A fractional Korn-type inequality

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of"directional"difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.


Introduction and statement of main results
The main focus of this paper is to study the function space of vector fields given by where Ω ⊂ R d is an open subset and the kernel K(z) is a nonnegative function with appropriate integrability. For a particular class of kernels, our main result states that X K,p (Ω) is equivalent to a Sobolev space. We use this identification and classical embedding estimates to obtain Sobolev regularity for solutions to a strongly coupled system of nonlocal equations with elliptic measurable coefficients. For p = 2, the space X K,2 (Ω) has been used in nonlocal continuum mechanics [21][22][23] where it appears as the energy space corresponding to the peridynamic strain energy in small strain linear models. Some basic structural properties of X K,p (Ω) have already been investigated in [5,13,14]. There it is shown that for any 1 ≤ p < ∞, the space X K,p (Ω) is a separable Banach space with , reflexive if 1 < p < ∞, and is a Hilbert space when p = 2. Conditions on the kernel K can be imposed so that a Poincaré-Korn type inequality holds over subsets that contain no nontrivial zeros of the semi-norm [·] Xρ,p . It is not difficult to see that [v] XK,p = 0 if and only if v is an affine map with skew-symmetric gradient. These functional analytic properties of the space were used to demonstrate well posednesss of some nonlocal variational problems using the direct method of the calculus of variations, see [14] for more.
As a difference-based function space, it may seem that X K,p (Ω) contains functions with some "differentiability." This is not in general true, however. Taking a radial K that is compactly supported and with the property that K(x) |x| p is integrable, it is shown in [14] that X K,p (Ω) = L p (Ω; R d ). In the event that the space X K,p (Ω) is a proper subset of L p (Ω; R d ), the fact that the semi-norm utilizes the "directional" or "projected" difference quotient v(y) − v(x) |y − x| · (y − x) |y − x| appears to make the space relatively big compared to those that use the full difference quotient. Nevertheless, by averaging the projected difference quotient over enough directions, it is reasonable to think that the semi-norm generated will be comparable with the one that is associated with the full difference quotient. However, this remains unclear in general. Finding general conditions on K and Ω so that equivalence holds is an open problem, and here we restrict our discussion on the special class of kernels K(|ξ|) = 1 |ξ| d+ps−p , 0 < s < 1 , 1 < p < ∞ .
We denote the corresponding space by X s p (Ω). These kernels are associated with the fractional Sobolev spaces W s,p (Ω; R d ) via the Gagliardo semi-norm, where W s,p (Ω; R d ) is given by The question is now if X s p (Ω) is the same as W s,p (Ω; R d ) for these fractional kernels. In a recent work [12], the second author answers the above question in the affirmative for the special case p = 2, and Ω = R d or R d + . When p = 2, and Ω = R d , the question is tractable because both spaces X s 2 (R d ), and W s,2 (R d ; R d ) can be characterized by Fourier symbols which made the camparison of norms more straightforward; see [5]. For functions defined over the half-space R d + and vanishing near the hyperplane x d = 0, one can use an appropriate extension operator to control the semi-norm [·] W s,2 (R d + ) by the semi-norm [·] X s 2 (R d + ) of vector fields in the dense class C 1 c (R d + ; R d ). In this paper we extend these results to any p ∈ (1, ∞) again providing an answer to the question of equivalence of spaces in the affirmative. Let us introduce the function space (1)X s p (Ω) = Closure of C ∞ c (Ω; R d ) in X s p (Ω). Theorem 1.1 (Fractional Korn's inequality). For any s ∈ (0, 1) and 1 < p < ∞, While the first inequality in (2) is trivial, the second inequality is the interesting one, as it gives a control of the integral norm of a pointwise larger function by the integral norm of a pointwise smaller function. We call the second inequality a fractional Korn's inequality for the following reason. For a smooth vector field f , the semi-norm [f ] W s,p uses the full difference quotient which locally behaves as while the semi-norm [f ] X s p uses the projected difference quotient and locally behaves as where (∇f (x)) sym = 1 2 (∇f (x) T + ∇f (x)) is the symmetric part of the gradient matrix. The connection between the projected difference quotient and ∇ sym runs deeper; multiplying the seminorm by the proper correcting constant (1 − s) it has been shown in [11], following the argument in [3], that the space X s p (Ω) "converges" to This association suggests that X s p (Ω) is the fractional analogue of W 1,p Sym (Ω; R d ). In turn, W 1,p sym (Ω; R d ) is known to coincide with W 1,p (Ω; R d ) via the classical Korn's inequality, a fundamental tool in the theory of linearized elasticity; see [4] for a complete proof. As such, establishing X s p (Ω) = W s,p (Ω; R d ) in the affirmative amounts to proving a version of Korn's inequality for fractional Sobolev spaces.
