{1} (G)},$

(2.3.126)

for some constant C, and every $u \in L^{1} (G)$ . We claim that there exists a constant $C^{'}$ such that

$\inf_{w \in R} {‖ u - w ‖}_{L^{A} (G)} ⩽ {‖ u - Π u ‖}_{L^{A} (G)} ⩽ C (A) \inf_{w \in R} {‖ u - w ‖}_{L^{A} (G)},$

(2.3.127)

for every $u \in L^{A} (G)$ and for every Young function A.

The left-wing inequality in (2.3.127) is trivial. As far as the right-wing inequality is concerned, given any $w \in R$ , and any u in $L^{A} (G)$ we set

$v = w + {(u - w)}_{G} .$

Since Π, restricted to $R$ , agrees with the identity map, we have that $Π v = v$ . As a consequence,

$u - Π u = (u - v) - Π (u - v) .$

Thus,

${‖ u - Π u ‖}_{L^{A} (G} ⩽ {‖ u - v ‖}_{L^{A} (G)} + {‖ Π (u - v) ‖}_{L^{A} (G)} .$

(2.3.128)

By the triangle inequality,

${‖ u - v ‖}_{L^{A} (G)} = {‖ u - w - {(u - w)}_{G} ‖}_{L^{A} (G)} ⩽ 2 {‖ u - w ‖}_{L^{A} (G)} .$

(2.3.129)

Since the range of Π is a finite dimensional space, where all norms are equivalent, there exists a constant $C^{″}$ such that

${‖ Π (u - v) ‖}_{L^{A} (G)} + {‖ \nabla Π (u - v) ‖}_{L^{A} (G)} ⩽ C^{″} {‖ Π (u - v) ‖}_{L^{1} (G)} .$

(2.3.130)

Inequality (2.3.126) ensures that

${‖ Π (u - v) ‖}_{L^{1} (G)} ⩽ C {‖ u - v ‖}_{L^{1} (G)} = C {‖ u - w - {〈 u - w 〉}_{G} ‖}_{L^{1} (G)} .$

(2.3.131)

Now, by the triangle inequality,

${‖ u - w - {〈 u - w 〉}_{G} ‖}_{L^{1} (G)} ⩽ 2 {‖ u - w ‖}_{L^{1} (G)} .$

(2.3.132)

On the other hand, our assumptions on G ensure that a Poincaré-type inequality holds in $W^{1, 1} (G)$ . Hence there exists a constant C such that

${‖ u - w - {〈 u - w 〉}_{G} ‖}_{L^{1} (G)} ⩽ C {‖ \nabla (u - w) ‖}_{L^{1} (G)} .$

(2.3.133)

Altogether, inequality (2.3.127) follows.

Step 2: the union of two sets.

Let A and B be Young functions. An open set G in $R^{d}$ , $d ⩾ 2$ , will be called admissible with respect to the couple $(A, B)$ if there exists a constant C such that

$\inf_{w \in R} {‖ u - w ‖}_{L^{A} (G)} ⩽ C {‖ ε (u) ‖}_{L^{A} (G)}$

(2.3.134)

for every $u \in E^{A} (G)$ . We will show that, under certain assumptions, the union of two admissible sets will be admissible. So, assume that $G_{1}$ and $G_{2}$ are bounded connected open sets in $R^{d}$ with Lipschitz boundary. Assume that each of them is admissible with respect to $(A, B)$ , and $G_{1} \cap G_{2} \neq \emptyset$ . Then we have

$the set G_{1} \cup G_{2} is admissible with respect to (A, B) .$

(2.3.135)

In order to prove (2.3.135) we consider a ball $B \subset G_{1} \cap G_{2}$ . Fix $ω \in C_{0}^{\infty} (B)$ . Denote by $P_{1}$ the space of polynomials of degree not exceeding 1, and by $Π_{2} u \in P_{1}$ the averaged Taylor polynomial of second-order with respect to ω of a function $u \in L^{1} (G_{1} \cup G_{2})$ – see [37]. The operator $Π_{2} : L^{1} (G_{1} \cup G_{2}) \to P_{1}$ is linear, and, by [37, Corollary 4.1.5], there exists a constant C such that

