On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the H\"older regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^\alpha_{\tau,\tau}(O), 1/\tau=\alpha/d+1/p, \alpha>0$. This leads to a H\"older-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.


Introduction
Let O ⊆ R d be a bounded Lipschitz domain, T ∈ (0, ∞) and let (w k t ) t∈[0,T ] , k ∈ N, be independent one-dimensional standard Wiener processes defined on a probability space (Ω, F, P). We are interested in the regularity of the solutions to parabolic stochastic partial differential equations (SPDEs, for short) with zero Dirichlet boundary condition of the form on Ω × (0, T ] × ∂O, where the indices i and j run from 1 to d and the index k runs through N = {1, 2, . . .}. Here and in the sequel we use the summation convention on the repeated indices i, j, k. The coefficients a ij and σ ik depend on (ω, t) ∈ Ω × [0, T ]. The force terms f and g k depend on (ω, t, x) ∈ Ω × [0, T ] × O. By the nature of the problem, in particular by the bad contribution of the infinitesimal differences of the Wiener processes, the second spatial derivatives of the solution may blow up at the boundary ∂O even if the boundary is smooth, see, e.g., [28]. Hence, a natural way to deal with problems of type (1.1) is to consider u as a stochastic process with values in weighted Sobolev spaces on O that allow the derivatives of functions from these spaces to blow up near the boundary. This approach has been initiated and developed by N.V. Krylov and collaborators, first as an L 2 -theory for smooth domains O (see [28]), then as an L p -theory (p ≥ 2) for the half space ( [33,34]), for smooth domains ( [23,27]), and for general bounded domains allowing Hardy's inequality such as bounded Lipschitz domains ( [26]). Existence and uniqueness of solutions have been established within specific stochastic parabolic weighted Sobolev spaces, denoted by H γ p,θ (O, T ) in [26]. These spaces consist of elementsũ of the form dũ =f dt +g k dw k t , whereũ,f andg k , considered as stochastic processes with values in certain weighted L p (O)-Sobolev spaces, are L p -integrable w.r.t. P ⊗ dt. We refer to Section 3 for the exact definition.
In this article we treat regularity issues concerning the solution u of problem (1.1) which arise, besides others, in the context of adaptive numerical approximation methods.
The starting point of our considerations was the question whether we can improve the Besov regularity results in [4] in time direction. In [4] the spatial regularity of u is measured in the scale of Besov spaces where p ≥ 2 is fixed. Note that for α > (p − 1)d/p the sumability parameter τ becomes less than one, so that in this case B α τ,τ (O) is not a Banach space but a quasi-Banach space. It is a known result from approximation theory that the smoothness of a target function f ∈ L p (O) within the scale ( * ) determines the rate of convergence that can be achieved by adaptive and other nonlinear approximation methods if the approximation error is measured in L p (O); see [5,Chapter 4], [13] or the introduction of [4]. Based on the L p -theory in [26], it is shown in [4] that the solution u to problem (1.1) satisfies u ∈ L τ (Ω × [0, T ], P, P ⊗ dt; B α τ,τ (O)), for certain α > 0 depending on the smoothness of u 0 , f and g k , k ∈ N. In general, the spatial regularity of u in the Sobolev scale W s p (O), s ≥ 0, which determines the order of convergence for uniform approximation methods in L p (O), is strictly less than the spatial regularity of u in the scale ( * ). It can be due to, e.g., the irregular behaviour of the noise at the boundary or the irregularities of the boundary itself; see [37,Chapter 4] for the latter case. This justifies the use of nonlinear approximation methods such as adaptive wavelet methods for the numerical treatment of SPDEs, cf. [3,2]. The proof of (1.2) relies on characterizations of Besov spaces by wavelet expansions and on weighted Sobolev norm estimates for u, resulting from the solvability of the problem (1.1) within the spaces H γ p,θ (O, T ). An obvious approach to improve (1.2) with respect to regularity in time is to try to combine the existing Hölder estimates in time for the elements of the spaces H γ p,θ (O, T ) (see [26,Theorem 2.9]) with the wavelet arguments in [4]. However, it turns out that a satisfactory result requires a more subtle strategy in three different aspects. Firstly, we need an extension of the L p -theory in [26] to an L q (L p )-theory for SPDEs dealing with stochastic parabolic weighted Sobolev spaces H γ,q p,θ (O, T ) with possibly different summability parameters q and p in time and space respectively. These spaces consist of elementsũ of the form dũ =f dt +g k dw k t , whereũ,f andg k , considered as stochastic processes with values in suitable weighted L p (O)-Sobolev spaces, are L q -integrable w.r.t. P ⊗ dt. Such an extension is needed to obtain better Hölder estimates in time in a second step. Satisfactory existence an uniqueness results concerning solutions in the spaces H γ,q p,θ (O, T ) have been established in [25] for domains O with C 1boundary. Unfortunately, the techniques used there do not work on general Lipschitz domains. Also, the L q (L p )-results that have been obtained in [47] within the semigroup approach to SPDEs do not directly suit our purpose: On the one hand, for general Lipschitz domains O the domains of the fractional powers of the leading linear differential operator cannot be characterized in terms of Sobolev or Besov spaces as in the case of a smooth domains O; see, e.g., the introduction of [4] for details. On the other hand, even in the case of a smooth domain O we need regularity in terms weighted Sobolev spaces to obtain the optimal regularity in the scale ( * ).
Secondly, once we have established the solvability of SPDEs within the spaces H γ,q p,θ (O, T ), we have to exploit the L q (L p )-regularity of the solution and derive improved results on the Hölder regularity in time for large q. For O = R d + this has been done by Krylov [32]. It takes quite delicate arguments to apply these results to the case of bounded Lipschitz domains via a boundary flattening argument.
Thirdly, in order to obtain a reasonable Hölder-Besov regularity result, it is necessary to generalize the wavelet arguments applied in [4] to a wider range of smoothness parameters. This requires more sophisticated estimates.
