Let $G\subset {\mathbb{R}}^d$ be open and $k\in \mathbb{N}$. We define

$\begin{array}{cc}\hfill C(G)& :=\{u:G\to \mathbb{R}:u\text{ is continuous}\},\hfill \\ \hfill C^k(G)& :=\{u:G\to \mathbb{R}:{\mathrm{\nabla }}^{i}u\text{ is continuous for }i=0,...,k\},\hfill \\ \hfill C^{\infty }(G)& :=\{u:G\to \mathbb{R}:{\mathrm{\nabla }}^{i}u\text{ is continuous for all }i\in {\mathbb{N}}_0\}.\hfill \end{array}$

In Definition 1.1.1 if we replace G by the closed set $\bar{G}$ we are considering functions whose derivatives are continuous up to the boundary of G.

Let $G\subset {\mathbb{R}}^d$ be open and $\alpha \in (0,1]$. We define

$\begin{array}{cc}\hfill C^{\alpha }(\bar{G})& :=\{u:G\to \mathbb{R}:\underset{x\ne y}{\sup }\frac{|u(x)-u(y)|}{{|x-y|}^{\alpha }}<\infty \},\hfill \\ \hfill C^{\alpha }(G)& :=\{u:G\to \mathbb{R}:u\in C^{\alpha }(\bar{K})\text{ for all }K⋐G\}\hfill \end{array}$

as the set of (locally) α-Hölder continuous functions.

In the case $\alpha =1$ we obtain the set of Lipschitz continuous functions.

Let $G\subset {\mathbb{R}}^d$ be open and $u:G\to \mathbb{R}$. We define the support of u by

$\mathrm{spt}u:=\overline{\{x\in G:u(x)\ne 0\}}.$

By $C_0^0(G)$ and $C_0^{\infty }(G)$ we denote subclasses of $C^0(G)$ and $C^{\infty }(G)$ whose elements satisfy $\mathrm{spt}u⋐G$.

Let $(\mathfrak{X},\mathrm{\Sigma },\mu )$ be a measure space. We define

$\begin{array}{cc}\hfill & L^p(\mathfrak{X},\mathrm{\Sigma },\mu ):=\{u:\mathfrak{X}\to \mathbb{R}:u\text{ is }\mu \text{-measurable},{\int }_{\mathfrak{X}}{\left|u\right|}^p\mathrm{d}\mu <\infty \},1⩽p<\infty ,\hfill \\ \hfill & L^{\infty }(\mathfrak{X},\mathrm{\Sigma },\mu ):=\{u:\mathfrak{X}\to \mathbb{R}:u\text{ is }\mu \text{-measurable},\underset{\mu (N)=0}{\inf }\underset{x\in \mathfrak{X}\setminus N}{\sup }|u(x)|<\infty \}.\hfill \end{array}$

The elements of $L^p(\mathfrak{X},\mathrm{\Sigma },\mu )$ are equivalence classes. Two functions u and v belong to the same equivalence class if $u=v$ μ-almost everywhere in $\mathfrak{X}$, i.e. if

$\mu \{x\in \mathfrak{X}:u(x)\ne v(x)\}=0.$

• $L^p(\mathfrak{X},\mathrm{\Sigma },\mu )$ is a Banach space together with the natural norm

$\begin{array}{cc}\hfill {\Vert u\Vert }_p& :={\Vert u\Vert }_{L^p(\mathfrak{X})}:={\left({\int }_{\mathfrak{X}}{\left|u\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\text{for}1⩽p<\infty ,\hfill \\ \hfill {\Vert u\Vert }_{\infty }& :={\Vert u\Vert }_{L^{\infty }(\mathfrak{X})}:=\underset{\mu (N)=0}{\inf }\underset{x\in \mathfrak{X}\setminus N}{\sup }|u(x)|.\hfill \end{array}$

