Change of variable in spaces of mixed smoothness and numerical integration of multivariate functions on the unit cube

In a recent article by two of the present authors it turned out that Frolov's cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness. Those cubature formulae go well together with functions supported inside the unit cube $[0,1]^d$. The question for the optimal numerical integration of multivariate functions with non-trivial boundary data, in particular non-periodic functions, arises. In this paper we give a general result that the asymptotic rate of the minimal worst-case integration error is not affected by boundary conditions in the above mentioned spaces. In fact, we propose two tailored modifications of Frolov's cubature formulae suitable for functions supported on the cube (not in the cube) which provide the same minimal worst-case error up to a constant. This constant involves the norms of a ``change of variable'' and a ``pointwise multiplication'' mapping, respectively, between the function spaces of interest. In fact, we complement, extend and improve classical results by Bykovskii, Dubinin and Temlyakov on the boundedness of change of variable mappings in Besov-Sobolev spaces of mixed smoothness. Our proof technique relies on a new characterization via integral means of mixed differences and maximal function techniques, general enough to treat Besov and Triebel-Lizorkin spaces at once. The second modification, which only tackles the case of periodic functions, is based on a pointwise multiplication and is therefore most likely more suitable for applications than the (traditional) ``change of variable'' approach. These new theoretical insights are expected to be useful for the design of new (and robust) cubature rules for multivariate functions on the cube.