In this article we tackle and solve the tasks described above. We organize the article as follows. In Section 2 we recall the definition and basic properties of the (deterministic) weighted Sobolev spaces H γ p,θ (G) introduced in [38] (see also [43,Chapter 6]) on general domains G ⊆ R d with non-empty boundary. In Section 3 we give the definition of the spaces H γ,q p,θ (G, T ) and specify the concept of a solution for equations of type (1.1) in these spaces. Moreover, we show that if we have a solution u ∈ H γ,q p,θ (G, T ) with low regularity γ ≥ 0, but f and the g k 's have high L q (L p )-regularity, then we can lift up the regularity of the solution (Theorem 3.8). In this sense the spaces H γ,q p,θ (G, T ) are the right ones for our regularity analysis of SPDEs. Section 4 is devoted to the solvability of Eq. (1.1) in H γ,q p,θ (O, T ), O ⊆ R d being a bounded Lipschitz domain. The focus lies on the case q > p ≥ 2 and we restrict our considerations to equations with additive noise, i.e. σ ik ≡ 0. In Subsection 4.2 we consider equations on domains with small Lipschitz constants and derive a result for general integrability parameters q ≥ p ≥ 2 (Theorem 4.2). We use an L q (L p )-regularity result for deterministic parabolic equations from [16] and an estimate for stochastic integrals in UMD spaces from [46] to obtain a certain low L q (L p )-regularity of the solution. Then the regularity is lifted up with the help of Theorem 3.8. In Subsection 4.2, we consider the stochastic heat equation on general bounded Lipschitz domains. Here we use the results from [47] on maximal L q -regularity of stochastic evolution equations (see also [48] and [46]) to derive existence and uniqueness of a solution with low regularity. A main ingredient will be to prove that the domain of the square root of the weak Dirichlet Laplacian on L p (O) coincides with the closure of the test functions C ∞ 0 (O) in the L p (O)-Sobolev space of order one (Lemma 4.5). This stays true only for a certain range of p ∈ [2, p 0 ) with p 0 > 3. Thus, so does our result (Theorem 4.4). In a second step, we again lift up the regularity by using Theorem 3.8. In both settings we derive suitable a-priori estimates.
In Section 5 we present our result on the Hölder regularity in time of the elements of H γ,q p,θ (O, T ) (Theorem 5.1). It is an extension of the Hölder estimates in time for the elements of H γ,q p,θ (T ) = H γ,q p,θ (R d + , T ) in [31] to the case of bounded Lipschitz domains. The implications for the Hölder regularity of the solutions of SPDEs are described in Theorem 5.3.
In Section 6 we pave the way for the analysis of the spatial regularity of the solutions of SPDEs in the scale ( * ). We discuss the relationship between the weighted Sobolev spaces H γ p,θ (O) and Besov spaces. Our main result in this section, Theorem 6.9, is a general embedding of the spaces H γ p,d−νp (O), γ, ν > 0, into the Besov scale ( * ). Its proof is an extension of the wavelet arguments in the proof of [4,Theorem 3.1], where only integer valued smoothness parameters γ are considered. It can also be seen as an extension of and a supplement to the Besov regularity results for deterministic elliptic equations in [10] and [7,8,9,11]. To the best of our knowledge, no such general embedding has been proven before. In the course of the discussion we also enlighten the fact that, for the relevant range of parameters γ and ν, the spaces H γ p,d−νp (O) act like Besov spaces B γ∧ν p,p (O) with zero trace on the boundary (Remark 6.7).
In Section 7 the results of the previous sections are combined in order to determine the Hölder-Besov regularity of the elements of the stochastic parabolic spaces H γ,q p,θ (O, T ) and of the solutions of SPDEs within these spaces. The related result in [4] is significantly improved in several aspects; see Remark 7.3 for a detailed comparison. We obtain an estimate of the form for certain α depending on the smoothness and weight parameters γ and θ and for certain κ depending on q and α (Theorem 7.4). Using the a-priori estimates from Section 4, the right hand side of the above inequality can be estimated by suitable norms of f and g if u is the solution to the corresponding SPDE (Theorem 7.5). Let us mention the related work [1] on the Besov regularity for the deterministic heat equation. The authors study the regularity of temperatures in terms of anisotropic Besov spaces of type B α/2,α τ,τ ((0, T ) × O), 1/τ = α/d + 1/p. However, the range of admissible values for the parameter τ is a priori restricted to (1, ∞), so that α is always less than d(1 − 1/p). In our article the parameter τ in ( * ) may be any positive number, including in particular the case where τ is less than 1 and where B α τ,τ (O) is not a Banach space but a quasi-Banach space. Notation and Conventions. Throughout this paper, O always denotes a bounded Lipschitz domain in R d , d ≥ 1, as specified in Definition 2.5 below. General subsets of R d are denoted by G. We write ∂G for their boundary (if it is not empty) and G • for the interior. N := {1, 2, . . .} denotes the set of strictly positive integers whereas N 0 := N ∪ {0}. Let (Ω, F, P) be a complete probability space and {F t , t ≥ 0} be an increasing filtration of σ-fields F t ⊂ F, each of which contains all (F, P)-null sets. By P we denote the predictable σ-field generated by {F t , t ≥ 0} and we assume that . .} are independent one-dimensional Wiener processes w.r.t. {F t , t ≥ 0}. For κ ∈ (0, 1) and a quasi-Banach space (X, · X ) we denote by C κ ([0, T ]; X) the Hölder space of continuous X-valued functions on [0, T ] with finite norm · C κ ([0,T ];X) defined by For 1 < p < ∞, L p (A, Σ, µ; X) denotes the space of µ-strongly measurable and p-Bochner integrable functions with values in X on a σ-finite measure space (A, Σ, µ), endowed with the usual L p -Norm.
For any distribution f on G and any ϕ ∈ C ∞ 0 (G), (f, ϕ) denotes the application of f to ϕ. Furthermore, for any multi-index α = (α 1 , . . . , By making slight abuse of notation, for m ∈ N 0 , we write D m f for any (generalized) m-th order derivative of f and for the vector of all m-th order derivatives of f . E.g. if we write D m f ∈ X, where X is a function space on G, we mean D α f ∈ X for all α ∈ N d 0 with |α| = m. We also use the notation f Lp(O) ) 1/p . If we have two quasi-normed spaces (X i , · X i ), i = 1, 2, X 1 ֒→ X 2 means that X 1 is continuously linearly embedded in X 2 . For a compatible couple (X 1 , X 2 ) of quasi-Banach spaces, [X 1 , X 2 ] η denotes the interpolation space of exponent η ∈ (0, 1) arising from the complex interpolation method. In general, N will denote a positive finite constant, which may differ from line to line. The notation N = N (a 1 , a 2 , . . .) is used to emphasize the dependence of the constant N on the set of parameters {a 1 , a 2 , . . .}. In general, this set will not contain all the parameters N depends on. A ∼ B means that A and B are equivalent.

