Collect. Math. https://doi.org/10.1007/s13348-018-0213-2

Wolff potential estimates for Cheeger p-harmonic functions Takanobu Hara1

Received: 7 July 2017 / Accepted: 17 January 2018 © Universitat de Barcelona 2018

Abstract In this note, we give a new proof of Wolff potential estimates for Cheeger psuperharmonic functions on metric measure spaces given by Björn et al. (J Anal Math 85:339– 369, 2001). Also, we extend the estimate to Poisson type equations with signed data. Keywords Nonlinear elliptic equations · p-Laplacian · Wolff potentials · Metric space · Doubling measure · Poincarë inequality · Potential theory Mathematics Subject Classification 31C45 · 35J62 · 31C15 · 31C05

1 Introduction If u is a (classical) superharmonic function in  ⊂ Rn , then, from the Riesz decomposition theorem, u is represented as  u(x) = G  (x, y) dν(y) + h(x), 

where G  (·, ·) is the Green function of , ν is a Radon measure on  satisfying ν = −u in the distribution sense and h is a harmonic function in . The representation formula has many applications. In particular, by combining a dual formulation of capacity, the formula yields a necessary and sufficient condition of boundary regularity of harmonic functions. The criterion is called the Wiener criterion. A sufficient condition of the Wiener criterion for p-harmonic functions was first given by Maz ya [24]. Lindqvist and Martio [21] gave a new proof for the sufficiency and gave the necessity for p > n − 1. The restriction was due to a problem of pointwise estimate of p-

This work was supported in part by JSPS KAKENHI Grant No. 17H01092.

B 1

Takanobu Hara [email protected] Department of Mathematics, Hokkaido University, Kita 8 Nishi 10, Sapporo 060-0808, Japan

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superharmonic functions. If p = 2, the necessity follows easily from the above representation theorem. Kilpeläinen and Malý [14,15] proved the two-sided Wolff potential bound   1 ν inf u + Wνp (x0 , 2R) W p (x0 , R) ≤ u(x0 ) ≤ C B(x0 ,R) C for any nonnegative p-superharmonic function u satisfying − p u = ν in B(x0 , 2R), where Wνp (x, R) is the Wolff potential of ν defined by Wνp (x, R) =



R

0



ν(B(x, s)) s n− p

1/( p−1)

ds . s

The right-hand side was first introduced in [23] in the context of the theory of ( p, l)-capacities. Using this estimate, they gave the necessity of the Wiener criterion for p-harmonic functions for all p > 1. Note that W2ν (x, R) is a truncated version of Newtonian potential. Therefore, the two-sided bound has given a natural extension of the above representation theorem. The proof of Kilpeläinen and Malý was due to skillful test functions and an iteration method. Trudinger and Wang [27] gave a new proof of this two-sided bound, using Poisson modification and Harnack’s inequality. Korte and Kuusi [20] gave another proof of the upper bound, using Poisson modification, Kilpeläinen and Malý’s iteration method and a weak Harnack inequality. Moreover, they refined the method of Trudinger and Wang. Duzaar and Mingione [7] gave a similar pointwise bound for quasilinear equations with signed data, using a completely different method. Very recently, the author [9,10] gave a new proof of the upper bound, using Poisson modification, Kilpeläinen and Malý’s iteration method and a local L ∞ estimate of subsolutions. The method can be applied to equations with signed data. Mikkonen [25] extended the results of Kilpeläinen and Malý to weighted equations and gave several new applications. Björn, MacManus and Shanmugalingam [4] extended Mikkonen’s result to the Cheeger p-harmonic functions on metric measure spaces endowed a doubling measure and a weak p-Poincar’e inequality (see, Theorem 2.6 and Definition 3.1). However, they omitted the detail of the proof of the Wolff potential estimates itself. The aim of this paper is to give a new proof of the two-sided Wolff potential bound for Cheeger p-harmonic functions, using relatively simple new methods. First, using the method in [10], we refine the pointwise upper bound in [4] as the following: Theorem 1.1 Suppose that ν is a signed Radon measure satisfying |ν| ∈ (N0 ())∗ . Let u be a weak solution to − p u = ν in . Then for any γ > 0, there exists a constant C depending only on the doubling constant, the data of p-Poincaré inequality and γ such that for μ-a.e. x0 ∈ , ⎧ ⎫ 1/γ ⎨  ⎬ ν γ − u ± (x0 ) ≤ C u ± dμ + W p± (x0 , 2R) ⎩ B\ 1 B ⎭ 1, p

2

whenever B = B(x0 , R)   and R < diam(X )/4. Theorem 1.1 improves the estimates in [4,25] in the following sense: (1) We treat signed measures ν. (2) The range of the exponent γ is extended from ( p − 1, ∞) to (0, ∞). (3) The meaning of the pointwise value of u is taken in the weak-L σ sense. (see the left-hand side of (25).)

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All of these improvements are based on the weak-type estimate in Theorem 4.3. It is natural to expect a weak-type estimate for measure data. In fact, when ν is a siged measure, a solution of − p u = ν satisfies only weak-L σ estimates in general, and so does the Wolff potential of a measure ν. See Remark 5.3. Also, adapting the method in [20] for metric measure spaces, we prove the Wolff potential lower bound (Theorem 4.7). Combining the two estimates, we get the following:

Theorem 1.2 Let u be a nonnegative Cheeger p-superharmonic function in , and let B = B(x0 , R) ⊂ 100λB ⊂ , C p (X \) > 0. Let ν be the Riesz measure of u. Then, there exists a constant C depending only on the doubling constant and the data of p-Poincaré inequality such that   1 ν ν (1) W (x0 , R) ≤ u(x0 ) ≤ C inf u + W p (x0 , 2R) . 2B C p

Organization of the paper In Sect. 2, we collect several properties of metric measure spaces. In particular, we introduce Newtonian spaces and the Cheeger gradients. In Sect. 3, we introduce the notion of the Cheeger p-minimizer and collect its properties. Also, using a divergence structure of the Cheeger p-Dirichlet integral, we prove several lemmas. In Sect. 4, we prove Wolff potential estimates for weak solutions of Cheeger p-Laplacian. This part is a key to this paper. First, we give a local version weak-type estimate for weak solutions. Next, iterating the local weak-type estimate, we derive a pointwise Wolff potential upper bound. Finally, we give a pointwise Wolff potential lower bound, using Harnack-type estimates. In Sect. 5, we extend Wolff potential estimates to Cheeger p-superharmonic functions. In particular, combining these estimates, we establish a two-sided Wolff potential bound.

