Solenoidal difference quotients and their application to the regularity theory of the p-Stokes system

We prove existence of a solution to the divergence equation satisfying a new Bogovski-type estimate for the difference quotients. This enables us to give an alternative proof of the interior regularity of the solution to the p-Stokes problem, completely avoiding the pressure. Moreover, as a key preliminary result we prove boundedness of Calderón–Zygmund operators with standard kernels in weighted Lebesgue and Orlicz spaces over a general domain.


Introduction
We are concerned with the question of interior regularity of the weak solution of the steady Stokes approximation for flows of shear thinning fluids. It is given by − div S(Du) + ∇π = f in Ω, div u = 0 in Ω, u = 0 on ∂Ω, (1.1) where Ω ⊂ R n , n ≥ 2, is a bounded domain. In here, u = (u 1 , . . . , u n ) denotes the unknown velocity vector field, and π the unknown scalar pressure, while the external body force f = ( f 1 , . . . , f n ) is given. The extra stress tensor S depends only on Du := 1 2 (∇u + ∇u ), the symmetric part of the velocity gradient ∇u. The relevant example we have in mind is with p ∈ (1, 2], δ ≥ 0, and μ > 0. Notice that despite various efforts the optimal global regularity of this problem is still open (see [1,4,8] for partial results). However, the interior regularity we focus on here is well-known (see [4,15,19]). The standard proof uses localized difference quotients in each direction. Due to the localization, the corresponding test function is not solenoidal anymore, therefore appropriate properties of the pressure have to be used.
Here we modify this approach and use a solenoidal version of the localized difference quotient. Thus, we can completely avoid the pressure π. This is a completely new approach even for the classical Stokes problem, i.e., p = 2 in (1.2). To make it possible, we show that the solution of the divergence equation obtained via the Bogovski formula (see [5,6]) satisfies an additional estimate for the difference quotient. The proof of this is based on estimates of singular Calderón-Zygmund operators generated by standard kernels in arbitrary domains Ω ⊂ R n (see Theorem 3.4). This result is of independent interest since it shows that an analogy of the classical Calderón-Zygmund estimates (cf. [7,18]) in the whole space also hold in arbitrary domains. To prove it, we employ ideas from [5,6,10,16,23], where the divergence equation is treated, and modify them for our purposes. An important feature of our result is a careful tracking of the dependence of the constants on various quantities, which is missing in the literature.