Weighted Sobolev spaces
We start by recalling the definition and some basic properties of the (deterministic and stationary) weighted Sobolev spaces H γ p,θ (G) introduced in [38]. These spaces will serve as state spaces for the solution processes u = (u(t)) t∈[0,T ] to SPDEs of type (1.1) and they will play a fundamental role in all the forthcoming sections.
For p ∈ (1, ∞) and γ ∈ R, let H γ p := H γ p (R d ) := (1 − ∆) −γ/2 L p (R d ) be the spaces of Bessel potentials, endowed with the norm where F denotes the Fourier transform. It is well known that if γ is a nonnegative integer, then Let G ⊆ R d be an arbitrary domain with non-empty boundary ∂G. We denote by ρ(x) := ρ G (x) := dist(x, ∂G) the distance of a point x ∈ G to the boundary ∂G. Furthermore, we fix a bounded infinitely differentiable function ψ defined on G such that for all x ∈ G, Note that any non-negative smooth function ζ ∈ C ∞ 0 (R + ) with ζ > 0 on [e −1 , e] satisfies (2.2). For x ∈ G and n ∈ Z, define ζ n (x) := ζ(e n ψ(x)).
For p ∈ (1, ∞) and γ ∈ R we write H γ p (ℓ 2 ) for the collection of all sequences g = (g 1 , g 2 , . . .) of distributions on R d with g k ∈ H γ p for each k ∈ N and Analogously, for θ ∈ R, a sequence g = (g 1 , g 2 , . . .) of distributions on G is in H γ p,θ (G; ℓ 2 ) if, and only if, g k ∈ H γ p,θ (G) for each k ∈ N and Now we present some useful properties of the space H γ p,θ (G) taken from [38], see also [29,30].
(ii) Assume that γ − d/p = m + ν for some m ∈ N 0 , ν ∈ (0, 1] and that i, j ∈ N d 0 are multi-indices such that |i| ≤ m and |j| = m. Then for any u ∈ H γ p,θ (G), we have (vi) There exists a constant c 0 > 0 depending on p, θ, γ and the function ψ such that, for all c ≥ c 0 , the operator ψ 2 ∆ − c is a homeomorphism from H γ+1 p,θ (G) to H γ−1 p,θ (G). Remark 2.2. Assertions (vi) and (iv) in Lemma 2.1 imply the following: If u ∈ H γ p,θ−p (G) and ∆u ∈ H γ p,θ+p (G), then u ∈ H γ+2 p,θ−p (G) and there exists a constant N , which does not depend on u, such that . A proof of the following equivalent characterization of the weighted Sobolev spaces H γ p,θ (G) can be found in [38,Proposition 2.2].
for some c > 1 and k 0 > 0, where the constant N (m) does not depend on n ∈ Z and x ∈ G. Then,

If in addition
then the converse inequality also holds.

Remark 2.4. (i) It is easy to check that both
where N and N (0) are as in (2.1). Then, the sequence {ξ n : n ∈ Z} ⊆ C ∞ 0 (G) defined by ξ n :=ζ(2 n ψ(·)), n ∈ Z, fulfils the conditions (2.3) and (2.4) from Lemma 2.3 with c = 2 and a suitable k 0 > 0. Furthermore, In this paper, O will always denote a bounded Lipschitz domain in R d . More precisely: there exists a Lipschitz continuous function µ 0 : R d−1 → R such that, upon relabeling and reorienting the coordinate axes if necessary, we have with equivalent norms. This follows from [35, Theorem 9.7] and Poincaré's inequality.