• We set $p^{\prime }=\frac{p}{p-1}$ for $1<p<\infty$, $p^{\prime }=\infty$ for $p=1$ and $p^{\prime }=1$ for $p=\infty$. Let $1⩽p<\infty$ and $v\in L^{p^{\prime }}(\mathfrak{X},\mathrm{\Sigma },\mu )$. Then the mapping

$L^p(\mathfrak{X},\mathrm{\Sigma },\mu )\ni u\mapsto {\int }_{G}uv\mathrm{d}\mu$

belongs to the dual space $L^p{(\mathfrak{X},\mathrm{\Sigma },\mu )}^{\prime }$. On the other hand each element of $L^p{(\mathfrak{X},\mathrm{\Sigma },\mu )}^{\prime }$ can be represented via a function $v\in L^{p^{\prime }}(\mathfrak{X},\mathrm{\Sigma },\mu )$. In particular, the mapping

$L^{p^{\prime }}(\mathfrak{X},\mathrm{\Sigma },\mu )\ni v\mapsto (L^p(\mathfrak{X},\mathrm{\Sigma },\mu )\ni u\mapsto {\int }_{\mathfrak{X}}uv\mathrm{d}\mu )\in L^p{(\mathfrak{X},\mathrm{\Sigma },\mu )}^{\prime }$

is an isomorphism.

• The space $L^p(\mathfrak{X},\mathrm{\Sigma },\mu )$ is reflexive if and only if $1<p<\infty$.

• If $\mu (\mathfrak{X})<\infty$ we have that $L^q(\mathfrak{X},\mathrm{\Sigma },\mu )\subset L^p(\mathfrak{X},\mathrm{\Sigma },\mu )$ for $1⩽p⩽q⩽\infty$. In particular, the following holds

${\Vert u\Vert }_{L^p(\mathfrak{X})}⩽c(d,p,q,\mu (\mathfrak{X})){\Vert u\Vert }_{L^q(\mathfrak{X})}$

for every $u\in L^q(\mathfrak{X},\mathrm{\Sigma },\mu )$.

• We can define vector- or matrix-valued $L^p$-spaces component-wise. For simplicity we do not mention this in the notation. However, when considering functions with values in an infinite dimensional spaces X we denote this by writing $L^p(\mathfrak{X},\mathrm{\Sigma },\mu ;X)$.

Let $G\subset {\mathbb{R}}^d$ be open. We define

$L_{loc}^p(G):=\{u:G\to \mathbb{R}:u\in L^p(K)\text{ for all }K⋐G\}\text{for}1⩽p⩽\infty .$

Let $G\subset {\mathbb{R}}^d$ be open and $u\in L_{loc}^1(G)$.

• u is called weakly differentiable with respect to the i-th variable if there is a function $v_i\in L_{loc}^1(G)$ such that

${\int }_{G}u{\partial }_i\phi \mathrm{d}z=-{\int }_G{v}_i\phi \mathrm{d}z\text{for all}\phi \in C_0^{\infty }(G).$

We call $v_i$ the weak derivative of u with respect to the i-th variable. If weak derivatives with respect to all variables exist, we call u weakly differentiable and denote by $\mathrm{\nabla }u=(v_1,...,v_n)$ the weak gradient of u.

• Let $\alpha \in N_0^d$ and $\left|\alpha \right|:={\alpha }_1+...+{\alpha }_n$. For a smooth function u we denote by $D^{\alpha }u:=\frac{{\partial }^{\left|\alpha \right|}u}{{\partial }^{{\alpha }_1}{x}_1...{\partial }^{{\alpha }_n}{x}_n}$ its α-th classical derivative. A function $u\in L_{loc}^1(G)$ is called k-times weakly differentiable if for all $\alpha \in N_0^d$ with $\left|\alpha \right|⩽k$ there is a function $v_{\alpha }\in L_{loc}^1(G)$ such that

${\int }_{G}u{D}^{\alpha }\phi \mathrm{d}z={(-1)}^{\left|\alpha \right|}{\int }_G{v}_{\alpha }\phi \mathrm{d}z\text{for all}\phi \in C_0^{\infty }(G).$

We call $D^{\alpha }u=v_{\alpha }$ the α-th weak derivative.

