Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces

We prove optimal integrability results for solutions of the $p(\cdot)$-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials maps $L^1$ to variable exponent weak Lebesgue spaces.

problem, whose solution is presented in Sect. 8. Consider appropriately defined weak solutions to the boundary value problem −div(|∇u| p(x)−2 ∇u) = f in Ω, u = 0 o n ∂Ω (1.1) when the data f is merely an L 1 function. We refer to [14] for an extensive survey of such equations with non-standard growth. Based on the constant exponent case and computations on explicit solutions, one expects in the L 1 -situation that u ∈ w-L (Ω), while Bögelein and Habermann [6] proved that it is in L n( p(·)−1) n−1 − loc (Ω), for any > 0. By elementary properties of weak spaces (Proposition 3.2), these two results are in fact equivalent. However, as (1.2) is the borderline case = 0, it has turned out to be hard to reach. As in the constant exponent case, when = 0 the inclusions into the (strong) Lebesgue space do not hold.
Our approach to this problem relies on the recent pointwise potential estimates for solutions and their gradients to problems with L 1 or measure data, see [11,12,25]. The case of equations similar to (1.1) is covered in [6]. The potential that appears in the nonlinear situation is the Wolff potential, given by At a given point x, a solution to (1.1) is controlled by W f 1, p(x) (x), and its gradient is controlled by W f 1/ p(x), p(x) (x). These estimates are the nonlinear counterparts of representation formulas, as properties of solutions may be deduced from the properties of the potentials. Our aim is to exploit this and establish a local version of (1.2) by proving that the Wolff potential W f α(x), p(x) (x) has the appropriate mapping properties. This answers the open problem posed by Sanchón and Urbano [28,Remark 3.3] and completes the generalization of the Wolff potential approach for (1.1) started by Bögelein and Habermann in [6].
The usual way to look at the mapping properties of the Wolff potential is to estimate it pointwise by the Havin-Maz'ya potential (see [23]), which is an iterated Riesz potential. Thus, we study the mapping properties of the Riesz potential as well. For (strong) Lebesgue spaces, these properties are well known, see [8,26,27] and [9, Section 6.1]. Here we deal with the novel case of weak Lebesgue spaces.
Our first result, Theorem 4.3, is the strong-to-weak estimate for the Riesz potential I α (·) . We show that for every log-Hölder continuous positive function q requires a separate proof. This proof is based on pointwise estimate between the Riesz potential and the Hardy-Littlewood maximal operator. Then, we study how the Riesz potential acts on weak Lebesgue spaces, as this situation will inevitably happen when dealing with the Wolff potential on L 1 . This turns out to be a difficult question because the weak Lebesgue spaces are not well behaved. We show that the weak Lebesgue space is an interpolation space (Theorem 5.1). This allows us to use real interpolation to get weak-to-weak boundedness of the maximal operator:

Theorem 1.1 Let p be a bounded measurable function with p
In particular, is bounded when p is log-Hölder continuous and p − > 1.
With a complicated application of Hedberg's trick, we then prove in Theorem 6.6 that We combine these results and obtain in Theorem 7.1 that A combination of (1.3) and the pointwise potential estimates now yields (1.2), provided that an appropriate notion of solutions to (1.1) is used. This requires some care, as L 1 (Ω) is not contained in the dual of the natural Sobolev space W 1, p(·) 0 (Ω). Here we use the notion of solutions obtained as limits of approximations, or SOLAs for short. The idea is to approximate f with more regular functions, prove uniform a priori estimates in a larger Sobolev space W 1,q(·) 0 (Ω), and then pass to the limit by compactness arguments. This way, one finds a function u ∈ W 1,q(·) 0 (Ω) such that (1.1) holds in the sense of distributions. See e.g. [4,5,17] for a few implementations of this basic idea, and [20,28] for equations similar to the p(·)-Laplacian. In fact, the same approximation approach is used in proving the potential estimates.
A representative special case of what comes out by combining nonlinear potential estimates and our results about the Wolff potential is the following theorem. Theorem 1.2 Let f ∈ L 1 (Ω), and let p be bounded and Hölder continuous with p − 2. Suppose that u is a SOLA to (1.1). Then In other words, (1.2) holds locally under suitable assumptions. Similar results also follow for the fundamental objects of nonlinear potential theory, the p(·)-superharmonic functions. Finally, by examining the counterpart of the fundamental solution (Example 2), we show that the exponents in Theorem 1.2 are sharp, as expected.