Stochastic parabolic weighted Sobolev spaces and SPDEs
In this section, we first introduce the stochastic parabolic spaces H γ,q p,θ (G, T ) for arbitrary domains G ⊆ R d with non-empty boundary in analogy to the spaces H γ,q p,θ (T ) = H γ,q p,θ (R d + , T ) from [31,32]. Then we show that they are suitable to serve as solution spaces for equations of type (1.1) in the following sense: If we have a solution u ∈ H γ,q p,θ (G, T ) with low regularity γ ≥ 0, but f and the g k 's have high L q (L p )-regularity, then we can lift up the regularity of the solution (Theorem 3.8).
From now on let in the sense of distributions. That is, for any ϕ ∈ C ∞ 0 (G), with probability one, the equality where the series is assumed to converge uniformly on [0, T ] in probability.
In this situation we write Du := f and Su := g. The norm in H γ,q p,θ (G, T ) is defined as If p = q we also write H γ p,θ (G, T ) instead of H γ,p p,θ (G, T ).

Remark 3.3.
Replacing G by R d and omitting the weight parameter θ and the weight function ψ in the definitions above, one obtains the spaces H γ,q , and H γ,q p (T ) as introduced in [32, Definition 3.5]. The latter are denoted by H γ,q p (T ) in [31]; if q = p they coincide with the spaces H γ p (T ) introduced in [29, Definition 3.1].
We consider initial value problems of the form on an arbitrary domain G ⊆ R d with non-empty boundary. We use the following solution concept.
Definition 3.4. We say that a stochastic process u ∈ H γ,q p,θ (G, T ) is a solution of Eq. (3.1) if, and only if, in the sense of Definition 3.2.

Remark 3.5.
Here and in the sequel we use the summation convention on the repeated indices i, j, k. The question, in which sense, for a bounded Lipschitz domain O ⊆ R d , the elements of H γ,q p,θ (O, T ) fulfil a zero Dirichlet boundary condition as in Eq. (1.1), will be answered in Remark 6.7.
We make the following assumptions on the coefficients in Eq. (3.1). Throughout this paper, whenever we will talk about this equation, we will assume that they are fulfilled.
We will use the following result taken from [31, Lemma 2.3].
Lemma 3.7. Let p ≥ 2, m ∈ N, and, for i = 1, 2, . . . , m, x r and L p (ℓ 2 ) := H 0 p (ℓ 2 ). The constant N depends only on m, d, p, δ 0 , and K. Now we are able to prove that if we have a solution u ∈ H γ+1,q p,θ (G, T ) to Eq. (3.1) and if the regularity of the forcing terms f and g is high then we can lift the regularity of the solution. Note that in the next theorem there is no restriction, neither on the shape of the domain G ⊆ R d nor on the parameters θ, γ ∈ R.
where the constant N ∈ (0, ∞) does not depend on u, f and g.
Proof. The case m = 1, i.e., p = q is covered by [26,Lemma 3.2]. Therefore, let m ≥ 2. According to Remark 2.2 it is enough to show that .
Using the definition of weighted Sobolev spaces from Section 2, we observe that (Here ζ −nx u x is meant to be a scalar product in R d .) Now we can use Jensen's inequality and Remark 2.4(i) to obtain An application of Lemma 2.1(iii) and (iv) leads to . Therefore, it is enough to estimate the first term on the right hand side, Tonelli's theorem together with the relation applied to ∆u (n i ) with u (n) := ζ −n u for n ∈ Z, show that we only have to handle where we denote Thus, it is enough to find a proper estimate for Applying (3.2) first, followed by Tonelli's theorem, then Hölder's and Young's inequality, leads to Using the definition of f (n) and arguing as at the beginning of the proof, we get Combining the last three estimates, we obtain for any ε > 0 a constant N (ε) ∈ (0, ∞), such that .
Using similar arguments we obtain , which finishes the proof.
Iterating this result has the following consequence.
Remark 3.10. An extension of the results above to the case where the coefficients depend on the space variable x ∈ G can be proved along the lines of [24,26]. Also, the symmetry of a ij can be dropped. To keep the expositions at a reasonable level, we do not discuss these cases.

Solvability of SPDEs within
In this section we prove existence and uniqueness of solutions to equations of type (3.1) on bounded Lipschitz domains O ⊆ R d in the spaces H γ,q p,θ (O, T ). We are mainly interested in the case q > p. The main ingredient will be Corollary 3.9 which allows us to lift up the regularity of the solution once we have established a certain low L q (L p )-regularity and if f and the g k 's have high L q (L p )regularity. In this section we restrict ourselves to equations of type (3.1) with σ ≡ 0 and vanishing initial condition, i.e., we consider the problem We expect, however, that the lifting argument in Corollary 3.9 can be used to derive similar results for general equations of type (3.1). We establish existence of solutions with low L q (L p )-regularity in two different ways which correspond to two different restrictions in our assumptions. First, in Subsection 4.1 we consider Lipschitz domains with sufficiently small Lipschitz constants. Here we use an L q (L p )-regularity result for deterministic PDEs and basic estimates for stochastic integrals in UMD Banach spaces to derive a result for general integrability parameters q ≥ p ≥ 2. Then, in Subsection 4.2 we consider the case of general bounded Lipschitz domains. Applying techniques from the semigroup approach to stochastic evolution equations in Banach spaces, we are able prove existence and uniqueness of solutions of the stochastic heat equation in H γ,q p,d (O, T ) for integrability parameters p ∈ [2, p 0 ) and q ≥ p.