${‖ Π_{2} u ‖}_{L^{1} (G_{1} \cup G_{2})} \leq C {‖ u ‖}_{L^{1} (B)}$

for every $u \in L^{1} (G_{1} \cup G_{2})$ . Let us denote by $Π_{R}$ the $L^{2}$ -orthogonal projection from $P_{1}$ into $R$ . We have that

${‖ Π_{R} p ‖}_{L^{1} (G_{1} \cup G_{2})} \leq c {‖ p ‖}_{L^{1} (G_{1} \cup G_{2})}$

for every $p \in P_{1}$ . Thus, the linear operator $Π = Π_{R} \circ Π_{2}$ maps $L^{1} (G_{1} \cup G_{2})$ into $R$ , and there exists a constant C such that

${‖ Π u ‖}_{L^{1} (G_{j})} ⩽ {‖ Π u ‖}_{L^{1} (G_{1} \cup G_{2})} \leq C {‖ u ‖}_{L^{1} (B)} ⩽ C {‖ u ‖}_{L^{1} (G_{j})} j = 1, 2$

(2.3.136)

for every $u \in L^{1} (G_{1} \cup G_{2})$ . Due to inequality (2.3.136), inequality (2.3.127) ensures that there exists a constant C such that

$\begin{matrix} \inf_{w \in R} {‖ u - w ‖}_{L^{A} (G_{1} \cup G_{2})} & ⩽ {‖ u - Π u ‖}_{L^{A} (G_{1} \cup G_{2})} ⩽ \sum_{j = 1, 2} {‖ u - Π u ‖}_{L^{A} (G_{j})} \\ ⩽ C \sum_{j = 1, 2} \inf_{w \in R} {‖ u - w ‖}_{L^{A} (G_{j})} \end{matrix}$

(2.3.137)

for every $u \in L^{A} (G)$ . The conclusion (2.3.135) follows from (2.3.137) and (2.3.134) applied with $G = G_{j}$ , for $j = 1, 2$ .

Step 3: The general cases.

Any open Lipschitz domain G is the finite union of open sets $G_{i}$ , $i = 1, \dots, k$ , starshaped with respect to a ball. Since G is connected, after, possibly, relabelling, we may assume that the sets $\cup_{i = 1}^{j - 1} G_{i}$ and $G_{j}$ have a non-empty intersection. The conclusion then follows from repeated use of (2.3.135). □

Remark 2.3.7

A proof of Corollary 2.3.1 can also be given based on the representation formula (2.3.123). After differentiating (2.3.123) the main part is a singular operator. It can be shown that it enjoys the properties (2.1.34)–(2.1.39). Hence it is continuous from $L^{A} (G)$ to $L^{B} (G)$ thanks to Theorem 2.1.9. We refer to [31] and [46] for details.

In the following we are concerned with the necessity of the conditions (2.3.107) and (2.3.108) for a Korn-type inequality.

Theorem 2.3.14

Let G be an open bounded set in $R^{d}$ . Let A and B be Young functions such that

${‖ \nabla u ‖}_{L^{B} (G)} ⩽ C {‖ ε (u) ‖}_{L^{A} (G)}$

(2.3.138)

for some constant C and for every $u \in W_{0}^{1, 1} (G) \cap E_{0}^{A} (G)$ . Then conditions (2.3.107) and (2.3.108) hold.

Our proof of inequality (2.3.107) is based on the technique of laminates as in [52]. In general a first order laminate is a probability measure ν on $R^{d \times d}$ given by

$ν = λ δ_{A} + (1 - λ) δ_{B}$

where $λ \in (0, 1)$ and $A, B \in R^{d \times d}$ with $rank (A - B) = 1$ . Here $δ_{X}$ denotes the Dirac measure concentrated on the matrix X. We say ν has average C if $λ A + (1 - λ) B = C$ . We obtain a second order laminate if we replace $δ_{A}$ (resp. $δ_{B}$ ) by a first order laminate with average A (resp. B). Iteratively we can define laminates of arbitrary order with a given average. For a detailed discussion we refer to [102] and [118].