Notation
Throughout the text, we use the symbols C, c to denote generic constants which may change from line to line but are independent of "crucial" quantities. In many cases, the dependence of such constants on various quantities will be explicitly specified. Furthermore, we write f ∼ g if there exist constants c, C > 0 such that c f ≤ g ≤ C f .
A set Q ⊂ R n is called an (open) cube if there exist a 1 , . . . , a n ∈ R and (Q) > 0 such that Q = x ∈ R n a i < x i < a i + (Q), i = 1, . . . , n .
The number (Q) is the side length of Q and the point c := (a 1 + 1 2 (Q), . . . , a n + 1 2 (Q)) is the center of the cube Q, which can be also denoted by Q = Q(c, (Q)). For α > 0 and a cube Q = Q(c, (Q)) we use the notation α Q := Q(c, α (Q)). Moreover, a cube Q is said to be dyadic if there exist j i , . . . , j n , k ∈ Z such that Notice that, in the above definition, we consider only cubes with faces parallel to the coordinate axes.
While working with function spaces, we do not distinguish between spaces of scalar, vector-valued or tensor-valued functions. However, we denote vectors by boldface lower-case letters (e.g., u) and tensors by boldface upper-case letters (e.g., S). For vectors u, v ∈ R n , the standard tensor product u ⊗ v ∈ R n×n is defined as (u ⊗ v) i j := u i v j , and the symmetric tensor product as u ). The scalar product of vectors is denoted by u · v = n i=1 u i v i and the scalar product of tensors is denoted by A · B := n i, j=1 A i j B i j . If E ⊂ R n is a measurable set, |E| denotes its Lebesgue measure. In the following definitions of function spaces, we always assume E to be a domain in R n , with a sufficiently smooth boundary, if needed. We use standard Lebesgue spaces (L p (E), · p ) and Sobolev spaces (W k, p (E), · k, p ). If the underlying domain E needs to be indicated, we denote the respective norms by · L p (E) and · W k, p (E) . If p ∈ (1, ∞), we denote by p the conjugated exponent p := p p−1 . Besides the standard L p spaces we will also consider their weighted variants. A weight is any measurable function ω : R n → [0, ∞). If p ∈ [1, ∞) and ω is a weight, the space L p ω (E) consists of all measurable functions f on E such that If p > 1, an A p weight is a weight such that ω and ω 1− p are locally integrable and where B ⊂ R n are balls. In this case, we write ω ∈ A p . If p = 1, a weight is called an A 1 weight if 0 < ω < ∞ a.e. and [ω] In here, M is the maximal operator (with respect to non-centered balls), defined by The symbol supp f denotes the support of a function f . The set of all compactly supported, smooth functions defined on E is denoted by 0 (E) may be equipped with the gradient norm ∇ · p , thanks to the Poincaré inequality. Next, we denote by W 1, p 0,div (E) the subspace of W 1, p 0 (E) consisting of solenoidal vector fields u, i.e., such that div u = 0. By L p ω,0 (E) we denote the subspace of L p ω (E) consisting of functions f with vanishing mean value, i.e., such that E f (x) dx = 0.
Orlicz and Sobolev-Orlicz spaces also appear frequently in this article. We briefly present their elementary properties here, for details we refer to [20,21].
An N -function is a continuous, nonnegative, strictly increasing and convex function ψ on [0, ∞) which additionally satisfies ψ(0) = 0, lim t→∞ Furthermore, we define If Δ 2 (ψ) < ∞, we say that ψ satisfies the Δ 2 -condition. From now on, let us assume that Δ 2 (ψ) < ∞ and Δ 2 (ψ * ) < ∞. Then we denote by L ψ (E) and W 1,ψ (E) the classical Orlicz and Sobolev-Orlicz spaces, respectively. More precisely, f ∈ L ψ (E) if the modular is finite, and f ∈ W 1,ψ (E) if both f and ∇ f belong to L ψ (E). When equipped with the Luxemburg norm the space L ψ (E) becomes a Banach space. The same holds for the space W 1,ψ (E) when equipped with the norm · ψ + ∇ · ψ . Notice that the dual space (L ψ (E)) * can be identified with the space L ψ * (E). Furthermore, by W . If E is bounded and sufficiently regular, the Poincaré inequality for Orlicz modulars (see [25,Lemma 3]) implies that W 1,ψ 0 (E) may be equipped with the gradient norm ∇ · ψ . By L ψ 0 (E) and C ∞ 0,0 (E) we denote the subspaces of L ψ (E) and C ∞ 0 (E), respectively, consisting of functions f such that E f (x) dx = 0.

Difference quotient
If k ∈ {1, . . . , n}, we denote by e k the k-th vector of the canonical basis of the Euclidean space R n , i.e., e k = (δ ik ) n i=1 . If E ⊂ R n , we denote E ± he k := x ∈ R n ∃y ∈ E : x = y ± he k , Let F : R n → R n×n be a measurable tensor field (or a vector field or a real-valued function) and h > 0. Then we define the difference quotients of F as follows: We will also use the notation Elementary calculations show that we have the following variant of the product rule for For F, G ∈ L 1 (E), extended by zero outside E, the partial integration formula The proof of this assertion in the case of the special N -function ψ(t) = t p can be found in [14, Theorem 3 (i) in Section 5.8]). In fact, replacing the special N -function t p by a general N -function ψ one can proceed exactly as outlined there to obtain (2.4). These comments also apply for the converse statement [14, Theorem 3 (ii) in Section 5.8]). More precisely, if

Operators with kernels
A measurable function K : R n × R n → R is called a kernel.
kernel with respect to E if there exists a constant κ 1 > 0 such that, for any x, y, z ∈ E satisfying the following conditions hold true: The set of all kernels satisfying the above conditions with a given constant κ 1 will be denoted by SK(E, κ 1 ).

Remark 2.2
As it is common in the literature, it is possible to replace the right-hand side in (2.9) and (2.10) by with δ > 0. Although the results could be obtained in this generalized setting as well, we restrict ourselves to the case δ = 1 to avoid further complications.