Introduction
The seminal papers by Korobov [16], Hlawka [15] and Bakhvalov [2] started the research for the efficient computation of multivariate integrals Ω f (x) dx via cubature formulae of type where X n := {x 1 , ..., x n } ⊂ Ω ⊂ R d denotes the set of given integration nodes and (λ 1 , ..., λ n ) denotes the vector of integration weights. This field developed into several directions and attracted a lot of interest in the past 50 years until present, see Temlyakov [28,29,30,32], Dubinin [7,8], Skriganov [26], Triebel [36], Hinrichs et al. [13,14], Novak and Woźniakowski [21], Novak, Krieg [17], Dick and Pillichshammer [6], Dũng, Ullrich [9], and Markhasin [19] to mention just a few. However, several fundamental problems, for instance the construction of optimal (and easy to generate) point sets X n in (1.1) is still subject of intense research. Frolov's construction [11] dates back to the 1970s and gives a relatively easy construction of such point sets which perform optimal for several function classes of mixed smoothness. The recent results in [38] will be the starting point for the investigations in this paper. The concept of dominating mixed smoothness does not only connect discrepancy theory and optimal numerical integration. Function classes built upon this type of multivariate regularity also play an important role in many real-world problems. Several applications are modeled in Sobolev spaces with dominating mixed smoothness H s mix , see for instance Yserentant's book [43] for regularity properties of solutions of the electronic Schrödinger equation. However, the Hilbert space model is not sufficient when it comes to numerical integration issues. We observed in [38,14] that it seems to be more appropriate to define smoothness function classes on L p instead of L 2 . The case p = 1 turns out to be particularly important. To see this let us consider multi-variate kink functions which often occur in mathematical finance, e.g., the pricing of a European call option, whose pay-off function possesses a kink at the strike price [12]. In general, one can not expect Sobolev regularity higher than s = 3/2. However, when considering Besov regularity we can achieve smoothness s = 2. In a sense, one sacrifices integrability for gaining regularity. Looking at the error bounds in [38,14] one observes that certain cubature rules benefit from higher Besov regularity while the integrability p does not enter the picture.
In the recent paper [38] two of the present authors studied the performance of the classical Frolov cubature formulae [11,37] in several settings of classes with dominating mixed smoothness. It turned out that in Besov (B s p,θ ) and Triebel-Lizorkin (F s p,θ ) spaces with dominating mixed smoothness F d :=Å s p,θ , where A ∈ {B, F}, of functions supported strictly inside the unit cube Ω := [0, 1] d , the above mentioned cubature formulae provide the optimal rate of convergence among all cubature formulae of type (1.1) with respect to the minimal worst-case integration error given by However, Frolov's method does not seem to be suitable for functions with non-trivial boundary data, like multivariate periodic functions F d := A s p,θ (T d ) or non-periodic functions on the cube F d := A s p,θ ([0, 1] d ), cf. Definitions 3.1, 3.2 below. The case of periodic Besov spaces of dominating mixed smoothness has been studied in [7,8,9,14]. What concerns optimal cubature in non-periodic function classes, which are actually most relevant for applications, there are only partial results known, see for instance [36,9] for the correct lower bounds of Int n (B s p,θ ). A classical approach towards upper bounds has been proposed by Bykovskii [3] by first performing a change of variable to obtain for some differentiable ψ : [0, 1] d → [0, 1] d , and use afterwards a cubature formula (1.1) for the right-hand integrand in (1.3). The main observation is the fact that this approach results in a modified cubature formula Arranging the kernel ψ in a way such that the function | det ψ ′ (x)|f (ψ(x)) preserves mixed smoothness properties and is supported in [0, 1] d one can now use the above mentioned (optimal) methods to approximate the integral. Thus, the remaining step in analyzing the method (1.4) is to prove the preservation of mixed smoothness properties under change of variable, or in other words, to establish the boundedness of mappings for suitably chosen ψ(x). A straight-forward choice is certainly given by assuming a tensor product structure. That is, e.g., given univariate functions ψ : R → R which are integrated C k (R) bump functions ϕ supported in [0, 1] and, with a slight abuse of notation, define ψ(x 1 , ..., x d ) := ψ(x 1 ) · ... · ψ(x d ). A natural and simple choice is given by the family of polynomials  [7,8] to the Besov spaces B s p,θ if 1 ≤ p, θ ≤ ∞ and s > 1/p , and by Temlyakov [32] to the spaces W s p , s ∈ N, 1 ≤ p ≤ ∞. Dubinin and Temlyakov analyzed the change of variable with a C ∞ -kernel instead of (1.6), namely (1.9) With this particular choice both authors were able to incorporate even the important case p = 1 (see the discussion above). Note, that the boundedness is still open for F s 1,θ , see Remark 4.3, (iii) below. However, the kernel in (1.9) has some obvious disadvantages. It involves terms which get very small and others that get very large at the same time. Therefore, motivated from numerical robustness issues one is interested in as simple as possible change of variable kernels with the least (smoothness) requirements. The functions in (1.6) are polynomials and therefore good for our purpose. However, from a tractability point of view, i.e., if one is interested in the dependence of the involved constants on the dimension, every method of this type will be most likely intractable. Some results on this issue in the Hilbert space setting can be found in Kuo, Sloan and Woźniakowski [18], see also Nuyens and Cools [22] for numerical experiments.
Our main intention is to extend the mentioned boundedness results on the operators T ψ k d to spaces A s p,θ , in particular Triebel-Lizorkin spaces of dominating mixed smoothness F s p,θ , for the largest possible range of parameters.