Lemma 2.3.1

[52], eq. (5)

Let ν be a laminate with average C, then there is a sequence of uniformly Lipschitz continuous functions $u_{i} : {(0, r)}^{d} \to R^{d}$ with boundary data Cx such that

$\int_{{(0, r)}^{d}} Φ (| \nabla u_{i} |) d x ⟶ r^{d} \int_{R^{d \times d}} Φ (| X |) d ν (X),$

(2.3.139)

for every continuous function Φ.

Proof of Theorem 2.3.14

Part 1: Inequality (2.3.108) holds.

Assume, without loss of generality, that the unit ball $B_{1}$ , centered at 0, is contained in G, and denote by $ω_{d}$ its Lebesgue measure. Let us preliminarily observe that inequality (2.3.138) implies that

$A dominates B near infinity .$

(2.3.140)

Indeed, given any nonnegative function $h \in L^{A} (0, ω_{d})$ , consider the function $v : B_{1} \to R^{d}$ given by

$v (x) = (\int_{| x |}^{1} h (ω_{d} r^{d}) d r, 0, \dots, 0) for x \in B_{1} .$

Then $v \in L^{A} (B_{1})$ , and

$| ε (v) (x) | ⩽ | \nabla v (x) | = h (ω_{d} | x |^{d}) for x \in B_{1} .$

An application of (2.3.138), with u replaced by v, shows that

${‖ h ‖}_{L^{B} (0, ω_{d})} = {‖ \nabla v ‖}_{L^{B} (Ω)} ⩽ C {‖ ε (v) ‖}_{L^{A} (Ω)} ⩽ C {‖ \nabla v ‖}_{L^{A} (Ω)} = {‖ h ‖}_{L^{A} (0, ω_{d})} .$

Thus $L^{A} (0, ω_{d}) \to L^{B} (0, ω_{d})$ , and (2.3.140) follows.

Now, given h as above, define the function $ρ : [0, 1] \to [0, \infty]$ by

$ρ (r) = \int_{r}^{1} \frac{h (ω_{d} t^{d})}{t} d t for r \in [0, 1],$

and the function $v : B_{1} \to R^{d}$ by

$u (x) = Q x ρ (| x |) for x \in B_{1},$

where $Q \in R^{d \times d}$ is any skew-symmetric matrix such that $| Q | = 1$ . We have that u is a weakly differentiable function, and

$\begin{matrix} ε (u) (x) & = \frac{Q x ⊙ x}{| x |^{2}} ρ^{'} (| x |) | x |, \\ \nabla u (x) & = Q ρ (| x |) + \frac{Q x \otimes x}{| x |^{2}} ρ^{'} (| x |) | x | \end{matrix}$

for a.e. $x \in B_{1}$ . Here, ⊙ denotes the symmetric part of the tensor product of two vectors in $R^{d}$ . Hence,

$\begin{matrix} | ε (u) (x) | & ⩽ | ρ^{'} (| x |) | | x | = h (ω_{d} | x |), \\ ρ (| x |) & ⩽ | \nabla u (x) | + | ρ^{'} (| x |) | | x | = | \nabla u (x) | + h (ω_{d} | x |) \end{matrix}$

for a.e. $x \in B_{1}$ . Thus, due to (2.3.138) and (2.3.140),

$\begin{matrix} {‖ \int_{s}^{ω_{d}} \frac{h (r)}{r} d r ‖}_{L^{B} (0, ω_{d})} = {‖ \int_{| x |}^{1} \frac{h (ω_{d} t^{d})}{t} d t ‖}_{L^{B} (B_{1})} = {‖ ρ (| x |) ‖}_{L^{B} (B_{1})} \\ ⩽ {‖ \nabla u ‖}_{L^{B} (B_{1})} + {‖ h (ω_{d} | x |^{d}) ‖}_{L^{B} (B_{1})} ⩽ C {‖ ε (u) ‖}_{L^{A} (B_{1})} + {‖ h (ω_{d} | x |^{d}) ‖}_{L^{A} (B_{1})} \\ ⩽ C^{'} {‖ h (ω_{d} | x |^{d}) ‖}_{L^{A} (B_{1})} = C^{'} {‖ h (s) ‖}_{L^{A} (0, ω_{d})} \end{matrix}$

(2.3.141)

for suitable constants C and $C^{'}$ . Thanks to the arbitrariness of h, inequality (2.3.141) implies, via Lemma 1.2.1, that (5.3.12) holds for some c and $t_{0}$ .