Notation
We write simply A B if there is a constant c such that A cB. We also use the notation A ≈ B when A B and A B. For compatible vector spaces, the space and abbreviate g + := g + U and g − := g − U . We say that g : U → R satisfies the local log-Hölder continuity condition if for all x, y ∈ U . We will often use the fact that g is locally log-Hölder continuous if and only if for some g ∞ 1, c > 0 and all x ∈ U , then we say g satisfies the log-Hölder decay condition (at infinity). If both conditions are satisfied, we simply speak of log-Hölder continuity. By the log-Hölder constant we mean max{c, c }.
By a variable exponent we mean a measurable function p : The set of variable exponents is denoted by P 0 (U ); P 1 (U ) is the subclass with 1 p − . By P log 0 (U ) and P log 1 (U ) we denote the respective subsets consisting of log-Hölder continuous exponents.
We define a modular on the set of measurable functions by setting The variable exponent Lebesgue space L p(·) (U ) consists of all the measurable functions f : U → R for which the modular L p(·) (U ) ( f ) is finite. The Luxemburg norm on this space is defined as Equipped with this norm, L p(·) (U ) is a Banach space. We use the abbreviation f p(·) to denote the norm in the whole space under consideration. The norm and the modular are related by the inequalities For open sets U , the variable exponent Sobolev space W 1, p(·) (U ) consists of functions u ∈ L p(·) (U ) whose distributional gradient ∇u belongs to L p(·) (U ). The norm More information and proofs for the above facts can be found for example from [9,Chapters 2,4,8,and 9].
By Ω we always denote an open bounded set in R n .
In auxiliary results, we use the convention that constants (implicit or explicit) depend on the assumptions stated in the result. For instance, in Proposition 3.2, the assumptions are that p, q ∈ P 0 (Ω) and ( p − q) − > 0, so in this case, the implicit constant (potentially) depends on p − , p + , q − , q + , ( p − q) − , and on the dimension n.

Basic properties of weak Lebesgue spaces
The inequalities (2.2) imply that the requirement in Definition 3.1 is equivalent with Another immediate consequence of (2.2) which we will use in the proofs below is that We immediately obtain the following two inclusions: -for bounded sets, the inclusion w-L p(·) (Ω) ⊂ w-L q(·) (Ω) holds when p q, since the inequality · p(·) · q(·) holds for the corresponding strong spaces.
The following result is from [28, Proposition 2.5]. We present a simpler proof here. .
Note that Proposition 3.2 works not only for bounded sets but also for every open set with a finite measure. It can be similarly proved that for all exponents p, q, r with ( p − q) − > 0 and r p.
However, the same is not true for the weak Lebesgue space. Indeed, in this case, the following property holds: Hence, by the definition of the weak space, we obtain that 1}. Now by a similar calculation as above, we The last claim, regarding the case of q constant, follows from a change of variables:

Strong-to-weak estimates for the Riesz potential
Let α : Ω → R be log-Hölder continuous with 0 < α − α + < n. We consider the Riesz potential in Ω, and write Because Ω is bounded and α is log-Hölder continuous we observe as in [15, p. 270] that I α(·) f (x) and are pointwise equivalent. Thus we obtain the following result from [9, Proposition 6.1.6].