For $k\in \mathbb{N}$ and $1⩽p⩽\infty$ we define

$\begin{array}{cc}\hfill W^{k,p}(G)& :=\{u:G\to \mathbb{R},D^{\alpha }u\in L^p(G),\text{ for all }\alpha \in N_0^d,\left|\alpha \right|⩽k\},\hfill \\ \hfill W_0^{k,p}(G)& :={\overline{C_0^{\infty }(G)}}^{W^{k,p}(G)}.\hfill \end{array}$

• $W^{k,p}(G)$ ($1⩽p⩽\infty$) is a Banach space together with the norm

$\begin{array}{cc}\hfill {\Vert u\Vert }_{k,p}:={\Vert u\Vert }_{W^{k,p}(G)}& :={\left(\sum _{\left|\alpha \right|⩽k}{\int }_G{\left|D^{\alpha }u\right|}^{p}dz\right)}^{\frac{1}{p}}\text{for}1⩽p<\infty ,\hfill \\ \hfill {\Vert u\Vert }_{k,\infty }:={\Vert u\Vert }_{W^{k,\infty }(G)}& :=\sum _{\left|\alpha \right|⩽k}\underset{{\mathcal{L}}^d(N)=0}{\inf }\underset{x\in G\setminus N}{\sup }|D^{\alpha }u(x)|.\hfill \end{array}$

• The space $W^{k,p}(G)$ is reflexive iff $1<p<\infty$.

• A sequence $(u_k)\subset W^{k,p}(G)$ ($1<p<\infty$) converges weakly to u in $W^{k,p}(G)$ if

${\int }_G{D}^{\alpha }{u}_{k}v\mathrm{d}z⟶{\int }_G{D}^{\alpha }uv\mathrm{d}z,k\to \infty ,$

for all $v\in L^{p^{\prime }}(G,{\mathbb{R}}^N)$ ($p^{\prime }:=\frac{p}{p-1}$) and for all $\alpha \in N_0^d$ with $\left|\alpha \right|⩽k$.

• The dual space of $W_0^{k,p}(G)$ will be denoted by $W^{-k,p^{\prime }}(G)$.

Let $G\subset {\mathbb{R}}^d$ be open and bounded with Lipschitz boundary (i.e. ∂Ω can be locally parametrized by Lipschitz continuous functions of $d-1$ variables) and $1⩽p<\infty$. Then $C^{\infty }(\bar{G})$ is dense in $W^{k,p}(G)$. In particular, for every $u\in W^{k,p}(G)$ there is $(u_m)\subset C_0^{\infty }(G)$ such that $u_m\to u$ in $W^{k,p}(G)$.

Let $G\subset {\mathbb{R}}^d$ be open.

a) The embeddings

$\begin{array}{cc}\hfill W_0^{1,p}(G)& \hookrightarrow L^{\frac{dp}{d-p}}(G)\mathit{\text{if}}p<d,\hfill \\ \hfill W_0^{1,p}(G)& \hookrightarrow C^{1-\frac{d}{p}}(\bar{G})\mathit{\text{if}}p>d,{\mathcal{L}}^d(G)<\infty ,\hfill \end{array}$

are continuous.

b) Let $G\subset {\mathbb{R}}^d$ be open and bounded with Lipschitz-boundary. The embeddings

$\begin{array}{cc}\hfill W^{1,p}(G)& \hookrightarrow L^{\frac{dp}{d-p}}(G)\mathit{\text{if}}p<d,\hfill \\ \hfill W^{1,p}(G)& \hookrightarrow C^{1-\frac{d}{p}}(\bar{G})\mathit{\text{if}}p>d,\hfill \end{array}$

are continuous.

Let $G\subset {\mathbb{R}}^d$ be open and bounded. The embedding

$W_0^{1,p}(G)\hookrightarrow L^q(G),q<\frac{pd}{d-p},$

is compact for all $p<d$.

