Besov regularity for solutions of p-harmonic equations

We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., divA(x, Du) = div F, whenA is a p-harmonic type operator, and under the assumption that x 󳨃→ A(x, ξ ) belongs to the critical Besov–Lipschitz space Bn/α,q. We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When div F = 0, we show that an analogous extra differentiability property for Du holds true under a Triebel– Lizorkin assumption on the partial map x 󳨃→ A(x, ξ ).

The aim of this paper is to extend the results of [1] to the case of p-harmonic type operators with p ≥ 2. More precisely, we will show that a fractional differentiability assumption for the operator A with respect to the x-variable yields a fractional differentiability for the solutions. In this case, the fractional differentiability of A(x, ⋅ ) transfers to V p (Du).
Our first result concerns the case of Triebel-Lizorkin coefficients, i.e., we assume that there exists a function g ∈ L n α loc (Ω) such that |A(x, ξ ) − A(y, ξ )| ≤ |x − y| α (g(x) + g(y))(μ 2 + |ξ | 2 ) See Section 2 for the definition of B α p,q and the meaning of locally. It is worth mentioning that there is a jump between (1.2) and (1.3) when stated in terms of the Triebel-Lizorkin scale F α p,q . The jump appears in the q index, when the order of differentiation becomes integer. Indeed, condition (1.2) fully describes equations with coefficients in the Sobolev space W 1,n , that is, the Triebel-Lizorkin space F 1 n,2 . In contrast, condition (1.3), for 0 < α < 1, says that A(x, ⋅ ) belongs to F α n/α,∞ . More explanations about this can be found in [15,Remark 3.3] and the references therein.
The proof of Theorem 1.1 does not seem to work when the coefficients are assumed to belong to F α n/α,q for finite values of q. As in the linear case (see [1]), the Besov setting fits better in this context. To be precise, given 0 < α < 1 and 1 ≤ q ≤ ∞, we assume that there exists a sequence of measurable non-negative functions and at the same time the following holds: for each ξ ∈ ℝ n , and almost every x, y ∈ Ω such that 2 −k ≤ |x − y| < 2 −k+1 . We will shortly write then that (g k ) k ∈ ℓ q (L n α ). If A(x, ξ ) = a(x)|ξ | p−2 ξ and Ω = ℝ n , then (A4) says that a belongs to B α n/α,q , see [15,Theorem 1.2].
Under (A4), we are able to deal with non-homogeneous equations and we prove that the extra differentiability of the solutions is related to the regularity of the datum and of the coefficients, both measured in the Besov scale. More precisely, we have the following result.
holds locally, provided that 0 < β < 1 and 1 ≤ q ≤ 2n n−2β . The parameter μ in assumption (A1) plays a very important role. When μ > 0, the equation is non-degenerate elliptic while the case μ = 0 corresponds to degenerate cases. For instance, a model case for μ > 0 is given by while a typical degenerate problem is the weighted p-Laplace equation In the degenerate case, the ellipticity assumption (A1) is lost when |ξ | approaches zero, and the estimates worsen even in the classical theory (see [22]). Actually, in this case we are not able to prove an extra fractional differentiability of the function V p (Du) completely analogous to our previous theorem. Instead, due to the degeneracy μ = 0, we have the following weaker result in the sense that the differentiability of the datum F still transfers to the function V p (Du), but with a loss in the order of differentiation, even assuming the datum in a Besov space slightly smaller than B β 2,q . More precisely we have the following theorem.
holds locally, provided that 1 ≤ qp 2 ≤ 2n n−2β . Note that, for p = 2, Theorems 1.2 and 1.3 both recover [1, Theorem 3] at the energy space and in the case α = β. Actually, when dealing with equations with linear growth, the natural degree of integrability of the gradient of the solutions as well as of their extra α fractional Hajlasz gradients is 2. Therefore, the higher fractional differentiability results at the energy space are those proving that Du belongs to B α 2,q or F α 2,q . In [1], extra fractional differentiability results for equations with linear growth have been established also in spaces different from the natural ones, i.e., it has been proven that Du belongs to B α s,q and F α s,q for some s ̸ = 2 sufficiently close to 2.
All our theorems rely on the basic fact that the Besov spaces B α n/α,q and the Triebel-Lizorkin space F α n/α,∞ continuously embed into the VMO space of Sarason (e.g., [11,Proposition 7.12]). Linear equations with VMO coefficients are known to have a nice L p theory (see [12] for n = 2 or [13] for n ≥ 2). A first nonlinear growth counterpart was found in [14] for A(x, ξ ) = ⟨A(x)ξ, ξ⟩ p−2 it with the Sobolev type embedding for Besov Lipschitz spaces to obtain the higher integrability of the gradient of the solutions of equation (1.1). Such higher integrability allows us to estimate the difference quotient of order α of the gradient of the solutions that yields their Besov type regularity. The paper is structured as follows. In Section 2 we give some preliminaries on Harmonic Analysis. In Section 3 we prove Theorem 1.1, and in Section 4 we prove Theorems 1.2 and 1.3.