Proposition 4.1 Let p ∈
Here M denotes the Hardy-Littlewood maximal function given by For a measurable function f and measurable set B, we use the notation f B for the mean integral of f over B.
We also need the following Jensen-type inequality. The lemma is a restatement of [9, Theorem 4.2.4] in our current notation, cf. also the proof of Lemma 4.3.6 in the same source.

Lemma 4.2 Let A ⊂ R n be measurable and p ∈
The next statement shows that the Riesz potentials behave as expected in the variable exponent weak space. We will use the exponent q to overcome the difficulty illustrated in Proposition 3.3.

Theorem 4.3 Suppose that p ∈
Proof By (3.1), it is enough to show that for every f ∈ L p(·) (Ω) with f p(·) 1 and every t > 0 we have By Proposition 4.1, for a suitable c > 0, By the definition of the maximal function, for every x ∈ E, we may choose Then Hence we have for every y ∈ B x that By the Besicovitch covering theorem, there is a countable covering subfamily (B i ) of {B x } with bounded overlap. Thus, we obtain by Lemma 4.2 that

Real interpolation and weak Lebesgue spaces
It is well known that real interpolation between the spaces L p and L ∞ gives a weak Lebesgue space in the limiting situation when the second interpolation parameter equals ∞. We shall prove that the same holds in the variable exponent setting. We recall that, for 0 < θ < 1 and 0 < q ∞, the interpolation space (A 0 , A 1 ) θ,q is formed from compatible quasi-normed spaces A 0 and A 1 by defining a norm as follows. For a ∈ A 0 + A 1 we set Here the Peetre K -functional is given by We saw in Proposition 3.3 that weak L p(·) -spaces are not very well behaved. Real interpolation in the variable exponent setting is even more challenging (cf. [3,16]). Fortunately, we can get quite far with the following special case, whose proof already is quite complicated.
Then by definition We assume without loss of generality that f, f 0 , f 1 0.
We start by proving that Then it remains to prove the second of the inequalities .
Hence, in the definition of f X , we may take the infimum over s > 0 and functions This completes the proof of the inequality f X f w-L p(·) . We show next that f X f w-L p(·) . By homogeneity, it suffices to consider the case where the right-hand side equals one. Thus, by (3.2), we can assume that for every λ > 0. Since We choose s := t θ −1 so that t −θ ts = 1. Thus, it suffices to show that for all t > 0. We next note that max{ for all z > 0. It is enough to show that the inequality holds for all z = 2 k 0 , k 0 ∈ Z. Define For z = 2 k , we observe that A k ⊂ { f > z 1−θ } and thus conclude from (5.1) that Substituting z = 2 k 0 in (5.2), we find that it is enough to prove that Hence it follows that which is the required upper bound.
The following feature is the main property of the real interpolation method [30, Proposition 2.4.1]: If T is a linear operator which is bounded from X 0 to Y 0 and from X 1 to Y 1 , then T is bounded from for θ ∈ (0, 1) and q ∈ (0, ∞]. If simple functions are dense in the spaces, then the claim holds also for sublinear operators (cf. [7, Theorem 1.5.11], or [10, Corollary A.5] for the variable exponent case; see also [2, Lemma 4.1] for a discussion in a general framework). This, together with Theorem 5.1 for X 0 = Y 0 = L p(·) (R n ) and X 1 = Y 1 = L ∞ (R n ) yields the following corollary.

Corollary 5.2 Assume that T is sublinear, T
L. Diening has shown that the boundedness of M : L p(·) (R n ) → L p(·) (R n ) implies the boundedness of M : L s p(·) (R n ) → L s p(·) (R n ) for some s < 1 [9, Theorem 5.7.2]. Furthermore, it is known that the maximal operator is bounded on L p(·) (R n ) when p ∈ P log 1 (R n ) and p − > 1 [9,Theorem 4.3.8]. In view of the previous result these facts immediately imply Theorem 1.1.