Our proof of Theorem 1.1 makes use of the classical characterization of functions in the fractional spaces in terms of their Poisson integrals. Given a vector function f , its Poisson integral is defined as u(x, t) = p t * f (x), where for each t > 0, the function p t (x) is the standard Poisson kernel. For s ∈ (0, 1), 1 < p < ∞ it is well-known [24,Proposition 7 Moreover, the semi-norm [f ] W s,p is equivalent with The key idea is the introduction of a "Poisson-type" integral U(x, t) of a vector field f . We construct U using a convolution with a modified "Poisson-type" matrix kernel whose components are some linear combination of convolutions of components of the vector field f . The structure of the Poisson-type kernel reveals that each component of U is related with components of the Poisson integral u via Riesz transforms leading to the norm relation Combining these inequalities with the characterization of W s,p (R d ; R d ) in terms of Poisson integrals we obtain the equivalence of spaces. Interestingly, this approach also leads to a characterization of the whole Besov scale Λ s p,q in terms of the newly defined Poisson-type integrals. These and other related results will be reported elsewhere.
As an application of Theorem 1.1 we show improved Sobolev regularity of weak solutions to the coupled system of nonlocal equations formally given as In the above Ω is a bounded subset of R d for d ≥ 2, the functions F, u : R d → R d , and the quantity D(u) is given by We also assume that s ∈ (0, 1), p ≥ 2, and that A(x, y) is a measurable function such that α 1 ≤ A(x, y) ≤ α 2 and symmetric in the sense that A(x, y) = A(y, x) for any x, y ∈ R d . Properly speaking, for a given vector field u ∈ X s p (Ω), the operator L s p,Ω (u) is a vector of distributions acting on test functions ϕ ∈ C ∞ 0 (R d ; R d ) via Let F ∈ [X s p (Ω)] * , the dual space of X s p (Ω), be given. We say u ∈ X s p (Ω) is a weak solution to the nonlocal system (3) if for all ϕ ∈ C ∞ 0 (R d ; R d ), L s p,Ω (u), ϕ = F, ϕ where ·, · is the duality pairing between [X s p (Ω)] * and X s p (Ω). For p = 2, the system of equations given in (3) is closely related to a nonlocal linearized continuum materials model via peridynamics [21][22][23]. In this case, the leading operator in the system is made up of weighted averages of some linear combinations of vectors of difference quotients.
See [5,13,14] for proper mathematical analysis for the linear case. The quantity what is known as the "linearized nonlocal strain" and has been used in nonlinear models of damage and fracture [7,8] as well. For any 1 < p < ∞ and A(y, x) = a(|y − x|), by using variational methods well posedness of the coupled system (3) has been established in [14] with appropriate volumetric conditions. Moreover, by exploiting the connection between the spaces X s p (Ω) and W 1,p Sym (Ω; R d ) it has been shown that (3) is a fractional analogue of a strongly coupled nonlinear system of partial differential equations of the type In fact, for specific variational problems, this relationship has been established via Γ-convergence in [14] in the event of vanishing nonlocality (that is, s → 1 − ). Regularity of solutions of the nonlinear system has been the subject of recent works, see [26]. The second main result of the paper is on the self-improving properties of solutions to the nonlocal nonlinear coupled system (3). The following is the precise statement we will prove. Theorem 1.2. Let s ∈ (0, 1), p ≥ 2, and Ω ⊂ R d be a bounded domain. Let F ∈ [X s−ε(p−1) p (Ω)] * , and let u be a weak solution to the coupled system of nonlocal equations (3) corresponding to F. Then there exists a positive constant ε 0 such that for all ε ∈ (0, ε 0 ) the weak solution u belongs to W s+ε,p loc (Ω). Moreover, for any η ∈ C ∞ c (Ω), there exists a positive constant C such that The implication of the regularity result in the theorem is that, with no additional smoothness conditions on the coefficient A(x, y), a weak solution to the coupled system (3) has improved fractional differentiability in response to improved regularity in the data. For scalar equations, this type of self-improving property of solutions is obtained in [6] using reverse Hölder inequalities and nonlocal Gehring-type lemmas, obtained in [19] via a commutator estimate and later obtained in [1] via a functional analytic approach. The main contribution of this paper is the extension of the self-improving properties of solutions obtained in the above cited works to the nonlocal nonlinear system (3). We should note that the application of appropriate embedding estimates imply both improved differentiability and higher integrability. For scalar nonlocal equations higher integrability (without improved differentiability) of weak solutions was established in [2] following classical techniques. The result in [2] is extended to hold for solutions to the nonlocal system (3) in the recent work [20].