Part 2: Inequality (2.3.107) holds.

Let us preliminarily note that, if $A (t) = \infty$ for large t, then (2.3.107) holds trivially. We may thus assume that A is finite-valued, and hence continuous. By (2.3.140), the function B is also finite-valued and continuous.

For ease of notations, we hereafter focus on case when $d = 2$ . An analogous argument carries over to any dimension along the lines of [52, Lemma 3]. Given $a, b \in R$ , we define the matrix $G_{a, b}$ as

$G_{a, b} = (\begin{matrix} 0 & a \\ b & 0 \end{matrix}),$

and set $δ_{a, b} = δ_{G_{a, b}}$ . Next, we define the sequence ${μ^{(m)}}$ of laminates of order 2m iteratively by

${\begin{matrix} μ^{(0)} = δ_{t, t}, \\ μ^{(m)} = \frac{1}{3} δ_{2^{- m} t, - 2^{- m} t} + \frac{1}{6} δ_{- 2^{1 - m} t, 2^{1 - m} t} + \frac{1}{2} μ^{(m - 1)} \end{matrix}$

(2.3.142)

for $m \in N$ . We claim that $μ^{(m)}$ is a laminate with average $G_{2^{- m} t, 2^{- m} t}$ for $m \in N$ . Indeed, the following holds

$μ^{(m)} = \frac{1}{4} δ_{- 2^{1 - m} t, 2^{1 - m} t} + \frac{3}{4} μ^{(m - 1)} .$

(2.3.143)

Since $rank (G_{- t, t} - G_{t, t}) = 1$ , the right-hand side of (2.3.143) is a laminate with average $G_{2^{- 1} t, t}$ for $m = 1$ . Hence, $μ^{(1)}$ is a laminate with average $G_{2^{- 1} t, 2^{- 1} t}$ . An induction argument then proves our claim. Now, note the representation formula

$μ^{(m)} = 2^{- m} δ_{t, t} + \sum_{k = 1}^{m} (\frac{1}{3} 2^{k - m} δ_{2^{- k} t, - 2^{- k} t} + \frac{1}{6} 2^{k - m} δ_{- 2^{1 - k} t, 2^{1 - k} t})$

(2.3.144)

for $m \in N$ . We remark that $δ_{t, t}$ is concentrated at a symmetric matrix, whereas the sum in (2.3.144) is concentrated at skew-symmetric matrices. We define the functions $Φ_{j} : R^{2 \times 2} \to [0, \infty)$ , for $j = 1, 2$ , by

$\begin{matrix} Φ_{1} (X) & = A (| X^{sym} - G_{2^{- m} t, 2^{- m} t} |), \\ Φ_{2} (X) & = B (C^{- 1} | X - G_{2^{- m} t, 2^{- m} t} |), \end{matrix}$

for $X \in R^{2 \times 2}$ . Here, $X^{sym} = \frac{1}{2} (X + X^{T})$ is the symmetric part of X, and C is the constant appearing in (2.3.138). Fix $m \in N$ . Without loss of generality, we may assume that $0 \in G$ . We choose $r > 0$ so small that ${(0, r)}^{2} \subset G$ . Let $m \in N$ be arbitrary. Due to Lemma 2.3.1 applied with $ν = μ^{(m)}$ , there exists a sequence ${u_{i}}$ of Lipschitz continuous functions $u_{i} : {(0, r)}^{2} \to R^{2}$ , such that $u_{i} (x) = G_{2^{- m} t, 2^{- m} t} x$ on $\partial {(0, r)}^{2}$ , and