Weak-to-weak estimates for the Riesz potential
As usual, we denote by p the Hölder conjugate exponent of p, taken in a pointwise sense, 1/ p(x) + 1/ p (x) = 1. Following Diening (and [9]), for exponents we use the notation p B to denote the harmonic mean of p over the measurable set B, The following claim is proved as part of the proof of [9, Lemma 6.1.5].
where B is a ball centered at x ∈ R n .
We next generalize this claim to slightly more general norms, which will appear below when we estimate in the dual of a weak Lebesgue space. We need the following auxiliary result.
However, if R 0 < δ, then the constrained minimum occurs at δ, in which case f (δ) = α β δ −β ≈ δ −β . Hence the estimate of the minimum equals ∈ (0, n) and let θ > 0 be so small that the infimum of r := (1 − θ) p is greater than 1. Then, where B is a ball centered at x ∈ R n .
Proof Let B := B(x, δ) and denote f (y) := |x − y| α−n χ R n \B (y). By the definition of the interpolation norm, On the other hand, the opposite inequality holds with constant 1, since we may choose f 1 = f χ A and f 2 = f χ R n \A in the first infimum. So we conclude that Since r > 1, the infimum is not achieved when sup{| f |χ A } > inf{| f |χ R n \A } (since in this case we can shift mass to decrease the L r (·) -norm while conserving the L 1 -norm). Assuming that |{ f = c}| = 0 for all c ∈ R, it follows that A must be of the form {| f | < c} for some c 0. In our case, f is radially decreasing and so A = R n \ B(R), for some R ∈ [δ, ∞]. This corresponds to the functions f 1 = |x − ·| α−n χ R n \B(R) and f 2 = |x − ·| α−n χ B(R)\B .
For simplicity we denote s := r # α . A straight calculation gives f 2 1 ≈ R α − δ α . Then it follows from Lemma 6.1 that Recall that s ∞ is the limit value of s at infinity, from the definition of log-Hölder continuity.
We further observe that If t 0 > 1 (so that δ < 1), we find that So in this case, Since p is log-Hölder continuous and x ∈ B = B(x, δ), we have δ s B ≈ δ s(x) . Thus, For t 0 1, we similarly conclude that f (L r (·) ,L 1 ) θ,1 ≈ δ − n p # B , using that δ s B ≈ δ s ∞ which holds by the log-Hölder decay since δ 1.
According to [29,Theorem 1.11.2], the duality formula Hence, we obtain the Hölder inequality In the following result, we generalize [9, Lemma 6.1.5] where the same conclusion was reached under the stronger assumption that f L p(·) 1.
Then, we obtain the following analog of [9, Theorem 6.1.9] using the previous lemma and Theorem 1.1: Proof We write t := p/ p # α . By a scaling argument, we may assume that f w-L p (·) 1. By Lemma 6.5, there exists h ∈ w- and so we obtain By assumption, (t 0 p # α ) − > 1, and hence, it follows from Theorem 1.1 that As was noted before, Furthermore, a log-Hölder continuous exponent in a domain can be extended to a variable exponent in the whole space, with the same parameters [9, Proposition 4.1.7]. Thus, we obtain the following corollary.

Corollary 6.7 Let p ∈
Note that a direct use of Theorem 6.6 leads to the assumption α + p + < n in the corollary. However, (αp) + < n if and only if the domain can be split into a finite number of parts in each of which the inequality α + p + < n holds, so in fact these conditions are equivalent.