A result for domains with small Lipschitz constant
We need the following result concerning existence and uniqueness of solutions to SPDEs of the form , for the case p = q. It is taken from [26], see Theorem 2.12 and Remark 2.13 therein. Note that it also holds under weaker assumptions on the parameters and for more general equations than stated here.
where the constant N depends only on d, p, γ, θ, δ 0 , K, T and O.
(ii) There exists p 0 > 2, such that the following statement holds: if p ∈ [2, p 0 ), then there exists a constant κ 1 ∈ (0, 1), depending only on d, p, δ 0 , K and O, such that for any θ ∈ Here is the main result of this subsection.  Therefore, settingū := u − w, we know that for all ϕ ∈ C ∞ 0 (O), with probability one, It follows thatū is the unique solution in We are going to consider (4.5) ω-wise and apply an L q (L p )-regularity result for deterministic PDEs from [16]. To this end, we have to check that in the present situation our notion of a solution fits to the one described therein. Sinceū, , we know that, for almost every ω ∈ Ω, the mappings t →ū(ω, t, ·) and t → w(ω, t, ·) belong to Standard arguments lead to where the integrals are H −1 2,d+2 (O)-valued Bochner integrals. We obtain . Using approximation arguments one can verify that (4.6) even holds for all test-functions h which belong to the space H 1 2,2 ((0, T ) × O) considered in [16] and which vanish on (0, T ) × ∂O in the sense that h(t) ∈W 1 2 (O) = H 1 2,d−2 (O) for almost all t ∈ (0, T ). Moreover, for almost all ω ∈ Ω,ū(ω) belongs to the spaceH 1 2,2 ((0, T ) × O) as defined in [16] and it vanishes on (0, T ) × ∂O. Thus, for almost all ω ∈ Ω,ū(ω) is the unique solution inH 1 2,2 in the sense of [16]. Now we can apply [16, Theorem 8.1] and use the fact that the coefficients a ij are uniformly bounded due to Assumption 3.6 to obtain for almost all ω ∈ Ω, where the constant N does not depend on ω. We remark that the assumption on the Lipschitz constant K 0 comes into play at this point: Theorem 8.1 in [16] implies that there exists a constant c = c(d, p, q, δ 0 , K) such that, if K 0 ≤ c, then estimate (4.7) holds. Integration w.r.t. P and Hardy's inequality yield  The term Dw H 0,q p,d (O,T ) can be estimated with the help of an inequality for stochastic integrals in UMD Banach spaces with type 2 taken from [46]. To this end, let γ(ℓ 2 , H 1 p,d−p (O)) denote the Banach space of γ-radonifying operators from ℓ 2 to H 1 p,d−p (O), see [45] for a survey on this class of operators. Furthermore, let {e k } k∈N be the standard orthonormal basis of the Hilbert space ℓ 2 . Then, the stochastic process

1(iii) and (iv). Using this and similar arguments as above we obtain
with D e i • b denoting the (point-wise) composition of the operators D e i and b, and

An L q (L p )-theory of the heat equation on general bounded Lipschitz domains
In this subsection we present a first L q (L p )-theory for the stochastic heat equation on general bounded Lipschitz domains O ⊆ R d . We start by presenting the main result of this subsection, which we will prove later on. Moreover, there exists a constant N ∈ (0, ∞), which does not depend on f and g, such that .