Notation In this paper, we use the following notation. Let X = (X, d, μ) be a metric measure space endowed with a nonnegative Radon measure μ. Let A ⊂ X be a μ-measurable set. Let f be a μ-measurable function on X , and let ν be a signed Radon measure on X .

• • • • • • • • •

B(x, R) := {y ∈ X : d(x, y) < R}. For B = B(x, R) and λ > 0, we denote λB := B(x, λR).

diam(A) := sup{d(x,

y) : x, y ∈ A}. − f dμ := μ(A)−1 A A f dμ. f + := max{ f, 0}, f − := max{− f, 0}. ν+ , ν− : the Jordan decomposition of ν. 1 A (x) : the indicator function of A. Lip(A) : the family of all Lipschitz functions on A. Lipc (A) : the family of all compactly supported Lipschitz functions on A.

For open sets U, U  , we write U   U if U  is compact and U  ⊂ U . For a Banach space X, we denote by X∗ the dual of X.

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2 Preliminaries 2.1 General assumptions Let X = (X, d, μ) be a complete metric space endowed with a metric d and a positive complete Borel measure μ such that 0 < μ(B) < ∞ for all balls B ⊂ X . We also assume that the measure is doubling, i.e. that there exists a constant Cμ > 0 such that for all balls B in X , μ(2B) ≤ Cμ μ(B). (2) The constant Cμ is called the doubling constant of μ. Definition 2.1 Suppose that f is an extended real valued function on X . A nonnegative Borel function g on X is called an upper gradient of f if  | f (x) − f (y)| ≤ g ds (3) γ

for every x, y ∈ X and every locally rectifiable curve γ joining x and y. Let 1 < p < ∞. A property is said to hold for p-almost every paths if the set of nonconstant paths for which the property fails is of zero p-modulus (for the definition of the p-modulus, see e.g. Chapter 5 in [11] or Section 1.5 in [1]). If g is a nonnegative measurable function on X and if (3) holds for p-almost every path, then g is a weak upper gradient of f . By saying that (3) holds for p-almost every path we mean that it fails only for a path family with zero p-modulus. If f has an upper gradient in L p (X ), then it has a minimal weak upper gradient g f ∈ L p (X ) in the sense that for every p-weak upper gradient g ∈ L p (X ) of f , g f ≤ g μ-almost everywhere. By [1, Theorem 2.50], the minimal weak upper gradient can be obtained by the formula  g f (x) := inf lim sup − g dμ, g

R→0

B(x,R)

where the infimum is taken over all upper gradients g ∈ L p (X ) of f . Definition 2.2 We say that X supports a weak p-Poincaré inequality if there exist constants C P I > 0 and λ ≥ 1 such that for every balls B ⊂ X and every integrable functions f on X ,  1/ p  p − | f − f B | dμ ≤ C P I diam(B) − g f dμ , (4) where f B =

− B

λB

B

f dμ.

Hereafter, we always assume that μ is doubling and X supports a weak p-Poincaré inequality for some 1 < p < ∞. Also, we often call the constants p, C P I and λ appearing the Poincaré inequality as the data of the Poincaré inequality.

2.2 Newtonian spaces Following Shanmugalingam [26], we introduce Newtonian spaces. Definition 2.3 (Newtonian space) For u ∈ L p (X ), let  1/ p  u N 1, p (X ) := |u| p dμ + inf g p dμ , X

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X

Wolff potential estimates for Cheeger p-harmonic functions

where the infimum is taken over all upper gradients of u. The Newtonian space on X is the quotient space N 1, p (X ) := {u : u N 1, p (X ) < ∞}/ ∼, where u ∼ v if and only if u − v N 1, p (X ) = 0. The Newtonian space is a Banach space under its norm and a lattice, moreover the set of all Lipschitz functions is dense (see e.g. [1,11]). We say that u belongs to the local Newtonian 1, p space Nloc (X ) if u ∈ N 1, p () for any   X . We also define the Sobolev capacity of a set E ⊂ X by C p (E) = inf u N 1, p (X ) , u

where the infimum is taken over all u ∈ N 1, p (X ) such that u ≥ 1 on E. We say that a property regarding points X holds quasieverywhere (q.e.) if the set of points for which it fails 1, p has capacity zero. By [1, Corollary 1.69], for any u ∈ Nloc (X ) and A  X ,   (5) ess sup u = inf k ∈ R : C p ({x ∈ A : u(x) > k}) = 0 . A 1, p

By [1, Theorem 5.62], if u ∈ Nloc (X ), then q.e. x ∈ X is a Lebesgue point of u. An extended real valued function u on X is quasicontinuous if for every  > 0 there is an open set G  1, p with C p (G  ) <  such that u| X \G is continuous. By [1, Theorem 5.21], every u ∈ Nloc () is quasicontinous in . To compare the boundary values of Newtonian functions, following [13], we introduce Newtonian spaces with zero boundary values. Let  and X  be a measurable subset of X . The Newtonian space with zero values on X  \ is the space   1, p N0 (; X  ) = u|∩X  : u ∈ N 1, p (X  ) and u = 0 q.e. on X  \ . (6) 1, p

1, p

We also let N0 () = N0 (; X ). 1, p The space N0 () equipped with the norm inherited from N 1, p (X ) is a Banach space. 1, p 1, p 1, p As in [1, Proposition 2.39], N0 () = N0 (, ). If C p (X \) = 0, then N0 () = N 1, p (). The following approximation lemma holds: 1, p