Notations and preliminary results
In this paper we follow the usual convention and denote by c a general positive constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts. The norm we use on ℝ n will be the standard euclidean one and it will be denoted by | ⋅ |. In particular, for the vectors ξ , η ∈ ℝ n , we write ⟨ξ, η⟩ for the usual inner product and |ξ | := ⟨ξ, ξ ⟩ In what follows, B(x, r) = B r (x) = {y ∈ ℝ n : |y − x| < r} will denote the ball centered at x of radius r. We shall omit the dependence on the center and on the radius when no confusion arises.
For the auxiliary function V p , defined for all ξ ∈ ℝ n as where μ ≥ 0 and p ≥ 1 are parameters, we record the following estimate (see the proof of [10, Lemma 8.3]).
Noticing now that for p ≥ 2, one has and combining this with Lemma 2.1, we find that there exists a constant c > 0 such that for every ξ, η ∈ ℝ n .

Besov-Lipschitz spaces
Given h ∈ ℝ n and v : . As in [21, Section 2.5.12], given 0 < α < 1 and 1 ≤ p, q < ∞, we say that v belongs to the Besov space Equivalently, we could simply say that v ∈ L p (ℝ n ) and ∆ h v |h| α ∈ L q ( dh |h| n ; L p (ℝ n )). As usually, if one simply integrates for h ∈ B(0, δ) for a fixed δ > 0, then an equivalent norm is obtained because Again, one can simply take the supremum over |h| ≤ δ and obtain an equivalent norm. By construction, One also has the following version of the Sobolev embeddings (a proof can be found in [11,Proposition 7.12], taking into account that L r = F 0 r,2 , with 1 < r < +∞).
Given a domain Ω ⊂ ℝ n , we say that v belongs to the local Besov space B α p,q,loc if φv belongs to the global Besov space B α p,q (ℝ n ) whenever φ belongs to the class C ∞ c (Ω) of smooth functions with compact support contained in Ω. The following lemma is an easy exercise.
Here the measure dh |h| n is restricted to the ball B(0, r B ) on the h-space. Proof. Let us fix a smooth and compactly supported test function φ. We have the pointwise identity

It is clear that
and therefore one always has ∆ h φ |h| α ∈ L q ( dh |h| n ; L p (ℝ n )). As a consequence, we have the equivalence

However, it is clear that
if and only if the same happens for every φ = χ B and every ball B ⊂ 2B ⊂ Ω. The claim follows.
As in [21, Section 2.5.10], we say that a function v : ℝ n → ℝ belongs to the Triebel- Equivalently, we could simply say that v ∈ L p (ℝ n ) and ∆ h v |h| α ∈ L p (dx; L q ( dh |h| n )). It turns out that Besov-Lipschitz and Triebel-Lizorkin spaces of fractional order α ∈ (0, 1) can be characterized in pointwise terms. Given a measurable function v : ℝ n → ℝ, a fractional α-Hajlasz gradient for v is a sequence (g k ) k of measurable non-negative functions g k : ℝ n → ℝ, together with a null set N ⊂ ℝ n , such that the inequality holds whenever k ∈ ℤ, and x, y ∈ ℝ n \ N are such that 2 −k ≤ |x − y| < 2 −k+1 . We say that (g k ) ∈ ℓ q (ℤ; L p (ℝ n )) if The following result was proven in [15].
, where the infimum runs over all possible fractional α-Hajlasz gradients for v.
, where the infimum runs over all possible fractional α-Hajlasz gradients for v.