(4.13)
For bounded C 1 -domains G ⊆ R d this result has been already proven in [25]. Unfortunately, the techniques used there will not work if the boundary is assumed to be just Lipschitz continuous. Therefore, we choose to take another way. We will mainly use the fact that the domain of the square root of the negative weak Dirichlet-Laplacian on L p (O) coincides with the closure of the test functions in the L p (O)-Sobolev space of order one, at least for the range of p allowed in our assertion. Before we prove this fact let us get more precise and introduce some notations and definitions.
Let O be a bounded Lipschitz domain in R d . As in [49, Definition 3.1], for arbitrary p ∈ (1, ∞), we define the weak Dirichlet-Laplacian ∆ D p,w on L p (O) as follows: where δ ij denotes the Kronecker symbol. If we fix p ∈ (p 0 /(p 0 − 1), p 0 ) with p 0 = 4 + δ when d = 2 and p 0 = 3 + δ when d ≥ 3 where δ > 0 is taken from [49, Proposition 4.1], then, the unbounded operator ∆ D p,w generates a strongly continuous, analytic semigroup S p (t) t≥0 of contractions on L p (O), see [49, Theorem 3.8 and Corollary 4.2]. Thus, (−∆ D p,w ) 1/2 , the square root of the negative of ∆ D p,w , can be defined as the inverse of the operator [41,Chapter 2.6]. Endowed with the norm Exploiting the fundamental results from [49] and [19], we can prove the following identity, which is crucial if we want to apply the results from [47] in our setting. By [49,Proposition 4.1] the semigroups S 2 (t) t≥0 and S p (t) t≥0 are consistent, i.e., for all t ≥ 0, Using (4.14), this leads to which implies implies the boundedness of the operator Consequently, for any g ∈ Range((−∆ D p,w ) −1/2 ) = D((−∆ D p,w ) 1/2 ), with a constant N independent of g. Embedding (4.15) follows. It remains to prove the converse direction, i.e., To this end, we first notice that the strongly continuous analytic contraction-semigroup S p (t) t≥0 on L p (O) is positive in the sense of [17, p. 353], see [49,Lemma 4.4]. Therefore, by [22,Corollary 5.2], (−∆ D p,w ) has a bounded H ∞ -calculus of angle less than π/2. Consequently, it has bounded imaginary powers. This implies Also, by the definition of the weighted Sobolev spaces one easily sees that Combining these results, we obtain . Therefore, we can apply [47,Theorem 4.5(ii)] and obtain the existence of a stochastic process u ∈ L q (Ω × [0, T ], P, P ⊗ dt; D(−∆ D p,w )) (4.17) solving the stochastic evolution equation in the sense of [47,Definition 4.2] with X 0 := L p (O). Moreover, there exists a versionũ of u, such that the following equality is fulfilled in L p (O) a.s. for all t ∈ [0, T ] at once: We can fix a continuous versions of the stochastic process (w(t)) t∈[0,T ] , so that by (4.10) a.s.
, (4.18) which implies estimate (4.13) for γ = 0. To this end we will use the fact that the stochastic process is a version of u, see [47,Proposition 4.4]. Since −∆ D p,w has the (deterministic) maximal regularity property (see [49,Proposition 6.1]) and 0 ∈ ρ(∆ D p,w ), we obtain where we used again Lemma 4.5 and [35,Theorem 9.7]. Simultaneously, notice that −∆ D p,w and g (respectively b) fulfil the assumptions of [48, Theorem 1.1]; we have already checked them in our explanations above. Thus, applying this result, we obtain . (4.20) The constants in (4.19) and (4.20) do not depend on f and g. Therefore, using the last two estimates we obtain the existence of a constant N , independent of f or g, such that .
Since V is just a version of the solution u, Eq. (4.18) follows. In order to prove the assertion for arbitrary γ ≥ 0 and 2 ≤ p ≤ q < ∞ we can argue as we have done at the end of the proof of Theorem 4.2.

Hölder-Sobolev regularity of elements of H γ,q p,θ (O, T ) and implications for SPDEs
In this section we analyse the temporal Hölder regularity of functions in H γ,q p,θ (O, T ), where O is a bounded Lipschitz domain in R d . Our main interest lies on the case q = p. As an application, we obtain Hölder-Sobolev regularity for the solutions to SPDEs presented in Section 4. In combination with the Sobolev type embeddings for the spaces H γ p,θ (O) from the Section 2, we also obtain assertions concerning the Hölder regularity in time and space for elements of H γ,q p,θ (O, T ) (Corollary 5.2). Here is the main result of this section.