Lemma 2.4 ([1, Theorem 5.46]) Lipc () is dense in N0 (). Moreover, if 0 ≤ u ∈ 1, p N0 (), then the Lipschitz approximations of u can be chosen to be nonnegative and bounded above by u. The following Sobolev’s inequality holds: Lemma 2.5 ([1, Theorem 5.51]) Let  Q/(Q − p) if Q > p, κ := any fixed constant > 1 if Q ≤ p,

(7)

where Q := log2 Cμ . Then, there exists a constant C > 0 such that for every ball B = 1, p B(x, R) ⊂ X with R < diam(X )/4 and every u ∈ N0 (B),  1/(κ p)  1/ p p − |u|κ p dμ ≤ C R − gu dμ . B

B

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2.3 Differentiable structure Cheeger [5] showed that if the measure of X is doubling and if X supports a p-Poincaré inequality for some p > 1, then there exists a “differentiable” structure on X compatible with the notion of an upper gradient. More precisely, the following theorem holds: Theorem 2.6 ([5, Theorem 4.38]) Let X be a metric measure space equipped with a doubling Radon measure μ. Assume that X admits a weak p-Poincaré inequality for some 1 < p < ∞. Then, there exists a countable collection (Uα , X α ) of measurable sets U α and Lipschitz “coordinate charts” X α : X → Rk(α) such that μ(X \ α Uα ) = 0, μ(Uα ) > 0 and for all α the following hold: α (1) The functions X 1α , . . . X k(α) are linearly independent on Uα and 1 ≤ k(α) ≤ N , where N is a constant depending only on the doubling constant and the data of Poincaré inequality. (2) If f : X → R is Lipschitz, then there exist unique bounded vector-valued functions d α f : U α → Rk(α) such that for μ-a.e. x ∈ Uα ,

| f (y) − f (x) − d α f (x), X α (y) − X α (x)| = 0, r →0+ y∈B(x,r ) r lim

sup

where ·, · is the usual inner product in Rk(α) . Without loss of generality, we can assume that the sets Uα are pairwise disjoint and that dα f is extended by zero outside Uα . Regard d α f (x) as vectors in R N , and take D f (x) = α α d f (x). By definition, the differential mapping D : f  → D f is linear. Since the Newtonian space N 1, p (X ) equals the closure of the collection of Lipschitz functions on X , the differential operator D can be uniquely extended to all of N 1, p (X ) so that 1 |D f (x)| ≤ g f (x) ≤ C|D f (x)| C

(8)

for all f ∈ N 1, p (X ) and μ-a.e. x ∈ X , where C ≥ 1 is a constant and | · | is the Euclidean norm in R N . By this differentiable structure, the Newtonian space N 1, p (X ) is reflexive (see, e.g. [11, Theorem 13.5.7]). The differential operator D satisfies the following product and chain rule: if v is a bounded Lipschitz function on X , u ∈ N 1, p (X ), and : R → R is continuously differentiable with bounded derivative, then uv and ◦ u belong to N 1, p (X ) and satisfy D(uv) = u Dv + v Du, D( ◦ u) =  (u)Du. For further information of the Cheeger gradients, see also Cheeger [5], Keith [12], and Chapter 13 in [11].

3 Cheeger p-Laplace equations 3.1 Basic facts of p-(super, sub)minimizers Using Cheeger gradients, we define Cheeger p-(super, sub)minimizers:

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Definition 3.1 (Cheeger p-(super, sub)minimizer) Let  ⊂ X be an open set. We say that a 1, p function u ∈ Nloc () is a Cheeger p-superminimizer in  if   |Du| p dμ ≤ |D(u + ϕ)| p dμ (9) 



for all nonnegative ϕ ∈ Lipc (). If −u is a Cheeger p-superminimizer, we say that u is a 1, p Cheeger p-subminimizer. We say that a function u ∈ Nloc () is a Cheeger p-minimizer in  if u is a p-superminimizer and a p-subminimizer. We also say that a function is Cheeger pharmonic if it is a continuous Cheeger p-minimizer. The De Giorgi–Moser theory for p-minimizers was developed by many authors (see e.g., [2,16–18]). Several properties of p-minimizers, which are defined by minimal upper gradients, still hold for Cheeger p-minimizers. In particular, the following Harnack type inequalities for Cheeger p-minimizers hold. See [1, Theorems 8.4, 8.34, and pp. 363–364]. Lemma 3.2 Let u be a Cheeger p-subminimizer in , and let B = B(x, R) ⊂ . Then for any γ > 0, there exists a constant C depending only on the doubling constant, the data of p-Poincaré inequality and γ such that  1/γ γ ess sup u + ≤ C − u + dμ . 1 2B

B

Lemma 3.3 Let u be a nonnegative Cheeger p-superminimizer in , and let 50λB ⊂ . Let κ be a constant given by (7). Then, for any positive constant 0 < γ < κ( p − 1), there exists a constant C depending only on the doubling constant, the data of p-Poincaré inequality and γ such that  1/γ − u γ dμ ≤ C ess inf u. B

B

From these estimates, it follows that if u is a Cheeger p-minimizer in , then u can be modified on a set of capacity zero so that it becomes locally α-Hölder continuous in  for some 0 < α < 1 (see, [1, Theorem 8.14]). By Lemma 3.2, every Cheeger p-superminimizer is essentially locally bounded from below. Therefore, the following lower semicontinous regularization of u can be defined. Definition 3.4 Let u be a measurable function on  which is locally bounded from below. We define the lsc-regularization u˜ of u as u(x) ˜ = lim ess inf u. R→0 B(x,R)

By definition, u˜ is lower semicontinuous. From Lemma 3.2 and the argument in [1, Theorem 8.22], if u is a Cheeger p-superminimizer in  and if x ∈  is a Lebesgue point of u, then  u(x) = lim −

R→0 B(x,R)

u dμ = lim ess inf u. R→0 B(x,R)

In particular, if u is a Cheeger p-superminimizer in , then u˜ = u q.e. in . Hence, by taking the lsc-regularization, we can assume that every Cheeger p-superminimizer is lower semicontinuous.

