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Notation

Formula	Meaning
C	Continuous functions
C ^α	α-Hölder continuous functions
C ^k	k-times continuously differentiable functions
C ^∞	∞-times continuously differentiable functions
spt(φ)	Support of φ
$C_{0}^{\infty}$ $C_{0}^{\infty}$	C^∞-functions with compact support
$C_{0}^{k}$ $C_{0}^{k}$	C^k-functions with compact support
$D^{'}$ $D^{'}$	Dual of $C_{0}^{\infty}$ $C_{0}^{\infty}$
$C_{0, div}^{\infty}$ $C_{0, div}^{\infty}$	$C_{0}^{\infty}$ $C_{0}^{\infty}$ -functions with vanishing divergence
$D_{div}^{'}$ $D_{div}^{'}$	Dual of $C_{0, div}^{\infty}$ $C_{0, div}^{\infty}$
$C_{0, ⊥}^{\infty}$ $C_{0, ⊥}^{\infty}$	$C_{0}^{\infty}$ $C_{0}^{\infty}$ -functions with vanishing mean value
X ^′	Dual space of X
‖⋅‖_X	Norm on X
〈⋅,⋅〉_X	Inner product on X
$L^{n}$ $L^{n}$	n-dimensional Lebesgue measure
$H^{s}$ $H^{s}$	s-dimensional Hausdorff measure
$B$ $B$	Borelian σ-algebra
L ^p	Lebesgue-space of p-integrable functions
$L_{l o c}^{p}$ $L_{l o c}^{p}$	Lebesgue-space of locally p-integrable functions
$L_{div}^{p}$ $L_{div}^{p}$	L^p-functions with vanishing divergence
$L_{⊥}^{p}$ $L_{⊥}^{p}$	L^p-functions with vanishing mean
p ^′	Dual exponent of p: $p^{'} = p / (p - 1)$ $p^{'} = p / (p - 1)$
L^A/A(L)	Orlicz-space generated by A
$L_{⊥}^{A} / A {(L)}_{⊥}$ $L_{⊥}^{A} / A {(L)}_{⊥}$	L^A-functions with vanishing mean
$\tilde{A}$ $\tilde{A}$	Young conjugate of A
u ^⁎	Decreasing rearrangement of u
u⁎v	Convolution of u and v
Mf	Hardy-Littlewood maximal operator applied to f
$B_{r}$ $B_{r}$	Ball with radius r
$κ B_{r}$ $κ B_{r}$	Ball with same center as $B_{r}$ $B_{r}$ and radius κr
Π^k	Averaged Taylor polynomial of order k
W ^k,p	Sobolev functions with differentiability k and integrability p
$W_{l o c}^{k, p}$ $W_{l o c}^{k, p}$	Sobolev functions with differentiability k and local integrability p
$W_{div}^{k, p}$ $W_{div}^{k, p}$	W^k,p-functions with vanishing divergence
$W_{0}^{k, p}$ $W_{0}^{k, p}$	W^k,p-functions with vanishing trace
W ^−k,p	Dual space of $W_{0}^{k, p}$ $W_{0}^{k, p}$
$W_{div}^{k, p}$ $W_{div}^{k, p}$	W^k,p-functions with vanishing divergence
$W_{0, div}^{k, p}$ $W_{0, div}^{k, p}$	$W_{0}^{k, p}$ $W_{0}^{k, p}$ -functions with vanishing divergence
$W_{div}^{- k, p}$ $W_{div}^{- k, p}$	Dual space of $W_{0, div}^{k, p}$ $W_{0, div}^{k, p}$
W ^1,A	Orlicz-Sobolev space generated by A
$W_{0}^{1, A}$ $W_{0}^{1, A}$	W^1,A-functions with vanishing trace
$R_{s y m}^{d \times d}$ $R_{s y m}^{d \times d}$	Symmetric d × d matrices