Then there exists a constant N , which does not depend on T and u, such that .

(5.2)
Theorem 5.1 and Lemma 2.1(ii) yield the following so-called interior Schauder estimates of functions in H γ,q p,θ (O, T ).
Combining Theorem 5.1 with the results of Section 4, we immediately obtain the following result on the Hölder-Sobolev regularity of solutions of SPDEs. Theorem 5.3. Let O be a bounded Lipschitz domain in R d . Let 2 ≤ p ≤ q < ∞, γ ∈ N 0 , θ ∈ R and 2/q <β < β ≤ 1.
where the constant N ∈ (0, ∞) does not depend on u, f and g.
For the case that the summability parameters in time and space coincide, i.e., q = p, a result similar to Theorem 5.1 has been proven in [26], see Theorem 2.9 therein. The proof in [26] is straightforward and relies on [32,Corollary 4.12], which is a variant of Theorem 5.1 on the whole space R d . However, we are explicitly interested in the case q > p since it allows a wider range of parametersβ and β, and therefore leads to better regularity results. Unfortunately, the proof technique used in [26, Proposition 2.9] does not work any more in this case. Therefore, we take a different path: We use [32, Proposition 4.1], which covers the assertion of Theorem 5.1 with R d + instead of O, and the Lipschitz character of ∂O to derive Theorem 5.1 via a boundary flattening argument. To this end, we need the following two lemmas whose proofs are postponed to the appendix.
Then, for any γ ∈ [−1, 1], there exists a constant N = N (d, γ, p, θ, φ) ∈ (0, ∞), which does not depend on u, such that in the sense that, if one of the norms exists, so does the other one and the above inequality holds.
Proof of Theorem 5.1. Let us simplify notation and write f := Du and g := Su throughout the proof. We will show that (5.1) is true by induction over γ ∈ N; estimate (5.2) can be proved analogously.
We start with the case γ = 1. Fix x 0 ∈ ∂O and choose r > 0 small enough, e.g., r := r 0 (10K 0 ) −1 with r 0 and K 0 > 1 from Definition 2.5. Let us assume for a moment that the supports (in the sense of distributions) of u, f and g are contained in B r (x 0 ) for each t and ω. With µ 0 from Definition 2.5, we introduce the function which fulfils all the assumptions of Lemma 5.4. Note that, since r has been chosen sufficiently small, holds for all ν,θ ∈ R andp > 1. Together with Lemma 5.4 we obtain for any ν ∈ [−1, 1], Thus, denotingū := u • φ −1 ,f := f • φ −1 andḡ := g • φ −1 , by Lemma 5.5 we know that on G (2) we have dū =f dt +ḡ k dw k t in the sense of distributions. Furthermore, since ρ G (2) (y) = ρ R d holds for any ν ∈ [−1, 1], where we identify v • φ −1 with its extension to R d + by zero. Therefore, by making slight abuse of notation and writingū,f andḡ for the extension by zero on R d + ofū,f and g respectively, we havē and dū =f dt +ḡ k dw k t is fulfilled on R d + in the sense of distributions. Thus, we can apply [32,Theorem 4.1] and use the equivalences above to get estimate (5.1) in the following way: . Now let us give up the assumption on the supports of u, f and g. Let ξ 0 , ξ 1 , . . . , ξ m , be a partition of unity of O, such that ξ 0 ∈ C ∞ 0 (O), and, for i = 1, . . . , m, for each i ∈ {0, . . . , m}. For i ≥ 1 one gets the required estimate as before, using the fact that C ∞ 0 (O)-functions are pointwise multipliers in all spaces H ν p,θ (O), ν,θ ∈ R,p > 1, see, e.g., [38,Theorem 3.1]. The case i = 0 can be treated as follows: Since ξ 0 has compact support in O, for all ν,θ ∈ R andp > 1, we have and consequently .
By [32,Theorem 4.11], a further application of (5.4) and the fact that C ∞ 0 (O)-functions are pointwise multipliers in all spaces H ν p,θ (O), we obtain .
This finishes the proof of estimate (5.1) for the case γ = 1.
Next, let us move to the inductive step and assume that the assertion is true for some γ = n ∈ N. Fix u ∈ H n+1,q p,θ (O, T ). Then v := ψu x ∈ H n,q p,θ (O, T ) and dv = ψf x dt + ψg k x dw k t (component-wise). Also, by Lemma 2.1 (iii) and (iv), .
Using the induction hypothesis and applying Lemma 2.1(iii) and (iv) once more, we see that the induction goes through.