To prove Theorem 1.2, we follow the approach in [19] and is close in spirit with the technique of "differentiating the equation," and finding relations between higher derivatives of solutions and test functions in order to estimate derivatives of the solution. This is possible for classical linear equations via integration by parts and transferring derivatives to test functions. For a special case of the nonlocal system at hand, for p = 2, K = 1, and Ω = R d we can demonstrate this easily. First notice that we can write the operator L s 2,R d in Fourier symbols as where l 1 and l 2 are positive constants, see [5,12]. Then for ε > 0 small, via Plancherel's theorem L s+ε 2,R d u, ϕ = L s 2,R d u, (−∆) ε ϕ , where for any α the operator (−∆) α ϕ is the α-fractional Laplacian. When working with the nonlinear "regional" operator L s p,Ω (u), such a clean transfer of derivatives to the test function is not possible. However, as has been done in [19] one can measure the price of transferring the derivatives by estimating the resulting commutator. Unlike [19], the estimates we establish are based on the smaller [·] X s p norm, leading us to write some arguments closely resembling those in [19]. Afterward, we use our first result (Theorem 1.1) to conclude that the estimates are also valid using the larger semi-norm [·] W s,p .
The rest of the paper is organized as follows. In the first part, we focus on proving Theorem 1.1. To that end, in the next section we recall the classical Poisson kernel and will present some preliminaries. We will also review how it is used in the characterization of functions belonging to the fractional Sobolev spaces. In Section 3 we introduce a Poisson-type kernel that is central to our result. Its properties as well the relationship between associated Poisson-type integrals and classical Poisson integral will be established. This relationship will be used to in Section 4 to prove the main result of the paper. In the second part of the paper we will prove Theorem 1.2.

Preliminaries: Poisson integrals and The Riesz transforms
We recall the classical Poisson kernel and some of its properties that we will use in this paper. We begin with the formula where ω d is the volume of the unit sphere in R d+1 . It is easy to check that p t is an approximation to the identity. Its Fourier transform is given by F (p t )(ξ) = e −2π|ξ|t for every t > 0, where the Fourier transform operator F is given by the formula It then follows from the Fourier transform expression that the Poisson kernel has the semigroup property p t1 * p t2 = p t1+t2 for every t 1 , t 2 > 0. Using the notation ∇ for the vector of partial for some constant c > 0. For any f ∈ L p , 1 ≤ p ≤ ∞, its Poisson integral is given by The Poisson integral is a C ∞ harmonic function in R d+1 For a vector field f its vector-valued Poisson integral u(x, t) = p t * f (x) will be defined where the convolution is taken component wise.
The Riesz transforms will be used frequently throughout this work. We recall that for 1 ≤ j ≤ d and f ∈ S(R d ) the class of Schwartz functions, the j th Riesz transform is an operator defined as . From this formula it is immediately clear that the Riesz transforms commute with partial differential operators ∂ xj . We recall also the celebrated result of L p boundedness (c.f. [24, Chapter III]), namely The Riesz transforms can be used to establish relations between the partial derivatives of functions.
