Conductor Sobolev-type estimates and isocapacitary inequalities

In this paper we present an integral inequality con- necting a function space (quasi-)norm of the gradient of a func- tion to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the esti- mates obtained by V. Maz'ya and S. Costea, and sharp capacitary inequalities due to V. Maz'ya in the case of the Sobolev norm. The inequality, obtained under appropriate convexity conditions on the function space, gives a characterization of Sobolev-type inequali- ties involving two measures, necessary and sufficient conditions for Sobolev isocapacitary-type inequalities, and self-improvements for integrability of Lipschitz functions.


INTRODUCTION
If Lip 0 (Ω) is the class of all Lipschitz functions with compact support in a domain Ω ⊂ R n , Wiener's capacity of a compact subset K of Ω, extended in the obvious way for any p ≥ 1 as the p-capacity was used in [M05] to obtain the Sobolev inequality where M t is the level set {x ∈ Ω : |f (x)| > t} (t > 0). This "conductor inequality" is a powerful tool with applications to Sobolevtype imbedding theorems, which for p > 1 plays the same role as the co-area formula for p = 1.
An interesting extension based on the Lorentz space L p,q (Ω) (1 < p < ∞, 1 ≤ q ≤ ∞) has been recently obtained in [CoMa], showing that Our aim in this paper is to extend these capacitary estimates when a general function space X substitutes L p (Ω) or L p,q (Ω) in the definition of Cap p and Cap p,q . It could seem that for improvements of integrability only truncations methods are needed. In [KO] it appears that inequalities of Sobolev-Poincaré type are improved to Lorentz-type scales thanks to stability under truncations, but there and also in [CoMa], p-convexity is implicitly used, since the proofs are based on the inequalities f p L p,q (Ω,µ) + g p L p,q (Ω,µ) ≤ f + g p L p,q (Ω,µ) (1 ≤ q ≤ p), f q L p,q (Ω,µ) + g q L p,q (Ω,µ) ≤ f + g q L p,q (Ω,µ) (1 < p < q) of the Lorentz (quasi-)norms, for disjointly supported functions. With use of the fact that the constant in the right-hand side of the inequalities is 1, they can be extended to an arbitrary set of disjoint functions, and L p,q satisfies lower estimates with constant 1 (see Section 2).
A perusal in the proofs also shows that the limitation of the usual techniques is that they allow us to cover only a certain particular kind of spaces because of the lower p-estimates with constant 1, and it does not apply to a wider class of spaces.
However, by means of new techniques, we will see that an extension is possible in the setting of (quasi-)Banach spaces with lower p-estimates, independently of the value of the constant. Our results can be applied to many examples, which include Lebesgue spaces, Lorentz spaces, classical Lorentz spaces, Orlicz spaces, and mixed norm spaces.
The organization of the article is as follows. Since certain convexity conditions on the space are needed, in Section 2 we recall some basic definitions and known results concerning these concepts and present the most classical examples of spaces, not necessarily rearrangement invariant, satisfying these kinds of properties, and we include some facts concerning capacities and submeasures that we will require in the development of our results.
In Section 3, using a result due to Kalton and Montgomery-Smith on submeasures satisfying an upper p-estimate, we prove our main results.
In Section 4 we characterize Sobolev-type inequalities in the setting of rearrangement invariant (r.i.) spaces. Under appropriate conditions on the space X (see Theorem 4.2) and for any 0 < p < ∞, we show the equivalence of the following properties: Lorentz space, defined in Section 4. Moreover, under the appropriate conditions on Y , we show that In the particular case when X = L p , p ∈ (1, n), and Y = L s with s = np/(n − p), we recover the well-known self-improvement of integrability of Lipschitz functions In Section 5 we derive necessary and sufficient conditions for Sobolev-type inequalities in r.i. spaces involving two measures, recovering results obtained in [CoMa], [M05] and [M06] for Lorentz spaces.
Finally, in Section 6, we include some connections with the theory of the capacitary function spaces studied in [Ce], [CMS], and [CMS1].
As usual, the symbol f ≲ g means that there exists a universal constant c > 0 (independent of all parameters involved) such that f ≤ cg, and f ≃ g means that f ≲ g ≲ f . (Ω, µ) be a measure space and L 0 (Ω) the vector space of all (equivalence classes of ) measurable real functions on Ω. We shall say that X is a quasi-Banach function space if it is a quasi-Banach linear subspace of L 0 (Ω) with the following properties:

Function spaces. Let
(i) (Lattice property) If g ∈ X and f ∈ L 0 (Ω) such that |f | ≤ |g|, then f ∈ X and f X ≤ g X . (ii) (Fatou property) If 0 ≤ f n ↑ f almost everywhere, then f n X ≤ f X . For 0 < p < ∞, recall that X is said to satisfy an upper p-estimate or a lower p-estimate if there exists a constant M so that, for all n ∈ N and for any choice of disjointly supported elements respectively. The smaller constant M is called the upper p-estimate constant or the lower p-estimate constant, and it will be denoted by M (p) (X) or M (p) (X), respectively.