$\lim_{i \to \infty} \int_{{(0, r)}^{2}} Φ_{j} (\nabla u_{i}) d x = r^{2} \int_{R^{2 \times 2}} Φ_{j} (X) d μ^{(m)} (X) for j = 1, 2 .$

(2.3.145)

We define the sequence ${v_{i}}$ of functions $v_{i} : G \to R$ by $v_{i} (x) = u_{i} (x) - G_{2^{- m} t, 2^{- m} t} x$ if $x \in {(0, r)}^{2}$ , and $v_{i} (x) = 0$ if $G ∖ {(0, r)}^{2}$ . Then $v_{i} \in W_{0}^{1, \infty} (G)$ , and, by (2.3.145),

$\lim_{i \to \infty} \int_{G} A (| ε (v_{i}) |) d x = \lim_{i \to \infty} \int_{{(0, r)}^{2}} A (| ε {(v)}_{i} |) d x$

(2.3.146)

$\begin{matrix} = r^{2} \int_{R^{2 \times 2}} A (| (X^{sym} - G_{2^{- m} t, 2^{- m} t}) |) d μ^{(m)} (X), \\ \lim_{i \to \infty} \int_{Ω} B (C^{- 1} | \nabla v_{i} |) d x & = \lim_{i \to \infty} \int_{{(0, r)}^{2}} B (C^{- 1} | \nabla v_{i} |) d x \\ = r^{2} \int_{R^{2 \times 2}} B (C^{- 1} | X - G_{2^{- m} t, 2^{- m} t} |) d μ^{(m)} (X) . \end{matrix}$

(2.3.147)

We have the following chain

$\begin{matrix} \int_{R^{2 \times 2}} A (| X^{sym} - G_{2^{- m} t, 2^{- m} t} |) d μ^{(m)} (X) \\ ⩽ \frac{1}{2} \int_{R^{2 \times 2}} A (2 | X^{sym} |) d μ^{(m)} (X) + \frac{1}{2} \int_{R^{2 \times 2}} A (2 | G_{2^{- m} t, 2^{- m} t} |) d μ^{(m)} (X) \\ = \frac{1}{2} 2^{- m} A (2 | G_{t, t} |) + \frac{1}{2} A (2 | G_{2^{- m} t, 2^{- m} t} |) \\ = \frac{1}{2} 2^{- m} A (2 | G_{t, t} |) + \frac{1}{2} A (2 2^{- m} | G_{t, t} |) \\ ⩽ 2^{- m} A (2 | G_{t, t} |) . \end{matrix}$

(2.3.148)

Here the first inequality holds since A is convex, the first equality holds due to (2.3.144) and to the fact that $μ^{(m)}$ is a probability measure, and the last inequality follows from (1.2.6). Combining (2.3.146) with (2.3.148) implies

$\lim_{i \to \infty} \int_{G} A (| ε (v_{i}) |) d x ⩽ r^{2} 2^{- m} A (2 | G_{t, t} |) .$

(2.3.149)

Since A is a continuous function, there exists $t_{m} \in (0, \infty)$ such that

$r^{2} 2^{- m} A (2 | G_{t_{m}, t_{m}} |) = \frac{1}{2} .$

(2.3.150)

Thanks to (1.2.6), there exists $t_{0} > 0$ , independent of m, such that

$t_{m} ⩽ t_{0} 2^{m} .$

(2.3.151)

Therefore, by neglecting, if necessary, a finite number of terms of the sequence ${v_{i}}$ , we can assume that

$\int_{G} A (| ε (v_{i}) |) d x ⩽ 1$

for $i \in N$ . Hence, ${‖ ε (v_{i}) ‖}_{A} ⩽ 1$ for $i \in N$ , and, by (2.3.138), ${‖ \nabla v_{i} ‖}_{B} ⩽ C$ for $i \in N$ . Thus,

$\int_{G} B (C^{- 1} | \nabla v_{i} |) d x ⩽ 1$

for $i \in N$ . Combining the latter inequality with equation (2.3.147) implies us that