VMO coefficients in ℝ n
In this section, we recall a regularity result, proven in [1], that will be crucial in our proofs. Let n ≥ 2 and let A : Ω × ℝ n → ℝ n be a Carathéodory function such that assumptions (A1)-(A3) hold. We also require a control on the oscillations, which is described as follows. Given a ball B ⊂ Ω, let us denote One can easily check that the operator A B (ξ ) also satisfies assumptions (A1)-(A3). Now set and so any reasonable VMO condition on a(x) requires that the mean value of V(x, B) on B goes to 0 as |B| → 0. Our VMO assumption on general Carathéodory functions A consists of a uniform version of this fact. Namely, we will say that x → A(x, ξ ) is locally uniformly in VMO if for each compact set K ⊂ Ω, we have that lim Here c(B) denotes the center of the ball B and r(B) its radius.
The following theorem, proved in [1], is a regularity result for weak solutions of p-harmonic equations with VMO coefficients.
holds for any ball B such that λB ⊂ Ω.

Difference quotient
We recall some properties of the finite difference operator that will be needed in the sequel. We start with the description of some elementary properties that can be found, for example, in [10].
Proposition 2.6. Let F and G be two functions such that F, G ∈ W 1,p (Ω), with p ≥ 1, and let us consider the set Then the following hold: (2) If at least one of the functions F and G has support contained in Ω |h| , then The next result about the finite difference operator is a kind of an integral version of the Lagrange theorem.
which is equivalent, by Proposition 2.6 (2), to the following equality: We can write (3.1) as follows: and therefore Using the ellipticity assumption (A1) in the left-hand side and Proposition 2.6 (1), the previous equality yields By virtue of assumption (A2) and Young's inequality, we have To estimate the integrals I 2 and I 3 , we write p−1 2 = p 4 + p−2 4 , and then we use assumption (1.3) and Young's inequality as follows: Collecting the estimates of I 1 , I 2 and I 3 , we obtain Choosing ε = ν 8 and reabsorbing the first integral in the right-hand side by the left-hand side, we obtain Using Hölder's inequality and the first estimate of Lemma 2.7 in the first integral on the right-hand side, and the fact that supp η ⊂ B R , we get Inserting the previous estimate in (3.2), we obtain Using Lemma 2.1 in the left-hand side of the previous estimate yields We now divide both sides of inequality (3.3) by |h| 2α , and use the fact that η ≥ χ B R/2 to obtain where c = c(ν, L, p, n). The homogeneity of the equation together with Theorem 2.5 yields that Du ∈ L s loc (Ω) for every finite s > 1, and so, in particular, Du ∈ L np n−2α (B R ). Therefore, by Hölder's inequality, Since the above inequality holds for every h, we can take suprema over h ∈ B(0, δ) for some δ < R and obtain In particular, this tells us that V p (Du) ∈ B α 2,∞ , locally.