Definition 2.3 We say that a linear operator T on C
holds whenever the right-hand side is well-defined. For a given kernel K and ε > 0 we define the truncated kernel K ε by and denote by T ε the operator generated by the kernel K ε .

Definition 2.4
Let K be a kernel and let E ⊂ R n be a domain. We call K a Calderón-Zygmund kernel with respect to E, if there exists a constant κ 2 ∈ (0, ∞) such that the function N : satisfies the following conditions: 14) The set of all kernels satisfying the above conditions with a given constant κ 2 will be denoted by CZ(E, κ 2 ).

Remark 2.5
One can replace (2.14) by where q ∈ (1, ∞), and still retain the relevant properties of the operator generated by the kernel. For the sake of simplicity, we use the condition only with q = 2.

(p, ı)-structure
Let us define what it means that a tensor field S has a ( p, δ)-structure. For details on this matter, see [11,22]. For a tensor P ∈ R n×n we denote its symmetric part by where R n×n sym := {P ∈ R n×n | P = P }.
We use the notation |P| 2 = P · P.
It is convenient to define a special N -function ϕ = ϕ p,δ , with p ∈ (1, ∞), δ ≥ 0, by The function ϕ satisfies, uniformly in t and independently of δ, the equivalences In case that the (one-sided) derivative ϕ (0) does not exist, we assume that ϕ (t) t is continuously extended by zero for t = 0. We define the shifted N -functions {ϕ a } a≥0 (cf. [11,12,22]) by Note that the family {ϕ a } a≥0 satisfies the Δ 2 -condition uniformly with respect to a ≥ 0, i.e., Definition 2.6 Let S : R n×n → R n×n sym be a tensor field satisfying are satisfied for all P, Q ∈ R n×n with P sym = 0 and all i, j, k, l = 1, . . . , n. The constants γ 0 , γ 1 , and p are called the characteristics of S.

Remark 2.7
An important example of a tensor field S having a ( p, δ)-structure is given by S(P) = ϕ (|P sym |)|P sym | −1 P sym . In this case, the characteristics of S, namely γ 0 and γ 1 , depend only on p and are independent of δ ≥ 0.
Suppose that a tensor field S has a ( p, δ)-structure. Then we define its associated tensor field F : R n×n → R n×n sym by The connection between S, F and {ϕ a } a≥0 is best explained by the following proposition (cf. [11,22]).

17)
uniformly in P, Q ∈ R n×n . Moreover, uniformly in Q ∈ R n×n , The constants depend only on the characteristics of S.
For a detailed discussion of the properties of S and F and their relation to Orlicz spaces and N -functions we refer the reader to [3,22]. In what follows, we shall work only with S(P) and F(P), where P is a symmetric tensor. Therefore, we can drop the superscript " sym " in the above formulas. If S has a ( p, δ)-structure, from Proposition 2.8 we easily obtain the following equivalences: The equivalence constants depend here only on the characteristics of S. All assertions from this section may be proved by easy manipulations of definitions, and we omit their proofs.

Calderón-Zygmund estimates
Our interest lies in estimates concerning Orlicz modulars. However, we are going to prove the results first in weighted L p spaces. The following known extrapolation principle (see [9,Theorem 4.15]) offers an elegant connection between the two settings.
holds for all ( f , g) ∈ F and all weights ω ∈ A p 0 . Then for every N -function ψ satisfying Δ 2 (ψ) < ∞ and Δ 2 (ψ * ) < ∞ there exists a constant C ∈ (0, ∞) such that the inequality  [9,Proposition 3.5]) as well as the boundedness of the maximal operator in Orlicz spaces (see [17,Theorem 2.2]). The latter may be expressed by the inequality where c is the weak (1, 1)-constant of the maximal operator, which depends on n only. In fact, one can show that In here, c is a constant depending on Δ 2 (ψ), Δ 2 (ψ * ), the exponent α depends on Δ 2 (ψ), and ρ ψ (M) is the constant in the modular estimate of the maximal operator (it can be estimated by Δ 2 (ψ * ) Δ 2 (ψ) β , where β depends on n).
The auxiliary result below is a simple version of the Whitney covering lemma (cf. [2, p. 348]).