⊙	Symmetric tensor product
E ^p	L^p-functions with symmetric gradient in L^p
$E_{0}^{p}$ $E_{0}^{p}$	E^p-functions with vanishing trace
$E_{0, div}^{p}$ $E_{0, div}^{p}$	$E_{0}^{p}$ $E_{0}^{p}$ -functions with vanishing divergence
E ^A	L^A-functions with symmetric gradient in L^A
$E_{0}^{A}$ $E_{0}^{A}$	E^A-functions with vanishing trace
$E_{0, div}^{A}$ $E_{0, div}^{A}$	$E_{0}^{A}$ $E_{0}^{A}$ -functions with vanishing divergence
BD	Functions of bounded deformation
$M$ $M$	Bounded signed measures
H^A/H^A(L)	L^A-functions with divergence in L^A
$H_{0}^{A} / H_{0}^{A (L)}$ $H_{0}^{A} / H_{0}^{A (L)}$	H^A-functions with vanishing normal trace
Bog	Bogovskiĭ-operator
Δ⁻¹	Solution operator to the Laplace equation
Δ⁻²	Solution operator to the bi-Laplace equation
curl⁻¹	Solution operator to the curl equation
Q _r	Parabolic cube with radius r
κQ _r	Cube with same center as Q_r and radius κr
$M^{α}$ $M^{α}$	α-parabolic maximal function
$M_{σ}^{α}$ $M_{σ}^{α}$	α-parabolic maximal function with power σ
$L^{p} (0, T; V)$ $L^{p} (0, T; V)$	Bochner measurable functions with values in $V$ $V$ and integrability p
$C ([0, T]; V)$ $C ([0, T]; V)$	Continuous functions with values in $V$ $V$
$C^{α} ([0, T]; V)$ $C^{α} ([0, T]; V)$	α-Hölder continuous functions with values in $V$ $V$
$C_{w} ([0, T]; V)$ $C_{w} ([0, T]; V)$	Weakly continuous functions with values in $V$ $V$
$W^{k, p} (0, T; V)$ $W^{k, p} (0, T; V)$	k-times weakly differentiable functions with values in $V$ $V$ and integrability p
$(Ω, F, P)$ $(Ω, F, P)$	Probability space with sample space Ω, σ-algebra $F$ $F$ and probability measure $P$ $P$
${(F_{t})}_{t ⩾ 0}$ ${(F_{t})}_{t ⩾ 0}$	filtration
$L^{p} (Ω, F, P; V)$ $L^{p} (Ω, F, P; V)$	Random variable over $(Ω, F, P)$ $(Ω, F, P)$ with values in $V$ $V$ and moments of order p
$M_{2}$ $M_{2}$	Quadratically integrable martingales
$M_{2}^{c}$ $M_{2}^{c}$	Quadratically integrable continuous martingales
$M_{2}^{c, l o c}$ $M_{2}^{c, l o c}$	Locally in time quadratically integrable continuous martingales
$L_{2} (H_{1}, H_{2})$ $L_{2} (H_{1}, H_{2})$	Hilbert-Schmidt operators from $H_{1} \to H_{2}$ $H_{1} \to H_{2}$
${({〈 X 〉}_{t})}_{t ⩾ 0}$ ${({〈 X 〉}_{t})}_{t ⩾ 0}$	Quadratic variation process of ${(X_{t})}_{t ⩾ 0}$ ${(X_{t})}_{t ⩾ 0}$
${({〈 X, Y 〉}_{t})}_{t ⩾ 0}$ ${({〈 X, Y 〉}_{t})}_{t ⩾ 0}$	Covariation process of ${(X_{t})}_{t ⩾ 0}$ ${(X_{t})}_{t ⩾ 0}$ and ${(Y_{t})}_{t ⩾ 0}$ ${(Y_{t})}_{t ⩾ 0}$
$A_{q}$ $A_{q}$	Stokes operator on L^q
$A_{q}$ $A_{q}$	$A$ $A$ -Stokes operator on L^q
D	Domain of an operator