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3.2 Consequences of differentiable structure From [4, Proposition 3.5], u is a Cheeger p-superminimizer in  if and only if there exists 1, p a nonnegative Radon measure ν[u] ∈ (N0 ())∗ such that   p−2 |Du| Du · Dϕ dμ = ϕ dν[u] 



for all ϕ ∈ Lipc (). The measure ν[u] is called the Riesz measure of u. We shall introduce the Cheeger p-Laplace operator − p using the above divergence structure of Cheeger pDirichlet integral: Definition 3.5 Let  ⊂ X be open. Let ν be a signed Radon measure in (N0 ())∗ . We 1, p say that a function u ∈ Nloc () is a weak solution to 1, p

− p u = ν in  



if



|Du| p−2 Du · Dϕ dμ =



ϕ dν

(10)

for all ϕ ∈ Lipc (). 1, p

From the density of Lipc (), the equation (10) also holds for all ϕ ∈ N0 (). Note that 1, p by the argument in [4, Lemma 3.8], if ν is a nonnegative Radon measure in (N0 ())∗ and if E ⊂  with C p (E) = 0, then ν(E) = 0. In particular, the right-hand side of (10) is 1, p well-defined for each ϕ ∈ N0 (). The following existence theorem holds. Lemma 3.6 Suppose that  is bounded and C p (X \) > 0. Let ν ∈ (N0 ())∗ and 1, p θ ∈ N 1, p (). Then, there exists u ∈ θ + N0 () satisfying − p u = ν in . 1, p

Proof If θ ≡ 0, then the lemma is nothing but [4, Proposition 3.9]. The general case can be proved by using a bounded, coercive, demicontinuous map on a reflexive Banach space as in [22, p. 177]. Let T be a map from a reflexive Banach space Y to its dual space Y ∗ . We say that T is coercive if (T yi , yi )/yi Y → +∞ whenever yi Y → +∞. We say that T is demicontinuous if (T yi , z) → (T y, z) for all z ∈ Y whenever yi → y in Y . We say that T is bounded if T (E) is bounded in Y ∗ whenever E is bounded subset of Y . We apply the general result [22, p.171] which asserts that if T is a bounded, coercive, demicontinuous map, then for all f ∈ Y ∗ the equation T y = f has a solution y ∈ Y . 1, p For u ∈ N0 (), we write |||u||| = Du L p () . By (8) and Sobolev’s inequality (Lemma 2.5), the quantity |||u||| is comparable to u N 1, p () . Consider the map T : N0 () → (N0 ())∗ such that  (T w, ϕ) = |D(w + θ )| p−2 D(w + θ ) · Dϕ dμ. 1, p

1, p



1, p 1, p The map is bounded on Y = (N0 (), |||·|||) in the above sense. Also, if wi → w in N0 (), p N p−2 p−2 then Dwi → Dw in L (, μ; R ): hence |Dwi | Dwi → |Dw| Dw weakly in L p/( p−1) (, μ; R N ), i.e., T is demicontinuous. Let us prove coercivity of T . Since

|||w||| = Dw L p () ≤ D(w + θ ) L p () + Dθ  L p () ,

123

(11)

Wolff potential estimates for Cheeger p-harmonic functions

choosing w sufficiently large, we can assume that D(w + θ ) L p () ≥ 2Dθ  L p () . Then, Dθ  L p () ≤ D(w + θ ) L p () − Dθ  L p () ≤ |||w|||. (12) Recall Young’s inequality ab ≤ a p + C( p, )b p/ p−1 for any a, b ≥ 0,  > 0. By this and Hölder’s inequality,  |D(w + θ )| p−2 D(w + θ ) · Dw dμ (T w, w) =   = |D(w + θ )| p−2 D(w + θ ) · D(w + θ ) dμ   (13) |D(w + θ )| p−2 D(w + θ ) · Dθ dμ − 

p

p−1

≥ D(w + θ ) L p () − D(w + θ ) L p () Dθ  L p () ≥

1 p p D(w + θ ) L p () − CDθ  L p () . 2

It follows from (11), (12) and (13) that   (T w, w) 1 1 p |||w||| p − CDθ  L p () ≥ |||w||| |||w||| 2 1 ≥ |||w||| p−1 − C. 2 The right hand side goes to +∞ as |||w||| → +∞. Hence, T is coercive. 1, p As was observed at the beginning of the proof, T w = ν in (N0 ())∗ has a solution 1, p w ∈ N0 (). Then u = w + θ is a solution of the original Dirichlet problem.   Since the map R N  z  → |z| p ∈ R is strictly convex and differentiable,   |z 1 | p−2 z 1 − |z 2 | p−2 z 2 · (z 1 − z 2 ) > 0

(14)

whenever z 1 , z 2 ∈ R N , z 1  = z 2 . Using this monotone structure, we can prove the following 1, p comparison principle. In particular, the above solution is unique in θ + N0 (). Lemma 3.7 Suppose that  is bounded and C p (X \) > 0. Assume that u, v ∈ N 1, p () 1, p and (u − v)+ ∈ N0 (). Assume also that − p u ≤ − p v in the distribution sense, i.e.,   |Du| p−2 Du · Dϕ dμ ≤ |Dv| p−2 Dv · Dϕ dμ 



for all nonnegative ϕ ∈ Lipc (). Then u ≤ v q.e. in . Proof By assumption, we have    |Du| p−2 Du − |Dv| p−2 Dv · (Du − Dv) dμ ≤ 0. 0≤ {x∈ : (u−v)(x)>0}

Then D(u − v) = 0 μ-a.e. in {u > v} ∩ , and as gu−v is comparable to |D(u − v)|, it follows that gu−v = 0 μ-a.e. in {u > v} ∩ . So, g(u−v)+ = 0 μ-a.e. in , and therefore by 1, p u − v ∈ N0 () it follows that (u − v)+ = 0 q.e. in .  