Let us display such relations for the Poisson integral of a function now. First note that for Further, for any x ∈ R d and t > 0 we have We can verify the above identities by taking the Fourier transform in the x variable as follows.
demonstrating the first relation in (7). Conversely, establishing the second identity in (7). The pointwise relation in (7) and the L p boundedness of the Riesz transforms implies that for every Schwartz vector field f ∈ S(R d ; R d ) and 1 < p < ∞, where ≈ represents equivalence of norms up to a constant independent of f . Using density of Poisson integrals can be used to give a characterization of the L p norm of a function; see [24, Chapter IV] for details. Given a function f we introduce the Littlewood-Paley g-function of f in terms of its Poisson integral u as .
, and its L p norm is comparable with that of f . Most important to our work is the usefulness of Poisson integrals in characterizing fractional Sobolev spaces W s,p (R d ; R d ).
Moreover, there exists constants C 1 and C 2 depending only on s, p, and d such that Proof. The inequality on the left-hand side in (10) is proved in [24,25], and the right-hand proved in [25]. However the inequality on the right-hand side is the one that we need later and so for completeness we present its proof here. We will prove it for scalar functions, and for the vector case it follows easily by making the comparison component wise. We let u(x, t) = p t * f (x). Let x, y be such that x and x + y are Lebesgue points of f . We choose t = |y| and write We estimate each of the integrals associated with the three differences separately. We denote these integrals by I, II and III. Using the mean value theorem, It then follows from Minkowski's inequality that Then using polar coordinates (t = |y ′ |) we get that Now we repeat the same argument for the second difference; using (12), Then Minkowski's inequality gives us Calculations similar to the one above along with a second application of Minkowski's inequality show that Changing variables t = τ r in the inner integral we obtain that The quantity f j (x) − u(x, |y|) can be estimated exactly the same way, and so we obtain We now invoke the comparison estimate (8) to conclude the proof.

Poisson-type integrals
In this section we introduce a Poisson-type matrix kernel P t (x) that we convolve with vector fields so that the resulting Poisson-type integral can be used to characterize X s p (R d ) in the same spirit as Proposition 2.1..

Definition of Poisson-type kernel and integral.
Poisson-type kernel. Denoting M J (R) to be the space of J × J matrices with real entries, we We notice that the (d + 1) × (d + 1) matrix P t (x) has the form where x is considered both a row and column vector in R d .
The convolution in the above equation is taken in the sense of matrix multiplication. That is the

3.2.
Properties of the Poisson-type Kernel. We next establish basic but fundamental properties of P t which are analogues of the properties of the classical Poisson kernel. We begin by Moreover, it is immediate from the definition to see that for every 1 ≤ p ≤ ∞, P ∈ L p (R d ; M d+1 (R)) with the pointwise estimate By the norm of |A| for a matrix A = (a ij ) we mean the Frobenius norm |A| = i,j |a ij | 2 . In the following lemma we prove that the matrix kernel P t is in fact an approximation to the identity. For The conclusion of the lemma can be deduced from [10,Lemma 3.3] where it is shown that similar properties are enjoyed by the Poisson kernel for the Lamé system. To be specific, given constants µ, λ satisfying 3µ + λ > 0, µ + λ > 0, the matrix kernel K : is shown to be the Poisson kernel for the Lamé system in the upper half space, see [9,Lemma 5.1].
In addition the scaled kernel K t (x) := t −d K(x/t) is shown be an approximation to the identity. As a consequence, to prove the above lemma it suffices to note that for every x ∈ R d , t > 0, where we recall that p t (x) the Poisson kernel of the Laplacian in the upper half space. For each t > 0, we can now integrate both sides of (24) to get The smoothness and the convergences of the matrix convolutions also follow from the same results for K and p t . It can also be easily verified using Minkowski's inequality and the Lebesgue dominated convergence theorem as follows: where τ y F = F(x−y). The integrand in the last term is bounded by the L 1 function C |P(y)| F p L p , and for each y the integrand converges to zero by continuity of translations in L p .
Remark 3.2. A connection between P t and the semi-norm | · | X s p is obtained through the following important relation that we use below. For any z, x ∈ R d , we have where the vector function P(x, t) is given by P(x, t) : d+3 2 x t . As a consequence of this and the approximation to the identity result, we see that if F = (f , 0), then The matrix Poisson kernel P t also satisfies a semigroup property as documented in the next lemma. The following arguments rely centrally on an explicit formula for the Fourier transform of P t . Lemma 3.3. For each t > 0, the Fourier transform of P t (x) is given by As a consequence P t satisfies the semigroup property: for every t 1 , t 2 > 0 where the convolution is understood in the sense of matrix multiplication.