Besov spaces and their relationship to weighted Sobolev spaces
We turn our attention to the scale of Besov spaces where p ≥ 2 is fixed and O ⊆ R d is a bounded Lipschitz domain. As pointed out in the introduction, our motivation for considering this scale is its close connection to nonlinear approximation theory.
The main result of this section, Theorem 6.9, is a general embedding of the weighted Sobolev spaces

Besov spaces: Definition and wavelet decomposition.
Our standard reference concerning Besov spaces and wavelets is the monograph [5]. Throughout this subsection, let G ⊆ R d be an arbitrary domain.
For a function f : G → R and a natural number n ∈ N let be the n-th difference of f with step h ∈ R d . For p ∈ (0, ∞), the n-th order L p -modulus of smoothness of f is given by One definition of Besov spaces that fits in our purpose is the following: Definition 6.1. Let s, p, q ∈ (0, ∞) and n ∈ N with n > s. Then B s p,q (G) is the collection of all functions f ∈ L p (G) such that These classes are equipped with a (quasi-)norm by taking Remark 6.2. For a more general definition of Besov spaces, including the cases where p, q = ∞ and s < 0 see, e.g., [44].
We want to describe B s p,q (R d ) by means of wavelet expansions. To this end let ϕ be a scaling function of tensor product type on R d and let ψ i , i = 1, . . . , 2 d − 1, be corresponding multivariate mother wavelets such that, for a given r ∈ N and some M > 0, the following locality, smoothness and vanishing moment conditions hold: for all i = 1, . . . , 2 d − 1, 3) We assume that where we used the abbreviations for dyadic shifts and dilations of the scaling function and the corresponding wavelets Further, we assume that there exists a dual Riesz basis satisfying the same requirements. More precisely, there exist functions ϕ and ψ i , i = 1, . . . , 2 d − 1, such that conditions (6.2), (6.3) and (6.4) hold if ϕ and ψ are replaced by ϕ and ψ i , and such that the biorthogonality relations ϕ k , ψ i,j,k = ψ i,j,k , ϕ k = 0 , ϕ k , ϕ l = δ k,l , ψ i,j,k , ψ u,v,l = δ i,u δ j,v δ k,l , are fulfilled. Here we use analogous abbreviations to (6.5) and (6.6) for the dyadic shifts and dilations of ϕ and ψ i , and δ k,l denotes the Kronecker symbol. We refer to [5,Chapter 2] for the construction of biorthogonal wavelet bases, see also [12] and [6]. To keep notation simple, we will write ψ i,j,k,p := 2 jd(1/p−1/2) ψ i,j,k and ψ i,j,k,p ′ := 2 jd(1/p ′ −1/2) ψ i,j,k , for the L p -normalized wavelets and the correspondingly modified duals, with p ′ := p/(p − 1) if p ∈ (0, ∞), p = 1, and p ′ := ∞, 1/p ′ := 0 if p = 1.
The following theorem shows how Besov spaces can be described by decay properties of the wavelet coefficients, if the parameters fulfil certain conditions. Theorem 6.3. Let p, q ∈ (0, ∞) and s > max {0, d (1/p − 1)}. Choose r ∈ N such that r > s and construct a biorthogonal wavelet Riesz basis as described above. Then a locally integrable function A simple computation gives us the following characterization of Besov spaces from the scale ( * ) on R d . Corollary 6.5. Let p ∈ (1, ∞), α > 0 and τ ∈ R such that 1/τ = α/d + 1/p. Choose r ∈ N such that r > α and construct a biorthogonal wavelet Riesz basis as described above. Then a locally integrable function f : and (6.10) is an equivalent (quasi-)norm for B α τ,τ (R d ).
It is well-known, see [35,Theorem 9.7], that for k ∈ N 0 , In what follows we give a detailed proof of the extension of (6.14) to arbitrary smoothness parameters γ > 0 instead of m ∈ N. To this end, let us fix some notations. We will use a wavelet Riesz-basis ϕ k , ψ i,j,k : Furthermore, we want to distinguish the indices corresponding to wavelets with support in the interior of the domain from the ones corresponding to wavelets which might have support on the boundary of O. To this end we write where j, m ∈ N 0 and k ∈ Z d . Later we will also use the notation The following lemma paves the way for proving (6.14) for arbitrary γ > 0 instead of just γ = m ∈ N. It establishes an estimate for, roughly speaking, a discretization of a weighted Sobolev norm in terms of the supports of the wavelets in the interior of O at a fixed scaling level j ∈ N 0 . For better readability, we place the quite technical proof in the appendix. Remember that we write A • for the interior of an arbitrary subset A of R d . Lemma 6.8. Let O be a bounded Lipschitz domain in R d . Let p ∈ [2, ∞), γ ∈ (0, ∞) and ν ∈ R with γ ≥ ν. Furthermore, assume u ∈ H γ p,d−νp (O). Then, for all j ∈ N 0 , the inequality holds, with a constant N ∈ (0, ∞) which does not depend on j and u.
Now we can prove the main result of this section. We use the convention '1/0 := ∞'.
Proof. Let us start with the case ν > γ. Then, for any 0 < α < γ, we have where we used Lemma 6.6 and standard embeddings for Besov spaces. Therefore, in this case the assertion of the theorem follows immediately. From now on, let us assume that 0 < ν ≤ γ. We fix α and τ as stated in the theorem and choose the wavelet Riesz-basis of L 2 (R) from above with r > γ.
We also fix u ∈ H γ p,d−νp (O). Due to Lemma 6.6 we have u ∈ B ν p,p (O see, e.g., [42]. In the sequel we will omit the E in our notation and write u instead of Eu. Theorem 6.3 tells us that the following equality holds on the domain O: where the sums converge unconditionally in B ν p,p (R d ). Furthermore, cf. Corollary 6.5, we have see also [15]. Hence, by Lemma 6.6, it is enough to prove that We start with (6.15). The index set Γ introduced above is finite because of the boundedness of O, so that we can use Jensen's inequality to obtain k∈Γ | u, ϕ k | τ ≤ N k∈Γ | u, ϕ k | p 1/p τ ≤ N u τ B ν p,p (O) .
In the last step we have used Theorem 6.3 and the boundedness of the extension operator. Now let us focus on inequality (6.16). To this end, we use the notations from above and split the expression on the left hand side of (6.16) into (i,j,k)∈Λ 0 u, ψ i,j,k,p ′ τ + (i,j,k)∈Λ\Λ 0 u, ψ i,j,k,p ′ τ =: I + II (6.17) and estimate each term separately. Let us begin with I. Fix (i, j, k) ∈ Λ 0 . As a consequence of Lemma 6.8, we know that u Q • j,k ∈ B γ p,p (Q • j,k ). By a Whitney-type inequality, also known as the Deny-Lions lemma, see, e.g., [14,Theorem 3.5], there exists a polynomial P j,k of total degree less than γ, and a constant N , which does not depend on j or k, such that u − P j,k Lp(Q j,k ) ≤ N 2 −jγ |u| B γ p,p (Q • j,k ) .
We denote S ⋆ j,n := k ∈ Λ ⋆ j : Q j,k ∈ S j−n , n ∈ N 0 . To prove this, we first note that, since k 1 ≥ 1 fulfils (A.4), Fix k ∈ Λ ⋆ j and let n * be the smallest non-negative integer such that Q j,k ∩ S j−n * = ∅, i.e., n * := inf n ∈ N : Q j,k ∩ S j−n = ∅ ≤ j.
Then, there are two possibilities: On the one hand, Q j,k might be contained completely in S j−n * , i.e., Q j,k ⊆ S j−n * . Then we are done. On the other hand, it might happen that Q j,k is not completely contained in the stripe S j−n * . In this case, we claim that Q j,k ⊆ S j−(n * +1) , i.e., ρ(x) ∈ 2 −j+n * +1 2 −k 1 , 2 −j+n * +1 2 k 1 for all x ∈ Q j,k .
Let us therefore fix x ∈ Q j,k . Then, since the length of the diagonal of Q j,k is 2 −j 2M √ d, we have ρ(x) ≤ ρ j,k + 2 −j 2M √ d.
It remains to show that ρ(x) ≥ 2 −j+n * +1 2 −k 1 . We argue as follows: Since Q j,k is not completely contained in S j−n * , there exists a point x 0 ∈ Q j,k such that ρ(x 0 ) > 2 −j+n * 2 k 1 . Therefore, since the length of the diagonal of Q j,k is 2 −j 2M √ d, we have In the last step we used (A.4).
Step 2. We rearrange the cubes supporting the wavelets in classes containing only cubes with disjoint interiors. More precisely, let e 1 , . . . , e d be the canonical orthonormal basis in R d . Since and for any fixed m ∈ {1, . . . , (2M ) d }, if Q j,k , Q j,ℓ ∈ R j,m with k = ℓ, then Q • j,k ∩ Q • j,ℓ = ∅. In the sequel, we write