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The following proposition can be obtained from a more general theorem (see, [4, Theorem 4.12]). However, for completeness of the next section, we give an alternative short proof using convergence properties of Cheeger p-superminimizers. Proposition 3.8 Assume that  is bounded and C p (X \) > 0. Let ν be a nonnegative 1, p 1, p measure in (N0 ())∗ , and let E ⊂  be closed. Then, ν E ∈ (N0 ())∗ . Proof Since ν is a nonnegative measure in (N0 ())∗ , by the definition of that dual space 1, p we have that for all ϕ ∈ N0 () we have      ϕ dν  ≤ ν 1, p ϕ N 1, p () .   (N ())∗ 1, p



0

0

1, p N0 (),

Then since ν E ≤ ν, we have for nonnegative ϕ ∈  ϕ dν E ≤ ν(N 1, p ())∗ ϕ N 1, p () . 

For general ϕ ∈

0

0

1, p N0 (),

     ϕ dν E  ≤ ν 1, p ϕ N 1, p () .   (N0 ())∗ 0 

Since ϕ±  N 1, p () ≤ ϕ N 1, p () , this implies that ν E (N 1, p ())∗ ≤ 2ν(N 1, p ())∗ and 0

ν E belongs to (N0 ())∗ . 1, p

0

0

0

 

4 Wolff potential bounds In this section, we derive pointwise potential bounds for weak solutions. First of all, we introduce Wolff potentials. Definition 4.1 (Wolff potential) Let ν be a Radon measure on X , and let R > 0. We define the Wolff potential of ν by   R ν(B(x, s)) 1/( p−1) ds Wνp (x, R) = sp . μ(B(x, s)) s 0 The Wolff potential Wνp (x, R) is lower semicontinous, because it is given by a monotone increasing limit of continuous functions.

4.1 Potential upper bounds To prove the potential upper bound, we recall properties of Lorentz spaces L q,r (). Definition 4.2 (Lorentz space) Let f be a measurable function on . For 0 < q < ∞, we define  f  L q,∞ () := sup tμ({x ∈  : | f (x)| ≥ t})1/q , t>0



 f  L q,1 () := q

∞

μ({x ∈  : | f (x)| ≥ t})1/q dt.

0

By L q,∞ () (resp., L q,1 ()), we denote the space of all measurable functions f on  such that  f  L q,∞ () < ∞ (resp.,  f  L q,1 () < ∞).

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The space L q,∞ () is also called weak-L q (). It is well-known that L q,1 () ⊂ L q () ⊂ Moreover, as in [8, Theorem 1.4.17], for 1 < q < ∞, the following Hölder type inequality holds:      f g dμ ≤  f  L q/(q−1),1 () g L q,∞ () . (15)  L q,∞ ().



By a simple calculation, for any measurable function f and any nonnegative constant l,  f +  L q,∞ () ≤ lμ()1/q + ( f − l)+  L q,∞ () .

(16)

σ = κ( p − 1),

(17)

Hereafter, we take

where κ is defined by (7). By definition, if p ≥ Q, then we can choose any positive finite number as σ . The exponent is closely related to the fundamental solution of p-Laplacian (see, Remark 5.3). 1, p Below, we treat a signed Radon measure ν such that |ν| ∈ (N0 ())∗ . If ν+ is the positive part of ν, then            ϕ dν+  =  ϕ+ dν+ − ϕ dν − +        ≤ ϕ+ d|ν| + ϕ− d|ν| 



≤  |ν| (N 1, p ())∗ ϕ N 1, p () 0

1, p

for all ϕ ∈ N0 (). The same inequality holds for the negative part of ν. Hence, |ν| ∈ 1, p 1, p (N0 ())∗ if and only if ν± ∈ (N0 ())∗ respectively. Theorem 4.3 Suppose that |ν| ∈ (N0 ())∗ . Let u be a weak solution to − p u = ν in . Let B = B(x, R)   with R < diam(X )/4. Then, for any γ > 0, there exist constants C1 and C2 depending only on the doubling constant, the data of p-Poincaré inequality and γ , such that 1/γ  1 γ u +  L σ,∞ ( 1 B) ≤ C1 − u + dμ 2 μ( 21 B)1/σ B\ 21 B (18) 1/( p−1)  ν (B) + + C2 R p . μ(B) 1, p

Proof Let A = B\ 21 B. By Theorem 4.32 in [1], there exists a curve γ such that γ (0) = x 1, p and γ (lγ ) ∈ X \B. The curve intersects with A, so A is not empty. Let v ∈ u + N0 (A) be the weak solution to − p v = −ν− in A. Then, by Lemma 3.7, v ≤ u q.e. in A. Extend v as v = u outside of A. From definition, 



0≤ A

ϕ dν+ =



 |Du| p−2 Du − |Dv| p−2 Dv Dϕ dμ

A

123

T. Hara 1, p

for any nonnegative ϕ ∈ N0 (A). Fix ψ ∈ Lipc ( 43 B) such that 0 ≤ ψ ≤ 1, and substitute ϕ = I (u − v)ψ into the above inequality, where I (t) =  −1 min{t, }. Then,     |Dv| p−2 Dv − |Du| p−2 Du · Dψ I (u − v) dμ A     |Du| p−2 Du − |Dv| p−2 Dv · D I (u − v) ψ dμ. ≤ A

Since D I (u − v) =  −1 (Du − Dv) on {D I (u − v)  = 0}, by (14),     |Du| p−2 Du − |Dv| p−2 Dv · D I (u − v) ψ dμ A    |Du| p−2 Du − |Dv| p−2 Dv · D I (u − v) dμ. ≤ A

By the definition of v, we also have     I (u − v) dν+ ≤ ν+ (A). |Du| p−2 Du − |Dv| p−2 Dv · D I (u − v) dμ = A

Hence,

A



 A

  |Dv| p−2 Dv − |Du| p−2 Du · Dψ I (u − v) dμ ≤ ν+ (A) ≤ ν+ (B).

Take the limit  → 0. Then, by Lebesgue’s dominated convergence theorem,    |Dv| p−2 Dv − |Du| p−2 Du · Dψ dμ ≤ ν+ (B). {x∈A : (u−v)(x)>0}

Note that Du = Dv μ-a.e. on {x ∈ A : (u − v)(x) = 0}. Since u ≥ v q.e. in A and u = v outside A, it follows that    |Dv| p−2 Dv − |Du| p−2 Du · Dψ dμ ≤ ν+ (B). B

On the other hand, since ψ has a compact support in B, we have   |Du| p−2 Du · Dψ dμ = ψ dν ≤ ν+ (B). B

B

Thus, combining the two inequalities, we obtain  |Dv| p−2 Dv · Dψ dμ ≤ 2ν+ (B).