Contribution and main results
The main goal of the present paper is to construct cubature formulae that • are simple, (to some extent) stable and universal (in the sense that the algorithm does not depend on the specific parameters of the spaces) • and that perform optimal in the sense of the minimal worst-case error (1.2) with respect to non-periodic function spaces A s p,θ on the unit d-cube.
The ingredient (1.6) is almost what we want to put into (1.4), provided we can get rid of the condition (1.8). This is possible as we will show in the theorems below. We are able to state the result for spaces A s p,θ on the unit d-cube, where the smoothness (or the parameter k in (1.6)) only depends on the smoothness s of the space (and not on the integrability p). Note, that the result below is not just a generalization of the results from [3,8,31,32] These results are useful in the sense that they "transfer" the optimal rate of the minimal worst-case error from the known situationÅ s p,θ to the more difficult situation A s p,θ . As a direct consequence we obtain the equivalence in the range of parameters given in Theorems 1.1, 1.2 whenever the classes A s p,θ consist of continuous functions. Let us mention the important special case for 1 < p < ∞ and s > 1/p taking into account that F s p,2 = W s p . Note, that this includes the case of small smoothness, which is present if 2 < p < ∞ and 1/p < s ≤ 1/2, see [30], [38].
In case, one is only interested in the numerical integration of periodic functions, we present an even simpler modification that satisfies the second point above but uses only a pointwise multiplication with a certain function. To our best knowledge this method has not been used before. In contrast to the considerations above we will now restrict to function spaces on the d-torus T d , see Subsection 3.2 below, i.e., we want to numerically integrate functions which are 1-periodic in every component. There are many results for W s p (T d ) and B s p,∞ (T d ) in this direction, see for instance [32] and the references therein. The recent contribution [14] especially deals with periodic Besov spaces of dominating mixed smoothness and analyzes cubature formulae on digital nets. However, the picture is still not complete. Except for the Fibonacci cubature rule [30,9] in d = 2 the discussed methods in [36,14] are either not optimal or the optimal methods only work for specific (restricted) sets of parameters, like for instance small smoothness s < 2 in [14]. In other words, the cubature formula is optimal for low smoothness but is not able to benefit from higher regularity. One goal of this paper is to get rid of such conditions. To gain this, we pursue a similar goal as above, namely the reduction of the problem to theÅ s p,θ setting via a tailored transformation. To this end we use a sufficiently smooth function ψ : (1.10) Clearly, we need Ω [0, 1] d . Then for any 1-periodic integrable function in each component it holds Starting with a cubature formula Q n of type (1.1) for functions with support in Ω we modify as follows where {x} = x − ⌊x⌋ has to be understood component-wise. In contrast to (1.5) the remaining step for analyzing (1.11) on 1-periodic function spaces is to establish the boundedness of pointwise multiplier mappings (1.12) There is a rich theory concerning pointwise multiplication for isotropic spaces B s p,θ (R d ), F s p,θ (R d ), see Runst, Sickel [23]. To our best knowledge the field is rather open for spaces of dominating mixed smoothness. In contrast to the theorems above on change of variable we are able to include the quasi-Banach case (min{p, θ} < 1) within this framework. This is why the natural lower smoothness restrictions enter, namely s > σ p and s > σ p,θ , respectively, where For the B-spaces the result looks as follows.
Both theorems immediately imply the two-sided relation in the above range of parameters if additionally A s p,θ ֒→ C(R d ) holds true.

Organization of the paper
The paper is organized as follows. In Section 2 we collect the required tools from Fourier analysis, especially maximal inequalities. Section 3 is devoted to the definition and properties of the function spaces under consideration. There we characterize Besov-Triebel-Lizorkin spaces by mixed iterated differences which turn out to be the crucial tool for proving our main results. Sections 4 and 5 are the heart of the paper in which we prove the change of variable and pointwise multiplication results already discussed in the introduction section. For the convenience of the reader we also discuss the one-dimensional versions of the results in Sections 4 and 5 in detail.

Notation
As usual N denotes the natural numbers, N 0 = N ∪ {0}, Z denotes the integers, R the real numbers, and C the complex numbers.
If X and Y are two (quasi-)normed spaces, the (quasi-)norm of an element x in X will be denoted by x X . The symbol X ֒→ Y indicates that the identity operator is continuous. For two sequences a n and b n we will write a n b n if there exists a constant c > 0 such that a n ≤ c b n for all n. We will write a n ≍ b n if a n b n and b n a n .

Tools from Fourier analysis
In this section we will collect the required tools from Fourier analysis.

Preliminaries
The space S ′ (R d ), the topological dual of S(R d ), is also referred to as the set of tempered distributions on R d . Indeed, a linear mapping f : S(R d ) → C belongs to S ′ (R d ) if and only if there exist numbers k, ℓ ∈ N and a constant c = c f such that and the corresponding inverse Fourier transform F −1 f (ξ) = Ff (−ξ). As usual, the Fourier transform can be extended to If ϕ, ψ ∈ S(R d ) then ϕ * ψ still belongs to S(R d ). In fact, we have ϕ * ψ ∈ S(R d ) even if ϕ ∈ S(R d ) and f ∈ L 1 (R d ). The convolution can be extended to ). It makes sense pointwise and is a C ∞ -function in R d .