Proof. To preserve the flow of the presentation in this section, the Fourier transform of P t is computed in the appendix. To prove the semigroup property of P t we use the property of convolution and the explicit Fourier transform formula given in (25). We carry out this calculation via matrix multiplication. For any positive t 1 , t 2 we have that where in the second equality we have multiplied the matrix of Fourier symbols and have also used the fact that = 0 , the zero matrix, which can be verified easily by computation.
The next lemma summarizes integrability properties of the first derivatives of P t that we will be using later. The proof is purely computational and can be done following similar calculations for the Poisson kernel. We omit it here. Lemma 3.4. For each j, k, and ℓ ∈ {1, . . . , d, d + 1} and for every t > 0 we have that ∂ t p jk t (x) ∈ L 1 (R d ) and ∂ x ℓ p jk t (x) ∈ L 1 (R d ). In addition we have the following pointwise estimates: There exists a constant c = c(d) > 0 such that for any j, k = 1, 2, . . . d + 1.

Norm equivalence of Poisson integrals.
We begin first by establishing relations between the Poisson integrals obtained from p t and P t .
. Then for every t > 0 both Poisson integrals u(x, t) = p t * F(x) and U(x, t) = P t * F(x) are in S(R d ; R d+1 ). Moreover we have the following relations between u and U.
• For any j = 1, . . . , d, we have • For any j = 1, . . . , d, we have where R j is the j th Riesz transform.
Proof. We prove first the identity (26). For a fixed t > 0, since all the functions involved are in S(R d ), it suffices to check that the Fourier transform of the right-hand side agrees with that of the left-hand side in (26). From the definition of U, we see that for any t > 0, F x (U(·, t))(ξ) = F (P t )(ξ)F (F)(ξ). Now from the particular form of F and using the explicit formula (25) for the Fourier transform of P t we see that for any j = 1, . . . , d we have and that after simplification F x (U d+1 )(ξ, t) = −ı2πt e −2π|ξ|t ξ · F (f )(ξ). To complete the proof of the identity (26) we notice that the first term in (29) is precisely F x (u j )(ξ, t), whereas the second term can be rewritten to obtain Let us proceed to show (27). Using a direct calculation and some rearrangement we get The identity follows easily once we realize that the first term of (30) can be rewritten as while the second term in (30) can also be rewritten as Next we prove the identity (28). Again, by a direct calculation We need to connect the last expression in (31) with ∂ t U d+1 . To do so, we observe from (30) that where the last term is a rewriting of the expression (−ı2π)e −2π|ξ|t ξ · F (f )(ξ) in (30). Substituting this into (31) we get the desired result.
. Then there exists a positive constant C = C(d, p) such that for any t > 0 we have and for each k = 1, . . . , d we have Proof. We prove both inequalities for f ∈ S(R d ; R d ) and then the general case follows by density. Both inequalities (33) and (34) follow from identity (26) in Proposition 3.5. Indeed, for j = 1, . . . , d, we can differentiate the equation (26) in t to obtain that for any x ∈ R d and t > 0 In a similar fashion, if we differentiate the equation (26) in x k we obtain that Both inequalities (33) and (34) now follow by taking the L p norm on both sides of the above two equations and summing over j = 1, . . . , d. Note that we have used both the fact that the Riesz transforms commute with differential operators and that the Riesz transforms are L p bounded. 0). Then apply Lemma 3.1, Lemma 3.4 and Young's inequality to see that ∂ t P t * F n converges to ∂ t P t * F in L p (R d ; R d ) and that ∇ x P t * F n converges to ∇ x P t * F in L p (R d ; R d ).

A characterization of fractional Sobolev spaces
4.1. Equivalence of spaces. In this subsection we prove one of the main results of this paper, which is the equivalence of the spaces X s p (R d ) and W s,p (R d ; R d ). We paraphrase it in the following theorem.