$r^{2} \int_{R^{2 \times 2}} B (C^{- 1} | X - G_{2^{- m} t_{m}, 2^{- m} t_{m}} |) d μ^{(m)} (X) ⩽ 1 .$

(2.3.152)

Next, one can make use of (2.3.144) and obtain the following chain

$\begin{matrix} r^{- 2} & ⩾ \int_{R^{2 \times 2}} B (C^{- 1} | X - G_{2^{- m} t_{m}, 2^{- m} t_{m}} |) d μ^{(m)} (X) \\ ⩾ 2^{- m} B (C^{- 1} (1 - 2^{- m}) | G_{t_{m}, t_{m}} |) + \sum_{k = 1}^{m} \frac{1}{3} 2^{k - m} B (C^{- 1} (2^{- k} - 2^{- m}) | G_{t_{m}, t_{m}} |) \\ + \sum_{k = 1}^{m} \frac{1}{6} 2^{k - m} B (C^{- 1} (2^{1 - k} - 2^{- m}) | G_{t_{m}, t_{m}} |) \\ ⩾ \sum_{k = 1}^{m - 1} \frac{1}{3} 2^{k - m} B (C^{- 1} (2^{- k} - 2^{- m}) | G_{t_{m}, t_{m}} |) \\ ⩾ \sum_{k = 1}^{m - 1} \frac{1}{3} 2^{k - m} B (C^{- 1} 2^{- k - 1} | G_{t_{m}, t_{m}} |) \\ ⩾ \sum_{k = 1}^{m - 1} \frac{1}{3} \frac{1}{2 C} 2^{- m} t_{m} \frac{B (\frac{1}{2 C} 2^{- k} t_{m})}{\frac{1}{2 C} 2^{- k} t_{m}} . \end{matrix}$

(2.3.153)

It follows from (2.3.150), (2.3.152) and (2.3.153) that

$2 \cdot 2^{- m} A (2 | G_{t_{m}, t_{m}} |) ⩾ \sum_{k = 1}^{m - 1} \frac{1}{3} \frac{1}{2 C} 2^{- m} t_{m} \frac{B (\frac{1}{2 C} 2^{- k} t_{m})}{\frac{1}{2 C} 2^{- k} t_{m}} .$

Hence, by (2.3.151),

$\begin{matrix} A (c^{″} t_{m}) & ⩾ c t_{m} \sum_{k = 1}^{m - 1} \frac{B (\frac{1}{2 C} 2^{- k} t_{m})}{\frac{1}{2 C} 2^{- k} t_{m}} ⩾ c^{'} t_{m} \int_{2^{- m} \frac{t_{m}}{4 C}}^{\frac{t_{m}}{4 C}} \frac{B (s)}{s^{2}} d s \\ ⩾ c^{'} t_{m} \int_{\frac{t_{0}}{2 C}}^{\frac{t_{m}}{4 C}} \frac{B (s)}{s^{2}} d s \end{matrix}$

(2.3.154)

for suitable positive constants c, $c^{'}$ $c^{″}$ . Since $\lim_{m \to \infty} t_{m} = \infty$ , one can find $\hat{t} ⩾ \frac{t_{0}}{2 C}$ such that, if $t > \hat{t}$ , then there exists $m \in N$ such that $t_{m} ⩽ t < t_{m + 1}$ . Moreover, $\hat{t}$ can be chosen so large that A is invertible on $[\hat{t}, \infty)$ and

$t_{m} = c_{1} A^{- 1} (c_{2} 2^{m})$

for some positive constants $c_{1}, c_{2}$ . By (1.2.7), the latter equation ensures that $t_{m + 1} ⩽ 2 t_{m}$ for $m \in N$ . Thus, due to inequality (2.3.154),

$\begin{matrix} A (2 c^{″} t) & ⩾ A (2 c^{″} t_{m}) ⩾ A (c^{″} t_{m + 1}) \\ ⩾ c^{'} t_{m + 1} \int_{\frac{t_{0}}{2 C}}^{\frac{t_{m + 1}}{4 C}} \frac{B (s)}{s^{2}} d s ⩾ c^{'} t \int_{\frac{t_{0}}{2 C}}^{\frac{t}{4 C}} \frac{B (s)}{s^{2}} d s \end{matrix}$

for $t ⩾ \hat{t}$ . Hence, inequality (2.3.107) follows for suitable constants c and $t_{0}$ . □