Maximal inequalities
Let us provide here the maximal inequalities for the Hardy-Littlewood and Peetre maximal functions, respectively, which are essential for the proofs of where the supremum is taken over all cubes with sides parallel to the coordinate axes containing x. A vector valued generalization of the classical Hardy-Littlewood maximal inequality is due to Fefferman and Stein [10].
We require a direction-wise version of (2.1) 1 2s x i +s We denote the composition of these operators by M e = i∈e M i , where e is a subset of [d] and M ℓ M k has to be interpreted as M ℓ • M k . The following version of the Fefferman-Stein maximal inequality is due Stöckert [27].
Theorem 2.2. For 1 < p < ∞ and 1 < θ ≤ ∞ there exists a constant c > 0, such that for Iteration of this theorem yields a similar boundedness property for the operator M [d] .
The following construction of a maximal function is due to Peetre, Fefferman and Stein. Let Let Ω ⊂ R d be a compact set. Let further a > 0 and α = (α 1 , ..., α d ) ∈ N d 0 . Then there exist two constants c 1 , c 2 > 0 (independently of f ) such that holds for all f ∈ L 1 (R d ) with supp(Ff ) ⊂ Ω and all x ∈ R d . The constants c 1 , c 2 depend on Ω.
We will need the vector-valued Peetre maximal inequality which is a direct consequence of Lemma 2.3 together with Theorem 2.2.

Function spaces of dominating mixed smoothness
In this section we introduce the function spaces under consideration, namely, the Besov and Triebel-Lizorkin spaces of dominating mixed smoothness. Note that the Sobolev spaces W s p of mixed smoothness appear as a special case of the Triebel-Lizorkin spaces, namely W s p = F s p,2 if 1 < p < ∞ and s > 0. There are several equivalent characterizations of these spaces, see e.g. [42]. We begin with the usual definition of the spaces using a dyadic decomposition of unity on the Fourier side, following Triebel [36]. Afterwards, we present an equivalent characterization by mixed iterated differences. To begin with, we introduce the concept of smooth dyadic decomposition of unity.

Spaces on R d
Let us start with the definition of the function spaces A s with the usual modification for θ = ∞.
with the usual modification for θ = ∞.
Remark 3.3. In the special case θ = 2 and 1 < p < ∞ we put W s p := F s p,2 which denotes the Sobolev spaces of dominating mixed smoothness. It is well-known (cf. [24,Chapt. 2] for d = 2), that in case s ∈ N 0 the spaces W s p can be equivalently normed by The next lemma collects some frequently used embedding properties of the spaces.
Lemma 3.5(ii) gives a sufficient condition for A s p,θ to consist only of continuous functions. This is of particular importance for the analysis of cubature rules since function evaluations have to be defined reasonably.
respectively. Note, that we replace the L p (R d )-norms by L p (T d )-norms. If we denote by {c k (f )} k∈Z d the Fourier coefficients of f , then we could write see [24, p. 150].
Next we turn to the subspace of functions which are supported in a compact set Ω. That is, we define for s > σ p = max{1/p − 1, 0} the spaces The (quasi-)norm on these spaces is usually given by