(19)

B

1, p

Since Lipc ( 43 B) is dense in N0 ( 43 B) (Lemma 2.4), this inequality also holds for any ψ ∈ 1, p N0 ( 43 B) such that 0 ≤ ψ ≤ 1. Since v is a Cheeger p-subminimizer in A, it follows from Lemma 3.2 that  1/γ γ ess sup v+ ≤ C − v+ dμ B(x,R/8)

∂ 43 B.

B(x,R/4)

∂ 34 B

whenever x ∈ The set is not empty by Theorem 4.32 in [1]. So, by taking the 3 supremum over x ∈ ∂ 4 B and using the doubling condition (2), it follows that  1/γ γ ess sup v+ ≤ C − v+ dμ , (20) S

123

A

Wolff potential estimates for Cheeger p-harmonic functions

where S =

 x∈∂ 43 B

B(x, R/8). By (5), (v+ − l)+ = 0 q.e. on S for l = ess sup S v+ .

So, (v+ − l)+ = 0 q.e. on ∂ 34 B. By (6) and [1, Proposition 2.39], this implies that (v+ − 1, p 1, p l)+ ∈ N0 ( 43 B), and hence k −1 min{(v+ − l)+ , k} ∈ N0 ( 43 B) for any positive constant −1 k. Substituting ψ = k min{(v+ − l)+ , k} into (19), we get  1 |D min{(v+ − l)+ , k}| p dμ ≤ 2ν+ (B). k 43 B On the other hand, by the Sobolev’s inequality (Lemma 2.5), 1/(κ p)  − | min{(v+ − l)+ , k}|κ p dμ 3 4

B

 1/ p p ≤ C R − |D min{(v+ − l)+ , k}| dμ . 3 4

B

Combining the two inequalities and using Chebyshev’s inequality, we get  1/κ 1 ν+ (B) p−1 μ({x ∈ 2 B : (v+ − l)+ ≥ k}) k ≤ CRp . 1 μ(B) μ( 2 B) Therefore, by taking the supremum over k > 0, 1 μ( 21 B)1/σ

 (v+ − l)+  L σ,∞ ( 1 B) ≤ C 2

R

p ν+ (B)

μ(B)

1/( p−1) .

(21)

Combining (16), (20) and (21), we obtain 1 μ( 21 B)1/σ

1 (v+ − l)+  L σ,∞ ( 1 B) 2 μ( 21 B)1/σ  1/γ  1/( p−1) γ p ν+ (B) ≤C − v+ dμ +C R . μ(B) B\ 21 B

v+  L σ,∞ ( 1 B) ≤ l + 2

 Since v = u in 21 B and v ≤ u in B, replacing v by u, we arrive at the desired inequality.  Theorem 4.4 Let |ν| ∈ (N0 ())∗ , and let u be a weak solution to − p u = ν in . Let B = B(x0 , R)   with R < diam(X )/4. Suppose that one of the following conditions holds: 1, p

(1) u is lower semicontinuous at x0 ; or q (2) x0 is a Lebesgue point of u + for some q > 0. Then, for any γ > 0, there exists a constant C depending only on the doubling constant, the data of p-Poincaré inequality and γ such that ⎧ ⎫ 1/γ ⎨  ⎬ ν+ γ − u + dμ + W p (x0 , 2R) . u + (x0 ) ≤ C ⎩ B\ 1 B ⎭ 2

In particular, for μ-a.e. x0 , this pointwise estimate holds.

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T. Hara q

q

Proof If x0 is a Lebesgue point of u + , then x0 is also a Lebesgue point of u +0 for any 0 < q0 ≤ q. So, we can assume that 1 < q < σ without loss of generality. Also, we may assume that the right-hand side is finite without loss of generality. For m = 0, 1, . . ., take Rm = 2−m R, Bm = B(x0 , Rm ). Also, we take a nondecreasing sequence l0 = 0, lm+1 = lm +

1

1

η1/σ

μ(Bm+1 )1/σ

(u − lm )+  L σ,∞ (Bm+1 ) ,

where σ is defined by (17) and η ∈ (0, 1) is a sufficiently small constant to be chosen later. By the definition of lm , for any m ≥ 1, (u − lm−1 )+  L σ,∞ (Bm ) = (lm − lm−1 )(ημ(Bm ))1/σ .

(22)

By the definition, if lm = lm−1 for some m ≥ 1, then lm−1 = lm = lm+1 = · · · . Assume that lm > lm−1 . Then, since μ({x ∈ Bm : u(x) ≥ lm }) = μ({x ∈ Bm : (u(x) − lm−1 )+ ≥ lm − lm−1 }) ≤ ημ(Bm ), it follows from (15) that  p−1 (u − lm )+ dμ Bm

≤ Cμ({x ∈ Bm : u(x) ≥ lm })τ ( p−1) (u − lm )+  L σ/( p−1),∞ (Bm ) p−1

≤ C(ημ(Bm ))τ ( p−1) (u − lm )+  L σ/( p−1),∞ (Bm ) , p−1

where τ = 1/( p − 1) − 1/σ > 0. So, we have 1/( p−1)  p−1 − (u − lm )+ dμ Bm

1

≤ Cητ

p−1 1/( p−1)

(u − lm−1 )+  L σ/( p−1),∞ (B ) m μ(Bm )1/σ 1 = Cητ (u − lm−1 )+  L σ,∞ (Bm ) . μ(Bm )1/σ

(23)

Applying Theorem 4.3 for u − lm , we get 1 (u − lm )+  L σ,∞ (Bm+1 ) μ(Bm+1 )1/σ 1/( p−1)  1/( p−1)  p−1 p ν+ (Bm ) ≤ C1 − (u − lm )+ dμ + C 2 Rm . μ(Bm ) Bm \Bm+1

(24)

Combining (22), (23) and (25), we get τ

(lm+1 − lm ) ≤ Cη (lm − lm−1 ) +

C η1/σ



p ν+ (Bm ) Rm μ(Bm )

1/( p−1) .