The Wolff potential
Let μ be a positive, locally finite Borel measure. The (truncated) Wolff potential is defined by with the full Wolff potential being W μ α, p (x) := W μ α, p (x, ∞). There are several ways in which this can be generalized to the variable exponent setting. The most straightforward way is to consider the pointwise potential x → W μ α(x), p(x) (x, R). In this case, we immediately obtain the following inequality from the constant exponent setting: This was observed in [6, Subsection 5.2]. As we have noted, in the bounded domain case the Riesz potentials I α(x) f (x) and I α(·) f (x) are comparable. Thus, we obtain that However, there is no immediate way to change the exponent 1 p(x)−1 . As far as we can see, the above inequality cannot be used to derive Theorem 8.3, and thus, the validity of the claims in this part of [6, Section 5.2] is in doubt. (Additionally, their claim that I α(·) : L p(·) (R n ) → L p # α (·) (R n ) is bounded is false, see [15,Example 4.1]; the claim only holds for bounded domains. Of course, the latter claim is what is actually needed.) The Wolff potential has also been studied by Maeda [24]. To state the result as clearly as possible, let us denote g(y) : , this implies the desired inequality, which can succinctly be stated as provided one keeps track of which dot is related to which operation. The right-hand side in this equation is called the Havin-Maz'ya potential which is denoted by V μ α(·), p(·) (x). The following result is now a consequence of Theorems 4.3 and 6.6, and (7.1). Theorem 7.1 Let α, r , and p be bounded and log-Hölder continuous, with p − > 1 and r − 1, 0 < α − α + < n, (αpr) + < n, and p(x) 1 + 1/r (x) − α(x)/n for every x ∈ Ω. If f ∈ L r (·) (Ω), then Proof By (7.1), it suffices to consider the Havin-Maz'ya potential V f α(·), p(·) instead of the Wolff potential. Denote s := nr( p−1) n−αr ; by assumption p 1 + 1/r − α/n so that s − 1. Choosing q := 1/( p − 1) in Theorem 4.3, we see that Since (αpr) + < n, we find that (αs) + < n and thus by choosing q := s/[( p − 1)s # α ] in Theorem 4.3 we obtain that Further, since (αpr) + < n, we can use Corollary 6.7 for the function (I α(·) f ) 1/( p(·)−1) to conclude that The claim follows from this since s # α = nr( p−1) n−αpr .

An application to partial differential equations
In this section, we discuss consequences of our results and pointwise potential estimates for solutions to the nonlinear elliptic equation where μ is a Borel measure with finite mass. The right quantity for estimating solutions to (8.1) and their gradients is the Wolff potential W μ α(x), p(x) (x). Recall that for right-hand side data, a Borel measure μ with finite mass or a function in L 1 , we use the notion of solutions obtained as limits of approximations, SOLAs for short. Gradient potential estimates for SOLAs follow by working with a priori more regular solutions, and then transferring the information obtained to the limit. In the case of general measures, the latter step requires some care, as the approximants converge only in the sense of weak convergence of measures. For this reason, the approximation argument is not done using the final potential estimates. Certain intermediate estimates, from which the actual potential estimates are then built, need to be used instead. See [6, Proof of Theorem 1.4, p. 668] for details.
An alternative point of view is to start with the fundamental objects of the nonlinear potential theory related to the p(·)-Laplacian, namely p(·)-superharmonic functions. See [22, Definition 2.1, p. 1068] for the exact definition of this class. For a p(·)-superharmonic function u, there exists a measure μ such that (8.1) holds. This is the Riesz measure of u. Important results in nonlinear potential theory are derived by employing measure data equations like (8.1). The leading example is the necessity of the celebrated Wiener criterion for boundary regularity, see [18].
The gradient potential estimates in [6] are local: one works in a fixed ball, compactly contained in Ω. Thus, the solution under consideration can be a local SOLA, i.e. it suffices to choose approximations in a fixed compact subset of Ω.
If μ is a signed measure, we use the notation where |μ| is the total variation of μ.
To extend the gradient potential estimate to p(·)-superharmonic functions, we need the fact that these functions are local SOLAs. This is the content of the following theorem. Proof This follows in the same way as in the constant exponent case, Theorem 2.7 in [19]. For the reader's convenience, we sketch the argument here with the appropriate references for various auxiliary results. The proof consists of two main steps. First, we prove the claim when u is a weak supersolution. The general case is then reduced to the case of supersolutions by an approximation argument using the obstacle problem.
Assume first that u is a weak supersolution. Then, u ∈ W In the general case, the fact that u ∈ W 1,q(·) (Ω ) follows by a refinement of [21,Theorem 4.4]. By [13,Theorem 6.5], we may choose a sequence ( u i ) of continuous weak supersolutions increasing to u. Arguing as in [13, proof of Theorem 5.1], we can show that ∇ min( u i , k) → ∇ min(u, k) pointwise almost everywhere for any k ∈ R. It follows that ∇ u i → ∇u pointwise a.e., and the pointwise convergences easily imply that u i → u in W 1,q(·) (Ω ). The proof is completed by applying the case of supersolutions to the functions u i , together with the convergence of u i → u in W 1,q(·) (Ω ), see the proof of Theorem 2.7 in [19].
The following pointwise potential estimates hold for local SOLAs and p(·)-superharmonic functions. See [6] for (8.2) and (8.3) in the case of SOLAs, and [22] for (8.2) for p(·)-superharmonic functions. Finally, the gradient estimate (8.3) holds also for p(·)-superharmonic functions by an application of Theorem 8.1. The Hölder continuity of p is required for the gradient estimate, since its proof uses Hölder estimates for the gradient of weak solutions (cf. [1]).