Characterization by mixed iterated differences
In this section we will provide a characterization of the above defined spaces which is actually the classical way of defining them, see for instance Nikol'skij [20], Amanov [1], Schmeißer, Triebel [24], Temlyakov [31] and the references therein. For us it will be convenient to replace the classical moduli of continuity by so-called rectangular means of differences, see (3.3) below. This is is the counterpart of the ball means of differences for isotropic spaces, see [33,Thm. 2.5.11]. Let us first recall the basic concepts. For univariate functions f : R → C the mth difference operator ∆ m h is defined by where Id f = f and ∆ m h i ,i is the univariate operator applied to the i-th coordinate of f with the other variables kept fixed. Whenever we write the "product" i∈e A i of operators A i we mean the concatenation of the A i , see also the small paragraph before Theorem 2.2 above. Now we define the so-called rectangular means of differences by For j ∈ N d 0 we put e(j) = {i : j i = 0} and e 0 (j) = [d]\e(j). We have the following theorems.
Theorem 3.7. Let 0 < p < ∞, 0 < θ ≤ ∞ and s > σ p,θ . Let further m ∈ N be a natural number with m > s . Then Since the characterizations of the spaces B s p,θ and F s p,θ that are given in Theorems 3.6 & 3.7 are, as far as we know, not proven in the literature, we will give a proof here for the convenience of the reader.
Proof of Theorems 3.6 & 3.7. The statements in Theorems 3.6 & 3.7 are "discretized versions" of the characterizations given in [39]. In addition, the range of the sum j∈N d 0 is crucial. Here, we mainly have to establish the relation f F s p,θ ≍ f (m) F s p,θ which is done via the characterization for the spaces B s p,θ and F s p,θ via local means, see [42], [41]. Given a function Ψ : R → C, we denote by L Ψ ∈ N the maximal number such that Ψ has vanishing moments, i.e., Let Ψ 0 ∈ S(R) be a function satisfying the following conditions We have the following proposition, see [42, Thm. 1.23] and [41].
(ii) Let 0 < p < ∞ and 0 < θ ≤ ∞. Then F s p,θ is the collection of all f ∈ S ′ (R d ) such that Now we are in a position to prove Theorem 3.7. The proof of Theorem 3.6 is similar but technically less involved.
Proof of Theorem 3.7. Step 1. We prove the inequality Discretizing the right-hand side of the above inequality we obtain which implies (3.5) .
Step 2. We prove the reverse inequality in (3.5). This time we rely on the intrinsic characterization of F s p,θ via local means, see Proposition 3.8.
then we have supp(η) ∈ (−1, 1) and moreover A simple computation gives for a univariate function g Substep 2.2. We define the function η(·) on R d as d-fold tensor product, i.e.,

From (3.6) and (3.7) we have for
Then we obtain In view of Proposition 3.8 we finish the proof.
Theorem 4.1. Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, s > 0 and ϕ ∈ C k 0 (R) as above with k > ⌊s⌋ + 2 if p > 1 and k > ⌊s⌋ + 3 if p = 1. Then Theorem 4.2. Let 1 < p < ∞, 1 < θ ≤ ∞, s > 0 and ϕ ∈ C k 0 (R) as above with k > ⌊s⌋ + 2. (ii) With the technique used in this paper we were not able to give a result in the case p < 1 for both, B and F-spaces, not even in the univariate situation, see Lemma 4.4 below. Note, that also the case p = 1 is open in the F-case. However, we strongly conjecture that we can prove boundedness of the change of variable mapping if 1/2 < p ≤ 1 in both cases.
For the convenience of the reader we will first comment on a corresponding univariate result.
Before we come to the proof of the univariate boundedness results we need some preparation. It will turn out that difference characterizations of the function spaces under consideration is a suitable approach to analyze the respective operators.

Proof of Lemma 4.4
Let us first prove a technical lemma on the boundedness of quotients of derivatives of ϕ. Such terms naturally appear if one bounds the L p -norm of T ψ 1 f . For the particular choice ϕ = ψ ′ k , with ψ k from (1.6), the lemma below was proved in in [31, p. 238].
Proof. Obviously, it is enough to prove the statement for p = 1, since due to the boundedness of ϕ. We write for some p 1 , p 2 with 1/p 1 + 1/p 2 = 1. Using Lemma 4.5 we know that we need k > rp 1 p 1 −1 + 1 for the boundedness of the first factor and k > αp 2 p 2 −1 + 1 for the boundedness of the second. First note that the statement of the corollary is trivial for r = α = 0. If only one of them is zero, say r = 0, then we set p 1 = 1 and p 2 = ∞; again the statement is obvious. Hence, we assume now that r = 0 and α = 0 and we choose p 1 = (r + α)/α and p 2 = (r + α)/r. This leads to the restriction k > r + α + 1 in both cases. Since we want a result for all r + α ≤ m we impose the restriction k > m + 1. This finishes the proof. Now we are in position to prove Lemma 4.4.