This inequality also holds when lm = lm−1 . Choose η > 0 such that Cητ ≤ 1/2. Then, summing over m = 1, 2, . . . M, we get  1/( p−1) M  1 p ν+ (Bm ) (l M+1 − l1 ) ≤ (l M − l0 ) + C . Rm 2 μ(Bm ) m=1

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Wolff potential estimates for Cheeger p-harmonic functions

Using Theorem 4.3 again, we get 1 1 u +  L σ,∞ (B1 ) η1/σ μ(B1 )1/σ ⎧  1/( p−1) ⎫ 1/γ  ⎬ 1 ⎨ γ p ν+ (B0 ) ≤ 1/σ C − u + dμ + C R0 . ⎩ ⎭ η μ(B0 ) B0 \B1

l1 =

Moreover, by a simple calculation, 

p ν+ (Bm ) Rm μ(Bm )

1/( p−1)

 ≤C

2Rm

 s

p ν+ (B(x, s))

1/( p−1)

μ(B(x, s))

Rm

ds s

for all m ≥ 0, and hence ∞ 



m=0

p ν+ (Bm ) Rm μ(Bm )

1/( p−1) ν

≤ CW p+ (x0 , 2R).

Therefore, combining these inequalities and taking the limit M → ∞, we get   1/γ 1 ν+ γ − l∞ ≤ C u + dμ + W p (x0 , 2R) , 2 B0 \B1 where l∞ = lim M→∞ l M . On the other hand, by (16) and the definition of lm , 1 1 u +  L σ,∞ (Bm ) ≤ l∞ + (u − l∞ )+  L σ,∞ (Bm ) μ(Bm )1/σ μ(Bm )1/σ 1 ≤ l∞ + (u − lm−1 )+  L σ,∞ (Bm ) μ(Bm )1/σ ≤ l∞ + Cη1/σ (lm − lm−1 ). The second term of the right-hand side goes to zero as m → ∞ because l∞ is finite. Hence, taking the upper limit both sides, we obtain 1 lim sup u +  L σ,∞ (Bm ) 1/σ m→∞ μ(Bm )   ≤C

−

B0 \B1

γ u + dμ

1/γ

By lower semicontinuity of u or (15), this implies that   u + (x0 ) ≤ C

−

B0 \B1

γ u + dμ



ν + W p+ (x0 , 2R)

1/γ

(25) .



ν + W p+ (x0 , 2R)

.

This completes the proof.

 

Proof of Theorem 1.1 The same estimate holds for negative parts u − and ν− .

 

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T. Hara

4.2 Potential lower bound To derive the potential lower bound, we introduce the relative Cheeger p-capacity. Definition 4.5 Suppose that  ⊂ X is bounded open and K ⊂  is compact. The relative Cheeger p-capacity of E with respect to  is defined by  |Du| p dμ, (26) Cap p (K , ) = inf u

where the infimum is taken over all u ∈

1, p N0 ()



such that u ≥ 1 on K .

For each  > 1, there exists a constant C depending only on  and the data of X such that ¯ B) ≤ Cr − p μ(B) Cap p ( B,

(27)

whenever B = B(x, r ) ⊂ X . Moreover, when r ≤ diam(X )/8, the opposite inequality holds up to a difference of constant. For further properties of relative p-capacity, see also Section 6.3 in [1]. Lemma 4.6 Let u be a weak solution to − p u = ν ≥ 0 in . Let B = B(x0 , R) ⊂ 100λB ⊂  and C p (X \) > 0. Then, there exists a constant C depending only on the doubling constant and the data of p-Poincaré inequality such that   1 ν(B) 1/( p−1) ≤ ess inf u − ess inf u. Rp B 100λB C μ(B) 1, p

Proof Let v ∈ N0 (100λB) be the weak solution to − p v = ν B in 100λB. From Proposition 3.8, this function can be defined. Set M = sup v and m = inf v. ∂ 23 B

∂ 23 B

The set ∂ 23 B is not empty by Theorem 4.32 in [1]. Since v is p-harmonic in 100λB\B, these constants are well defined. Indeed, by Lemma 3.2, we have,  1/γ 1/γ  v γ dμ ≤ C  − v γ dμ M ≤C − 2B\B

2B

for all γ > 0 Moreover, since v is a nonnegative p-superminimizer in 100λB, by Lemma 3.3, 1/γ  γ − v dμ ≤ C ess inf v ≤ C ess inf v = Cm 2B

2B

3 2

B

for all 0 < γ < κ( p − 1). In particular, we have m ≤ M ≤ Cm.

(28)

Let v¯ = min{v, M}. By Lemma 3.7, v¯ ≥ m q.e. in B; therefore,    v¯ dν ≤ |Dv| p−2 Dv · D v¯ dμ = mν(B) ≤ 100λB

123

100λB

100λB

|D v| ¯ p dμ.

(29)

Wolff potential estimates for Cheeger p-harmonic functions

Let ϕ be the Cheeger p-potential of the condenser ( 23 B, 100λB). Since ϕ is Cheeger pharmonic in 100λB\ 23 B (see, [4, Lemma 3.2]), by Lemma 3.7, Mϕ − v¯ ≥ 0 q.e. in 100λB. Since v¯ is a Cheeger p-superminimizer in 100λB (see, e.g. [4, Remark 3.3]), we have  0≤ |D v| ¯ p−2 D v¯ · D(Mϕ − v) ¯ dμ, 100λB

and hence,



 |D v| ¯ p dμ ≤ 100λB

100λB

By (27), this implies that



|D(Mϕ)| p dμ = M p Cap( 23 B, 100λB).

|D v| ¯ p dμ ≤ C M p (R − p μ(B)).

(30)

100λB

Combining (28), (29) and (30), we obtain  1/( p−1) 1 p ν(B) ≤ m. R C μ(B) On the other hand, by Lemma 3.7, v ≤ u − ess inf u q.e. in 100λB. 100λB

 

Combining the two inequalities, we arrive at the desired estimate.