Corollary 2.3.2

Let G be a Lipschitz domain in $R^{d}$ , $d ⩾ 2$ . Let A and B be Young functions. Assume that there exists a constant C such that

${‖ u - u_{G} ‖}_{L^{B} (G)} ⩽ C {‖ \nabla u ‖}_{W^{- 1, A} (G)}$

(2.3.155)

for every $u \in L^{1} (G)$ . Then conditions (2.3.107) and (2.3.108) hold.

Proof

Let $u \in W_{0}^{1, A} (G)$ . As in the proof of Theorem (2.3.119) we can show the following chain

$\begin{matrix} {‖ \nabla u - {(\nabla u)}_{G} ‖}_{L^{B} (G)} = {‖ \nabla u - {(\nabla u)}_{G} ‖}_{L^{B} (G)} ⩽ C \sum_{i, j = 1}^{d} {‖ \frac{\partial u_{i}}{\partial x_{j}} - {(\frac{\partial u_{i}}{\partial x_{j}})}_{G} ‖}_{L^{B} (G)} \\ ⩽ C \sum_{i, j = 1}^{n} {‖ \nabla \frac{\partial u_{i}}{\partial x_{j}} ‖}_{W^{- 1, A} (G)} ⩽ C \sum_{i, j, k = 1}^{d} {‖ \frac{\partial^{2} u_{i}}{\partial x_{k} \partial x_{j}} ‖}_{W^{- 1, A} (G)} \\ ⩽ C \sum_{i, j, k = 1}^{d} ({‖ \frac{\partial {(ε (u))}_{i j}}{\partial x_{k}} ‖}_{W^{- 1, A} (G)} + {‖ \frac{\partial {(ε (u))}_{i k}}{\partial x_{j}} ‖}_{W^{- 1, A} (G)} + {‖ \frac{\partial {(ε (u))}_{j k}}{\partial x_{i}} ‖}_{W^{- 1, A} (G)}) \\ ⩽ C \sum_{i, j = 1}^{d} {‖ \nabla {(ε (u))}_{i j} ‖}_{W^{- 1, A} (G)} ⩽ C \sum_{i, j = 1}^{n} {‖ {(ε (u))}_{i j} - {({(ε (u))}_{i j})}_{G} ‖}_{L^{A} (G)} \\ ⩽ C {‖ ε (u) - {(ε (u))}_{G} ‖}_{L^{A} (G)} = C {‖ ε (u) ‖}_{L^{A} (G)} . \end{matrix}$

Here the second inequality in line two holds by (2.3.155). The first inequality in line four is true for every Young function by the very definition of the negative norm. The conclusion follows via Theorem 2.3.14, due to the arbitrariness of u. □

Corollary 2.3.3

Let G be a bounded domain in $R^{d}$ , $d ⩾ 2$ , which is starshaped with respect to a ball, and let ${Bog}_{G}$ be the Bogovskiĭ operator on G (see Section 2.1). Let A and B be Young functions such that

${‖ \nabla {Bog}_{G} f ‖}_{L^{B} (G, R^{d})} ⩽ C {‖ f ‖}_{L^{A} (G)}$

(2.3.156)

for some constant C, and for every $f \in C_{0, ⊥}^{\infty} (G)$ . Then conditions (2.0.11) and (2.0.12) hold.

Proof

A close inspection of the proof of Theorem 2.0.5 reveals that inequality (2.3.156) implies inequality (2.3.155). The conclusion thus follows from Corollary 2.3.2. □

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Table of Contents for Chapter 2: Fluid mechanics & Orlicz spaces

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2.1 Bogovskiĭ operator

2.2 Negative norms & the pressure

2.3 Sharp conditions for Korn-type inequalities

Table of Contents for
Chapter 2: Fluid mechanics & Orlicz spaces