Proposition 3.3
Let Ω ∈ R n be a domain. Then there exists a sequence {Q j } j∈N of dyadic cubes satisfying: Proof Let x k k∈N be a dense sequence of points in Ω. Define Q 1 as the largest 1 dyadic cube Q such that x 1 ∈ Q and dist (Q, ∂Ω) > 4 diam Q. Next, suppose that Q 1 , . . . , Q j−1 are defined and let k ∈ N be the smallest index such that x k / ∈ j−1 i=1 Q i . Then define Q j as the largest dyadic cube Q such that x k ∈ Q and dist (Q, ∂Ω) > 4 diam Q. The sequence Q j j∈N obviously satisfies (ii) and (iii). To verify (i), let x ∈ Ω be arbitrary. There exists a dyadic cube Q such that x ∈ Q and dist (Q, ∂Ω) > 4 diam Q. By density, x k ∈ Q for some k ∈ N. Then necessarily Q ⊆ Q j for some j ∈ N, hence x ∈ j∈N Q j .
The next theorem justifies the definition of Calderón-Zygmund operators with standard kernels by the Cauchy principal value of the integral (2.11) and shows their boundedness in L p ω (Ω) and L ψ (Ω). Although the result is well-known, in standard literature it appears only in the setting Ω = R n . We prove it below for any domain Ω ⊂ R n .
Proof of Theorem 3. 4 We are going to prove the theorem only for the operator T . The proof for T ( * ) follows the same reasoning. At first, let us assume that where  N (x, y)

η(x)
satisfies the conditions (2.12) and (2.13). Moreover, it satisfies (2.14) globally, i.e., with E = R n . Thus, K ∈ CZ(R n , κ 2 ). Therefore, by [7,Theorem 2], is defined for a.e. x ∈ R n and all g ∈ C ∞ 0 (R n ) and it admits a bounded extension from L 2 (R n ) to L 2 (R n ). It follows from the proofs and comments in [7], in particular p. 295 and a remark on p. 306, that the corresponding operator norm satisfies T L 2 →L 2 ≤ c κ 2 , where c > 0 is a fixed constant depending only on the dimension n. Since T ε g(x) = T ε (gη)(x) holds for all ε > 0, g ∈ C ∞ 0 (R n ) and a.e. x ∈ R n , we may draw the same conclusions about the operator In particular, we get T g(x) = T (gη)(x), and thus (3.5) Let us next show that K is a standard kernel with respect to R n . Suppose that x, y, z ∈ R n satisfy (2.7). Then We will distinguish two cases. At first, assume that x, z ∈ Ω and dist (y, Ω) ≤ diam Ω. (3.7) Then obviously y ∈ Ω, and since K ∈ SK( Ω, κ 1 ), we have In the second and third condition we used the estimate which holds due to (3.6) and (3.7), and the property ∇η ∞ ≤ 2(diam Ω) −1 . Now suppose that (2.7) holds but (3.7) does not. Then one of the following situations occurs: ∈ supp η and, moreover, we get Hence, which shows that z / ∈ supp η. (iii) dist (y, Ω) ≤ diam Ω and dist (z, Ω) > 4 diam Ω. Then z / ∈ supp η and, moreover, we get Hence, which shows that x / ∈ supp η.
In each of the three cases we get K (x, y) = K (z, y) = K (y, x) = K (y, z) = 0, hence the conditions (2.8)-(2.10) are satisfied trivially. Altogether, we have now verified that K ∈ SK(R n , κ 0 ) with the constant κ 0 =: (1 + 3 · 2 n+2 )κ 1 . Thanks to this and the estimate (3.5), [18,Theorem 7.4.6] yields that for every p ∈ (1, ∞) there exists a nondecreasing function holds for all f ∈ C ∞ 0 (R n ) and all ω ∈ A p . It has to be noted that the desired monotone dependence of C p on [ω] A p is not mentioned in [18] but it can be verified through a careful inspection of the proofs leading to the result in there. Notice also that C p is independent of κ 1 , κ 2 and Ω.
In the next step, let ψ be an N -function with Δ 2 (ψ) < ∞ and Δ 2 (ψ * ) < ∞. Due to (3.8) Theorem 3.1 provides the existence of a positive constant C ψ such that To obtain this estimate we have performed the extrapolation with respect to the family Hence, (3.9) and (3.8) yield (3.2) and (3.3), respectively, for f ∈ C ∞ 0 (Ω). Since C ∞ 0 (Ω) is dense in L p ω (Ω) as well as in L ψ (Ω), the operator T admits corresponding bounded extensions such that (3.2) and (3.3) hold for any f ∈ L p ω (Ω) and f ∈ L ψ (Ω), respectively. So far, we have proved the theorem under the stronger assumption (3.4). To prove it in full generality, let Ω ⊂ R n be an arbitrary domain and K ∈ SK(Ω, κ 1 ) ∩ CZ(Ω, κ 2 ). By Proposition 3.3 there exists a sequence {Q j } j∈N of pairwise disjoint dyadic cubes such that Ω = j∈N Q j and Hence, for any j ∈ N, we have K ∈ SK( Q j , κ 1 ) ∩ CZ( Q j , κ 2 ), and using the first part of the proof we get for all ω ∈ A p and f ∈ L p ω (Ω). In here, it is important that C p p [ω] A p does not depend on Q j . Similarly, we obtain (3.3) for any f ∈ L ψ (Ω). Note also that Remark 3.5 is justified.