Theorem 4.7 Let u be a nonnegative weak solution to − p u = ν ≥ 0 in . Let x0 be a Lebesgue point of u, and let B = B(x0 , R) ⊂ 100λB ⊂ , C p (X \) > 0. Then, there exists a constant C depending only on the doubling constant and the data of p-Poincaré inequality such that 1 u(x0 ) ≥ Wνp (x0 , R). (31) C Proof Let χ := (100λ)−1 . Iterating Lemma 4.6, we get 1/( p−1)  ∞  ∞ m 1  m p ν(χ B) ≤ inf u − inf u (χ R) χm B C μ(χ m B) χ m−1 B m=0

m=0

≤ lim inf u m m→∞ χ B

≤ u(x0 ). On the other hand, since     χm R  ν(B(x0 , s)) 1/( p−1) ds ν(χ m B) 1/( p−1) , ≤ C (χ m R) p sp μ(B(x0 , s)) s μ(χ m B) χ m+1 R we have Wνp (x0 , R) ≤ C

∞   m=0

(χ m R) p

ν(χ m B) μ(χ m B)

1/( p−1)

By combining the two inequalities, the desired lower bound follows.

.  

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T. Hara

5 Extension to p-superharmonic functions In this section, we extend the Wolff potential bounds to Cheeger p-superharmonic functions. For applications of the Wolff potential bounds, see also [3,4,19].

5.1 Cheeger p-superharmonic functions Following [16,19], we define the Cheeger p-superharmonic functions. Definition 5.1 (Cheeger p-superharmonic function) A function u :  → (−∞, ∞] is called Cheeger p-superharmonic if (1) u is lower semicontinuous in , (2) u  ≡ ∞ and (3) u satisfies the comparison principle on each subdomain   : If h ∈ C( ) is a Cheeger p-harmonic function in  and u ≥ h on ∂ , then u ≥ h in  . If u is a Cheeger p-superminimizer, then its lower semicontinuous regularization is a Cheeger p-superharmonic function. Every Cheeger p-superharmonic function is a pointwise limit of a monotone increasing sequence of Cheeger p-superminimizers. Indeed, if u  ≡ ∞ is a lower semicontinuous, then u is Cheeger p-superharmonic if and only if the truncations u k = min{u, k} are Cheeger p-superminimizers (see, [16] or Chapter 9 in [1]). From Proposition 6.8 in [19], for any Cheeger p-superharmonic function, there exists a unique nonnegative Radon measure ν supported on  such that   | Du| p−2 Du · Dϕ dμ = ϕ dν 



for any nonnegative ϕ ∈ Lipc (), where Du is the very weak gradient of u defined by Du = lim Du k . k→∞

The measure ν is also called the Riesz measure of u. From definition, ν[u k ] converges to ν[u] weakly as k → ∞. Conversely, from Proposition 6.9 in [19], for each nonnegative finite Radon measure ν, there exists a nonnegative Cheeger p-superharmonic function u such that 1, p ν[u] = ν and u k ∈ N0 () for any k > 0. Proof of Theorem 1.2 First, we derive the upper bound of u. By an argument similar to [1, Theorem 8.4], we can remove the restriction R < diam(X )/4. Applying Theorem 4.3 for u k , we get 1/γ  1 γ σ,∞ (B) ≤ C 1 − (u )  (u ) dμ k + k L + μ(B)1/σ 2B\B  1/( p−1) p ν[u k ](2B) + C2 R , μ(2B) where 0 < γ < κ( p − 1) is a constant appeared in Lemma 3.3. Since ν[u k ] converge ν weakly, for any compact subset K ⊂ , lim sup ν[u k ](K ) ≤ ν(K ). k→∞

Moreover, by the Fatou property of Lorentz norm (see, [8, p.14]), u +  L σ,∞ (B) =  lim inf (u k )+  L σ,∞ (B) ≤ C(σ ) lim inf (u k )+  L σ,∞ (B) . k→∞

123

k→∞

Wolff potential estimates for Cheeger p-harmonic functions

Hence, by the monotone convergence theorem, the above estimate is extended to u. Since u is lower semicontinuous, by the same argument as in Theorem 1.1, ⎧ ⎫ 1/γ ⎨  ⎬ − u γ dμ + Wνp (x0 , 2R) . u(x0 ) ≤ C ⎩ B\ 1 B ⎭ 2

On the other hand, by Lemma 3.3 and the monotone convergence theorem,  1/γ − u γ dμ ≤ C inf u. 2B

2B

Combining the two estimates, we obtain the desired upper bound. Next, we shall prove the lower bound of u. Applying Theorem 4.7 for u k , we have 1 ν[u k ] (x0 , R). W C p Since ν[u k ] converges to ν weakly, for any open subset O ⊂ , u(x0 ) ≥ u k (x0 ) ≥

lim inf ν[u k ](O) ≥ ν(O). k→∞

Thus, it follows from Fatou’s lemma that k ] (x , R). Wνp (x0 , R) ≤ lim inf Wν[u 0 p

k→∞

 

Therefore, (31) also holds for u. From the above proof, we have the following weak-type integrability estimate:

Theorem 5.2 If u is a Cheeger p-superharmonic function in , then u ∈ L σ,∞ loc (), where σ is defined by (17). Remark 5.3 It is known that by using Lemma 3.3, every p-superharmonic function is locally L q -integrable for 0 < q < σ = κ( p − 1). However, the above theorem is  κ( p−1),∞ q sharper, because L loc ()  q<κ( p−1) L loc () in general. The exponent κ( p − 1) is sharp in unweighted Rn . Indeed, u(x) = |x|( p−n)/( p−1) satisfies − p u = c(n, p)δ0 κ( p−1)

in Rn , where δ is the dirac measure supported on 0, but u ∈ / L loc (Rn ). Moreover, δ0 ( p−n)/( p−1) W p (x, ∞) = c(n, p)|x| . See also, [6,17] and [1, Remark 9.56 and Theorem 9.59]. Acknowledgements The author wishes to thank Professor Hiroaki Aikawa for suggesting this problem. The author would like to thank the reviewers for carefully reading the manuscript and the helpful comments.

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