Proof. In the above (EQ 1 ) is in (10) proved in Proposition 2.1 and (EQ 2 ) follows from the pointwise-in-t estimate proved in Proposition 3.6. What remains is the proof of the inequality (EQ 3 ). We prove it as follows. Recalling thatˆR To reveal the connection with the integrand in the semi-norm [f ] X s p we compute the derivative ∂ t P t (y) in the above convolution directly. For j = 1, . . . , d the j th term is given by A similar computation also shows that Notice that the expressions inside the integrals in (36) and (37) are linear combinations of the ∂ t p jk t after factoring the unit vector y |y| . As a result, these expressions enjoy the same pointwise estimates as ∂ t p jk t stated in the Lemma 3.4. That is, the expressions are majorized by t −d−1 as well as by |y| −d−1 . We will make use of these pointwise estimates below.
By splitting the convolution integrals in (36) and (37) into an integral over B t (0) and ∁B t (0), the complement of B t (0), we obtain that for any t > 0 and x ∈ R d

Now, using Minkowski's integral inequality we obtain that
The remaining part of the argument that estimates the right-hand side of the above inequality by the semi-norm [f ] X s p follows that of [24,Page 152] where it was done for classical Besov spaces. We repeat it here for clarity. Changing to polar coordinates, write y = rw ∈ R d , with r = |y| and w ∈ S d−1 . Define Then the last inequality in (38) can be rewritten in terms of Ψ(r) to obtain Multiply both sides by t 1−s and estimate the norm in L p ((0, ∞); t −1 dt) on both sides to obtain, using Hardy's inequalities [24, Appendix A], that (f (· + rw) − f (·)) · w p L p (R d ) dσ(w) and so we have where the last C depends only only on s, p, and d. This completes the proof of the theorem.

4.2.
Applications. One may now use the equivalence of spaces we have established to obtain inequalities that are important in application. The simplest of all is the fractional Poincaré-Korn inequality which we will need in the next section.
Another corollary of Theorem 4.1 is a fractional Sobolev embedding [16] that uses the seminorm X s p (R d ). Corollary 4.1.2. Let s ∈ (0, 1) and p ∈ (1, ∞) such that sp < d. Then there exists a constant C = C(d, p, s) such that for any measurable and compactly supported vector field f : where p * s = dp We will also use Theorem 4.1 to prove fractional Korn-type inequalities for functions defined on the half space R d + . The argument to prove such results is standard. We first extend vector fields to be defined over R d such that the norm of the extended vector field is controlled by the original one. Such an extension theorem is recently proved in [12], which we state below.
. The theorem is proved in [12]. We emphasize that the proof of the above extension theorem is nontrivial as the commonly used reflection across the hyperplane x d = 0 would not preserve the semi-norm | · | X s p . Extending by zero is also not appropriate, since it is not clear how to control the norm of the extended function. Rather we use an extension operator that has been used by J. A. Nitsche in [17] in his simple proof of Korn's second inequality. In showing the boundedness of the extension operator with respect to the semi-norm | · | X s p we need to first establish the fractional Hardy-type inequality. See [12] for more details. With extension at hand the proof of the result below is standard.
In future work we hope to report on the natural next step of establishing the equivalence of the X s p (Ω) with W s,p (Ω; R d ) defined over domains with sufficiently regular boundary. 5. Self improving properties for a coupled system of nonlocal equations 5.1. Preliminaries. Given a ball B ⊂ R d with radius r, κB represents a ball with the same center but with radius κr. Note that for a given u ∈ X s p (2B) and η ∈ C ∞ c (2B) with η ≡ 1 in B, the function ηu ∈ X s p (R d ). Moreover, for a constant C independent of u.
We also recall that for u ∈ X s where the last inequality follows from a Poincaré-Korn type inequality, Corollary 4.1.1, where C depends on the support set 4B, p, and ε. Combining the above two inequalities and using the fractional Korn's inequality we proved we have that, To estimate the second term II we proceed as follows. .
We repeat the argument used to bound I and get that where again we have applied the Poincaré-Korn and fractional Korn inequalities. Thus (45) is proved.