Proof of Theorems 1.2 and 1.3
We first prove that if A satisfies (A1)-(A4), then it is locally uniformly in VMO. This result is a straightforward extension to the case of operators A with (p − 1)-growth of [1, Lemma 18], which refers to operator with linear growth. We report it here for the sake of completeness.
The last term above is bounded by The first sum is very easy to handle, since Concerning the second, we see that In order to get the VMO condition, it just remains to prove that on every compact set K ⊂ Ω. To this end, we can fix r > 0 small enough and observe that the function x → ‖g k ‖ ℓ q (L n α (B(x,r))) is continuous on the set {x ∈ Ω : d(x, ∂Ω) > r}, as a uniformly converging series of continuous functions. As a consequence, there exists a point x r ∈ K (at least for small enough r > 0) such that sup x∈K ‖g k ‖ ℓ q (L n α (B(x,r))) = ‖g k ‖ ℓ q (L n α (B(x r ,r))) . Now, from ‖g k ‖ L n α (B(x,r)) ≤ ‖g k ‖ L n α (B(x r ,r)) and since this belongs to ℓ q , we can use dominated convergence to say that Each of the limits on the term on the right-hand side are equal to 0, since the points x r cannot escape from the compact set K as r → 0. This completes the proof.
Proof of Theorem 1.2. We first assume that α ≤ β. Let us fix a ball B R such that B 2R ⋐ Ω and a cut off function as a test function in equation (1.5). Using Proposition 2.6 (1), we obtain which, by Proposition 2.6 (2), is equivalent to (4.1) We can write (4.1) as follows: and therefore Now, using assumption (A1) in the left-hand side and Proposition 2.6 (1), the previous equality yields The integrals I 1 , I 2 and I 3 can be estimated exactly as we did in the proof of Theorem 1.1. After doing this for 2 −k ≤ |h| < 2 −k+1 , the above inequality reads as Now we estimate I 4 and I 5 . By using Young's inequality, we get since μ > 0, where ε > 0 will be chosen later. Similarly, where we used the first estimate in Lemma 2.7. Inserting the estimates of I 4 and I 5 in (4.2), we have Choosing ε = 3μ p−2 ν 8 , reabsorbing the last integral in the right-hand side of the previous estimate by the lefthand side, using Lemma 2.1, the fact that η ≡ 1 on B R/2 , and dividing both side by |h| 2γ , γ = min{α, β} = α, we conclude that and so ( ∫ where c = c(ν, L, p, n). Taking the L q norm with the measure dh |h| n restricted to the ball B(0, δ) on the h-space, we obtain that where c = c(ν, L, R, p, n, δ). For the estimate of J 1 , one can easily check that since γ < 1. The term J 2 can be controlled by the B β 2,q -seminorm of F, which is finite thanks to our assumption. Before estimating J 3 , recall that, by virtue of assumption (A4), for every k, one has g 2 k ∈ L n 2α . Also, since q ≤ 2 * β , by Lemma 2.2, we have that F ∈ L 2n n−2β loc , and so, by Theorem 2.5, we have that Du ∈ L 2n(p−1) n−2β loc . Now, from p ≥ 2 and α ≤ β, we easily see that and so we can proceed as follows. We write the L q norm in the integral J 3 in polar coordinates, assuming without loss of generality that δ = 1, so h ∈ B(0, 1) if and only if h = rξ for some 0 ≤ r < 1 and some ξ in the unit sphere S n−1 on ℝ n . We denote by dσ(ξ ) the surface measure on S n−1 . We bound the term J 3 by where we set r k = 1 2 k . Now, from (4.3), we see that Du ∈ L np n−2α loc . This, together with the assumption g k ∈ L n α , gives us that On the other hand, we note that for each ξ ∈ S n−1 and r k+1 ≤ r ≤ r k , Summarizing, we have with C = C(μ, α, β, p, q, n, ν, L). Lemma 2.3 now guarantees that V p,μ (Du) ∈ B γ 2,q , locally, and this concludes the proof.
When α > β, we have the embedding B α n/α,q ⊂ B β n/β,q ⊂ VMO. Thus, we can assume that (A4) holds with α replaced by β, and then repeat the previous proof. The claim follows.
Proof of Theorem 1.3. The proof goes exactly as that of Theorem 1.2 until estimate (4.2). We proceed now with the estimates of the integrals I 4 and I 5 . Assume first that α ≤ βp 2 . By using Hölder's and Young's inequalities, we get where ε > 0 will be chosen later. Similarly, using also Lemma 2.7, Inserting the estimates of I 4 and I 5 in (4.2) and recalling that μ = 0, we have and, by the elementary inequality (2.1), we get We now choose ε = 1 2c(p,ν) , use Lemma 2.1, reabsorb the last integral in the right-hand side of the previous estimate by the left-hand side, and divide both side by |h| 2α . We conclude that and so for a constant c that depends also on R and δ, and where we used that p ≥ 2. Taking the L q norm with the measure dh |h| n restricted to the ball B(0, δ) on the h-space, we obtain that where c = c(ν, L, R, p). For the estimate of J 1 , one can easily check that We now write the L q norm in the integral J 3 in polar coordinates, assuming without loss of generality that δ = 1, so h ∈ B(0, 1) if and only if h = rξ for some 0 ≤ r < 1 and some ξ in the unit sphere S n−1 on ℝ n . We denote by dσ(ξ ) the surface measure on S n−1 . We bound the term J 3 by where we set r k = 1 2 k . Now, since Du ∈ L np n−2α loc and g k ∈ L n α , Hölder's inequality implies ‖(τ rξ g k + g k ) .