Proof of Lemma 4.4.
Step 1. Let η j denote a smooth dyadic compactly supported decomposition of unity (for instance the one given at the beginning of Subsection 3.1). In particular, 1 = j∈N 0 η j (t) for all t ∈ R and η j (t) = 0 if |t| > 2 j+1 . We decompose f as follows where we put f j ≡ 0 in case j < 0 . Note, that with this definition, we can clearly write also f = ℓ∈Z f j+ℓ for every fixed j ∈ N 0 . Using the characterization by rectangular means of differences of F s p,θ , see Theorem 3.7, and the decomposition (4.1) we have Here we choose m = ⌊s⌋ + 1.
Step 2. Consequently, we obtain for |h| Where in the second inequality we have applied Theorem 2.1. Now we continue with for some C α . We distinguish two cases. First let us deal with α = m. We choose the number a such that max{1/p, 1/θ} < a ≤ m. This is possible since max{1/p, 1/θ} < 1 and m ≥ 1.
Since ϕ ∈ C k 0 (R), k > ⌊s⌋ + 2 = m + 1, Corollary 4.6 yields that for all r, α with r + α ≤ m, which leads to |ϕ (α) (t)Q γ (t)| |ϕ(t)| 1/p . Consequently, We consider the term By Lemma 2.3 and a homogeneity argument we have for any a > 0 Putting this into (4.7) with a > max{1/p, 1/θ} and using Theorem 2.4 afterwards yields (4.9) From (4.5), (4.7), and (4.9) we obtain (4.10) Step 3. It remains to deal with ℓ≥0 . We expand the difference and estimate We consider the term (4.12) Here again we use the Hardy-Littlewood maximal function. From Theorem 2.1 again we get We performed a change of variable in the second step and used p > 1 and ϕ ∈ C k 0 (R). With (4.11), (4.12), and (4.13) we obtain where we used s > 0 in the last inequality. Altogether we obtain from (4.10) and (4.14) In view of (4.2) we conclude that which finishes the proof.

Proof of Theorems 4.1 and 4.2
Proof of Theorem 4.2.
Step 1. We aim at adapting the proof of Lemma 4.4 to the multivariate mixed situation by applying the arguments direction-wise. The proper tool will be the characterization by rectangular means of differences, see Theorem 3.7. Let {η j } j∈N d 0 be a tensorized compactly supported smooth dyadic decomposition of unity and where we put f j ≡ 0 if j / ∈ N d 0 . Analogously, to (4.2) we estimate the quantity where 2 −j = (2 −j 1 , ..., 2 −j d ). Again we choose m = ⌊s⌋ + 1. Let us introduce some further notation first. If e ⊂ [d] and j ∈ N d 0 we denote For each ℓ ∈ Z d we denote Using Leibniz's formula we estimate (4.20) Here the sum on the right-hand side is taken over all α ∈ N d 0 with 0 ≤ α i ≤ m if i ∈ e 2 (j, ℓ), otherwise α i = 0 and β = m e 2 (j,ℓ) − α. We continue estimation by dividing e 2 (j, ℓ) = e 1 2 (j, ℓ) ∪ e 2 2 (j, ℓ) where α i < m if i ∈ e 1 2 (j, ℓ) and α i = m if i ∈ e 2 2 (j, ℓ). We have see (4.6), which leads to Changing variable we obtain Here [D (β) f j+ℓ ](x, ψ e 2 2 (j,ℓ) ) = [D (β) f j+ℓ ](z 1 , ..., z d ) with z i = ψ(x i ) if i ∈ e 2 2 (j, ℓ) otherwise z i = x i . Using the same scaling argument as in (4.4) and (4.8) yields with m ≥ a. Now we obtain from (4.20), (4.21) and (4.22) with a > max{1/p, 1/θ}. This is because i ∈ e 0 (j) implies ℓ i ≥ 0. Then Theorem 2.4 yields Step 3. In a view of (4.16) we conclude that The proof is complete.
Proof of Theorem 4.
Step 2. The case p > 1. We put m = ⌊s⌋ + 1. The proof is similar in F-spaces but less technical. Here we use the classical Hardy-Littlewood maximal inequality (the scalar version of Theorem 2.1) and the scalar version of Theorem 2.4. These inequalities do not depend on the parameter θ. Hence the result in the B-spaces can be extended to 0 < θ ≤ ∞.
Step 3. The case p = 1. Using the convolution representation of the mth order difference in e 2 (j, ℓ) we obtain from (4.19) Here y = (y 1 , ..., y d ) with y i = 0 if i ∈ e 2 (j, ℓ). We used the same notation in the proof of Theorem 4.2, see (4.17) and (4.18). Inserting this into R e(j) m (T ψ d f, 2 −j , ·) 1 and then changing the order of integration with the fact that Next step is carried out as F-spaces. Note that in this case we choose m = ⌊s⌋ + 2, since there must exist a such that 1 < a ≤ m for the inequality (4.22) to hold. The proof is complete.
Remark 4.7. The last step of the proof for p = 1 shows that, based on our method, we have to guarantee that max{1, s} < m. That's why we need the more restrictive condition k > m + 1 = ⌊s⌋ + 3 in Theorem 1.1. Under the additional assumption s ≥ 1 we can relax this condition to k > ⌊s⌋ + 2 as in the case p > 1. Another possibility to relax this condition is to use Nikol'skij inequality at the appropriate point, as it was already done in [8].