We now use the following algebraic identity: for a, b, c, d real numbers x, y)) . Now, by adding and subtracting the appropriate quantities and splitting the integral accordingly, we have [u] p X s p (B) ≤ L s p,4B u, ϕ + I 1 + I 2 , where Note that since the vector field ϕ ∈ X s,p (R d ) and that supp ϕ ⋐ 2B, by the density Lemma 5.2 where the supremum is over all φ ∈ C ∞ c (2B; R d ) and [φ] X s p (R d ) ≤ 1. As for I 1 , using the estimate that ∇η L ∞ ≤ C 1 diam(B) , Let t 2 = 1 − s. Then d + sp − 1 = d + s(p − 2) + s − t 2 , and Hölder's inequality with q = p p − 2 , q ′ = p 2 implies that Applying Cauchy-Schwarz to the last integral, Using again (43), the final estimate of I 1 is The integral I 2 can be estimated the same way as I 1 . Therefore, where the supremum is over all φ ∈ C ∞ c (2B; R d ) and [φ] X s p (R d ) ≤ 1. From the last estimate we apply Young's inequality to obtain the result and conclude the proof.

5.2.
Higher differentiability of solutions. In this section we prove the second main result of the paper. Before presenting the proof, we state a commutator estimate that is an adaptation of the commutator estimate established in [19]. The proof of the theorem is essentially identical to the result given in [19], and we omit it here.
For a certain normalizing constant c depending on s, p, and ε denote the commutator Then there exists a constant C ε = C(s, p, ε, n, Λ) > 0 such that Moreover, C ε is monotone increasing in ε. That is C ε ≤ C ε0 for any 0 < ε < ε 0 .
Step 1. In this step we establish that there exists ε 0 ∈ (0, 1 − s) such that for any 0 < ε < ε 0 , we have We apply the technique and the argument in [19]. First notice that since the support ofũ is contained in Ω 1 , we have that ũ p−1 (Ω) . Next, find finitely many balls (B k ) N k=1 ⊂ We also may assume that This is because the second term on the second line and the second term on the third line have disjoint support in the integrals. When using the constant C(ε) we are emphasizing that the constant depends on ε, and it may also depend on other quantities.
Using Lemma 5.3 and the fact that the union of the finite number of balls B k cover no more than Ω, we get for any δ > 0 Choosing δ sufficiently small we can estimate Now, the disjoint support of (1 − ρ k ) and φ implies, via Lemma 5.1 item 1), that where we have used the Poincaré-Korn and fractional Korn inequalities. As a consequence of (45) and (46), an application of Hölder's inequality gives us With the above, adding and subtracting L s p,8B kũ , Φ and using the properties of Φ shown above, our estimate becomes Lastly, we need to transform the support of the operator L from 8B k to Ω 2 . Since supp ψ ⊂ 6B k , the disjoint support of the integrals gives (using Hölder's inequality and then the Poincaré and Korn inequalities on the ψ integral) . The proof is complete.
Appendix A. The Fourier Transform of the Poisson-Type Kernel P t Here we obtain the Fourier transform of the Poisson-type kernel P t that has been used to establish relations between various Poisson integrals. Recall that the (d + 1) .
The function P d+1 t : R d → R d is a vector valued function, which we consider both a row and column vector, given by Finally the (d + 1) × (d + 1) entry is given by the function p d+1,d+1 We compute the Fourier transform of each of these functions and put those transforms together to obtain the Fourier transform of P t . We begin by writing some useful Fourier transform formulas. Rather than write the calculations explicitly in-line each time during a proof, we instead reference the formulas in their full generality. For completeness they are listed here and their proofs can be found in many textbooks, for example [15].
Proof of Item 1). Let j, k ∈ {1, 2, . . . , d}. Since P t = (p jk t ) ∈ L p (R d ; M d (R)) for every 1 ≤ p ≤ ∞ we have that p jk t ∈ S ′ (R d ) for every t > 0. Thus, its Fourier transform is a well-defined object in S ′ (R d ) and agrees with its Fourier transform as a function in L 1 (R d ). The plan is to make use of partial Fourier transforms. Specifically, we will compute F x p jk t (ξ) in S ′ (R d ) by using the Fourier transform of p jk in S ′ (R d+1 ). To that end, we first compute the Fourier transform of in S ′ (R d+1 ). We use several properties of the Fourier transform.