Let $u\in C^1(\bar{G})$ with $G\subset {\mathbb{R}}^d$ open and bounded with Lipschitz–boundary. Then we have

${\Vert \mathrm{tr}u\Vert }_{L^p(\partial G)}⩽c(d,p,G){\Vert u\Vert }_{W^{1,p}(G)}.$

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be open and bounded with Lipschitz boundary. Then we have

$W_0^{1,p}(G)=\{u\in W^{1,p}(G);\mathrm{tr}u=0\}$

for $1⩽p<\infty$.

Let A and B be Young functions, and let $L\in (0,\infty ]$.

(i) There exists a constant C such that

${\Vert \frac{1}{s}{\int }_0^{s}f(r)dr\Vert }_{L^B(0,L)}⩽C{\Vert f\Vert }_{L^A(0,L)}$

for every $f\in L^A(0,L)$ if and only if either $L<\infty$ and there exist constants $c>0$ and $t_0⩾0$ such that

$t{\int }_{t_0}^t\frac{B(s)}{s^2}ds⩽A(ct)\mathit{\text{for}}t⩾t_0,$

or $L=\infty$ and (1.2.14) holds with $t_0=0$. In particular, in the latter case, the constant C in (1.2.13) depends only on the constant c appearing in (7.0.1).

(ii) There exists a constant C such that

${\Vert {\int }_s^{L}f(r)\frac{dr}{r}\Vert }_{L^B(0,L)}⩽C{\Vert f\Vert }_{L^A(0,L)}$

for every $f\in L^A(0,L)$ if and only if either $L<\infty$ and there exist constants $c>0$ and $t_0⩾0$ such that

$t{\int }_{t_0}^t\frac{\tilde{A}(s)}{s^2}ds⩽\tilde{B}(ct)\mathit{\text{for}}t⩾t_0,$

or $L=\infty$ and (1.2.16) holds with $t_0=0$. In particular, in the latter case, the constant C in (1.2.15) depends only on the constant c appearing in (1.2.16).

Let A be a Young-function satisfying the ${\mathrm{\Delta }}_2$-condition and $G\subset {\mathbb{R}}^d$ open.

• The following holds

$W_0^{1,A}(G)={\overline{C_0^{\infty }(G)}}^{W^{1,A}(G)}=W^{1,A}(G)\cap W_0^{1,1}(G).$

• For every $u\in W^{1,A}$ there is a sequence $(u_k)\in C^{\infty }(G)\cap W^{1,A}(G)$ such that $u_k\to u$ in $W^{1,A}(G)$.

Let A be a Young function satisfying the ${\mathrm{\Delta }}_2$- and the ${\mathrm{\nabla }}_2$-condition. Then $L^A(G)$ is reflexive with

$L^A{(G)}^{\prime }\cong L^{\tilde{A}}(G).$

a) Let $v\in L_{loc}^1({\mathbb{R}}^d)$ and $\lambda >0$. The level-set $\{x\in {\mathbb{R}}^d:|M(v)(x)|>\lambda \}$ is open.

b) The strong-type estimate

${\Vert M(v)\Vert }_{L^p({\mathbb{R}}^d)}⩽c_p{\Vert v\Vert }_{L^p({\mathbb{R}}^d)}\forall v\in L^p({\mathbb{R}}^d)$

holds for all $p\in (1,\infty ]$.

c) The weak-type estimate

${\mathcal{L}}^d(\{x\in {\mathbb{R}}^d:|M(v)(x)|>\lambda \})⩽c_p\frac{{\Vert v\Vert }_p^p}{{\lambda }^p}\forall v\in L^p({\mathbb{R}}^d)$

holds for all $p\in [1,\infty )$ and all $\lambda >0$.

d) We have the estimate

${\int }_{\{M(v)>\lambda \}}|v{|}^p\mathrm{d}x⩽c_p{\int }_{\{|v|>\lambda /2\}}|v{|}^p\mathrm{d}x\forall p\in [1,\infty )$

for all $\lambda >0$.