Divergence equation
We proceed with proving the new estimate concerning the Bogovski solution of the divergence equation. We prove the result in the weighted-L p setting. The variant for Orlicz modulars will be then obtained as a corollary by extrapolation.
holds for all ω ∈ A p and all f ∈ C ∞ 0,0 (Q). The function C p is independent of Q. (iii) For every p ∈ (1, ∞) there exists a nondecreasing function C p : holds for all h > 0, ω ∈ A p and f ∈ C ∞ 0,0 (Q). The function C p is independent of Q.
holds for all h > 0 and all f ∈ L ψ 0 (Q). In here, f and B f are extended by zero outside Q so that their difference quotients are defined a.e. in R n . The constant C ψ is independent of Q.
Before we prove Theorem 4.1, let us show the following simple technical proposition.

Lemma 4.3
Let ∈ C ∞ (R n , R) and a, b, w ∈ R n . Then Proof of Theorem 4.1 At first suppose that Q = − 1 2 , 1 2 n . Choose a fixed function ∈ (Q) and it satisfies (4.1) in Q. From now on, we will use the notation 0 (R n ) and kernels K whenever the limit on the right-hand side exists. Part (ii) is proved in [13,23] without the monotone dependence of C p on [ω] A p . Note that this dependence of the constant is mentioned in [24]. For the sake of the self-consistency of this paper we sketch its proof here. The technique used here is the same as in part (iii), where full details will be given. Let i, j ∈ {1, . . . , n}. Since Q is convex and it contains supp , by [16, p. 119] we have for any f ∈ C ∞ 0,0 (Q). In here, and U i j is a kernel such that |U i j (x, y)| ≤ c 1,∞ |x − y| 1−n , where c is a positive constant. From [10, Lemma 6.1] we get that J i j ∈ SK(Q, κ) ∩ CZ(Q, κ), where κ depends only on 1,∞ and n. By Theorem 3.4 there exists a positive nondecreasing function c p such that for all f ∈ C ∞ 0,0 (Q). Furthermore, we have (see [10, p. 218]) for f ∈ C ∞ 0,0 (Q) and x ∈ Q. Hence, by [18, Theorem 7.1.9(b)] there exists a positive constant C (possibly depending on p, n) such that for all f ∈ C ∞ 0,0 (Q). Obviously, we also have Therefore, by combining (4.6), (4.7) and (4.8), we have shown that there exists a positive nondecreasing function C p such that (4.2) holds for all f ∈ C ∞ 0,0 (Q). Recall however that so far we have assumed Q = − 1 2 , 1 2 n . The result for a general cube will be obtained by rescaling at the end of the proof of part (iii). We continue with part (iii). We are going to prove it only for the difference quotient d + h,k . The proof for d − h,k is fully analogous. We are still assuming that Q = − 1 2 , 1 2 n and is as above. Moreover, suppose that f ∈ C ∞ 0,0 (Q), x ∈ Q and h > 0. We get the following representation: x − y |x − y| n ∞ |x−y| y + he k + ξ x − y |x − y| ξ n−1 dξ dy kernel with respect to Q with a constant independent of h. Let x, y, z ∈ Q be such that x = y and |x − z| ≤ 1 2 |x − y|. It is easy to see that and (4.13) Moreover, we have