Pointwise multiplication
We will first comment on the result for F-spaces in Theorem 1.4. Again, we make use of the useful characterization by differences in Theorem 3.7. The technique used in the proof might not be "optimal" in the sense of Remarks 5.2, 5.3 below. However, it is quite transparent and works well for spaces with dominating mixed smoothness A s p,θ where almost nothing is known in this direction. Recall the multiplication operator T ψ d is defined by for some sufficiently smooth and compactly supported function ψ. Similar to Lemma 4.4 we will proof a one-dimensional version of Theorem 1.4 first.

Remark 5.2 (Small values of s).
(i) We believe that the conditions s > σ p,θ and s > σ p in Lemma 5.1 and Theorems 1.3, 1.4 are only technical and caused by our proof technique. On the one hand s > σ p,θ (and s > σ p in the B-case) is necessary for the characterization of Triebel-Lizorkin spaces by means of differences, see Theorems 3.6, 3.7 and [4], and on the other hand our proof below requires this condition. One may relax (or even remove) this condition by using the characterization of Triebel-Lizorkin spaces via local means, see Proposition 3.8 above, [34,Thm. 4.2.2] and [42,Thm. 1.25]. Note that removing the condition s > σ p,θ will extend the range of parameters for which (1.13) holds. In fact, if 1/θ − 1 > 1/p then the case 1/p ≤ s < 1/θ − 1 for the F-scale in (1.13) is not covered by Theorem 1.4, although it is reasonable since F s p,θ ֒→ C(R d ). In other words, we would obtain Proof. Assume first that b = 1. By the mean value theorem from calculus we get where we used Leibniz' rule in the second estimate. Let us define What follows is a simple homogeneity argument to deal with b > 1 and supp = c m,a · c m,ψ · |bh| m max{1, |bh| a } · sup ξ∈R |f (t − ξ)| (1 + |bξ|) a = C m,a,ψ |bh| m max{1, |bh| a }P b,a f (t) .
Remark 5.6. The proof of Theorem 1.3 (B-case) is similar but technically less involved. As in the F-case we choose 0 < λ < p and a > 1/p such that s − a(1 − λ) > 0 if p ≤ 1 otherwise we put λ = 1. Again, one can do so because of s > σ p . Note, that the weaker condition s > σ p comes from the fact that the Peetre maximal function (2.2) is bounded in L p (T d ), 0 < p < ∞, if a > 1/p, which is the scalar version of Theorem 2.4 .
been initiated, and at the conference "Approximation Methods and Function Spaces" in Hasenwinkel, 2015. The authors would particularly thank D. Bazarkhanov for pointing out reference [8] to us. Moreover, Tino Ullrich gratefully acknowledges support by the German Research Foundation (DFG) and the Emmy-Noether programme, Ul-403/1-1.