Theorem 8.2 Let p be log-Hölder continuous with p −
2. Let u be positive p(·)-superharmonic or a local SOLA to (8.1). Then, there exists γ > 0 such that for all sufficiently small R > 0. For positive p(·)-superharmonic functions, the assumption p − > 1 suffices instead of p − 2. Suppose next that p is Hölder continuous. Then, for all sufficiently small R > 0.
The restriction p − 2 in the gradient estimates is related to the fact that there are substantial differences in gradient potential estimates in the cases p < 2 and p > 2 even with constant exponents, see [11]. For simplicity, we focus on the prototype case (8.1) here, but this result, and hence also Theorem 8.3 below, hold for more general equations of the form −div(a(x, ∇u)) = μ under appropriate structural assumptions on a(x, ξ). The interested reader may refer to [6,22] for details.
The following result is an immediate consequence of Theorems 7.1 and 8.2.
If r ≡ 1, μ can be a measure with finite mass instead of a function. Each of the inclusions comes with an explicit estimate.
Theorem 1.2 is of course contained in the above theorem when r ≡ 1. The interesting case in these results is when r − = 1; if r − > 1, we can use the pointwise inequality (7.1) and the strong-to-strong estimate for the Riesz potential to get estimates in strong Lebesgue spaces with the same exponents.
When r ≡ 1, the above inclusions are sharp for constant p on the scale of w-L q spaces. This is a special case of the following examples.
Example 1 Let B be the unit ball in R n , and assume that the exponent p is smooth and radial. where the last estimate follows from the log-Hölder continuity of q.
The last integral is finite if (8.5) holds. Starting from the first inclusion in (8.9), we get a similar lower bound. Hence u ∈ w-L q(·) (B) if and only if (8.5) holds. Repeating the same argument using (8.8), we obtain the condition (8.6).

Example 2
The right-hand side of the differential equation in the previous example is a delta measure, which is not an L 1 -function. However, the example can be modified to yield a function in L 1 . Denote by u the function from the previous example and define v r (x) := a r − b r |x| when |x| r, u(x) otherwise.
The constants a r and b r are chosen so that v r ∈ C 1 . Then a r = |∇u(r )| ≈ r − n−1 p(0)−1 by the computations above. A direct calculation shows that −div(|∇v r | p(x)−2 ∇v r ) = a p(x)−1 r n − 1 |x| + p (x) log a r .
If we suppose that p is Lipschitz continuous, then n − 1 |x| + p (x) log a r ≈ n − 1 |x| for small enough r and so the right-hand side of this equation is positive in B(0, r ). Furthermore, the right-hand side is in L 1 uniformly and v r u as r → 0, so we see that the conclusions from the previous example hold also for the L 1 case.