Some examples.
For the sake of the reader's convenience, let us present some examples of spaces satisfying these kinds of properties.
As usual, if f ∈ L 0 (Ω), f * will denote the non-increasing rearrangement of f defined by and f * * (t) := t −1 t 0 f * (s) ds the average function. Recall that a quasi-Banach function space X on Ω is said to be rearrangement invariant (r.i.) if f ∈ X, g ∈ L 0 (Ω) and g * ≤ f * imply g ∈ X and g X ≤ f X .

]).
It is well known that Λ p (w) is p-convex with constant 1 when w is decreasing and p-concave with constant 1 when w is increasing (see [KM]).
More generally, a positive function b is said to be slowly varying q,b (Ω) (see, e.g., [Nev]). Example 2.3 (Γ p (w)). Suppose that the weight w satisfies the nondegeneracy conditions 1 For 0 < p ≤ 1 and r ≥ 1, or for 1 < p < r < ∞, Γ p (w) is r -concave if and only if For details see [KM1]. Example 2.4 (Orlicz spaces). Let φ be a Young function, and consider the Luxemburg norm defined by for all λ ≥ 1 and u ≥ u 0 ≥ 0, then L φ (Ω) is r -concave. • If µ(Ω) = ∞, then the above inequalities need to be satisfied for all u ≥ 0. For details see [K1] and [K2]. Function spaces that are not rearrangement invariant may also be considered: satisfies a lower pq-estimate with constant 1.
Indeed, if f and g are two disjointly supported functions, it follows from [BP,Theorem 1 Similarly, in the case L p n (µ n )[. . . [L p 1 (µ 1 )] ] of n parameters, we have a lower p 1 · · · p n -estimate with constant 1.
Example 2.6 (Mixed norm weighted Lorentz spaces). Suppose 1 ≤ p, q < ∞, and, for a measurable function f on Ω = Ω 1 × Ω 1 , let f * y (x, t) denote the decreasing rearrangement of f with respect to the second variable y, when the first variable x is fixed (see [BK]).
Let u and v be weights on Ω 1 and Ω 2 , u such that U(x) := x 0 u(t) dt is quasi-superadditive. Then the space Λ q (v) [Λ p (u)] defined by the condition also satisfies a lower pq-estimate.
Let 0 ≤ a ≤ 1. An application of Hölder's inequality gives It follows that, since Λ p (u) satisfies a lower p-estimate (see [CS,Lemma 3.2]), if f , g ∈ Λ p (u) are disjointly supported, then Finally, choosing we obtain Observe that, if U is superadditive, then M (p) (Λ p (u)) = 1. Remark 2.7. In Example 2.2, if w is decreasing (respectively, increasing), then for all q > p, Λ p (w) is q-concave with constant 1 (respectively, q-convex with constant 1 for all 0 < q < p) if and only if Λ p (w) is isometric to L p . See [KP,Corollary 4].

Capacities.
Let Ω be a domain of R n endowed with the Lebesgue measure m n , Lip(Ω) be the class of Lipschitz functions on Ω, and Lip 0 (Ω) = {u ∈ Lip(Ω) : sup u compact in Ω}.
From now on, X = X(Ω) denotes a quasi-Banach function space on Ω. Given a compact set K ⊂ Ω and an open set G ⊂ Ω containing K, we call the couple (K, G) a conductor and denote Each conductor has an X-capacity defined by Cap X (K, G) := inf{ ∇u X : u ∈ W (K, G)} that for X = L p,q recovers the capacity Cap X = Cap 1/p p,q from [Co]. From the definition (see [M85], [M05] and [Co]) we have the following statements: • If Ω 1 ⊂ Ω 2 are open and K is a compact subset of Ω 1 , then and K is a compact subset of Ω 1 , then We will write Cap X (·) = Cap X (·, Ω) if Ω has been fixed.

Submeasures. If
A is an algebra of subsets on Ω, a set-function φ : A → R is said to be monotone if it satisfies φ(∅) = 0 and φ(A) ≤ φ(B) whenever A ⊂ B, and φ is said to be normalized when φ(Ω) = 1. A monotone set-function φ is a submeasure if For any 0 < p < ∞, we say that a monotone set-function φ satisfies an upper p-estimate if φ p is a submeasure, and a lower p-estimate if φ p is a supermeasure.
In the proof of our main result, Theorem 3.1, we shall use [KMo,Theorem 2.2], where it is shown that if 0 < p < 1 and ϕ is a normalized supermeasure which satisfies an upper p-estimate, then there exists a measure µ on Ω such that ϕ ≤ µ and µ(Ω) ≤ K p , where For a more complete treatment, see [KMo] and the references quoted therein.

SOBOLEV CAPACITARY INEQUALITIES
In this section we will present our extensions of [CoMa,Theorem 4.2] to an arbitrary parameter p, 0 < p < ∞, and X a Banach or quasi-Banach function space which satisfies a lower p-estimate.
Theorem 3.1. Suppose 0 < p < ∞, and let a > 1 be a constant. If X is a Banach function space that satisfies a lower p-estimate, then where c is a constant that depends on a, p and M (p) (X). In particular, where c depends on p and M (p) (X).
Since X is a Banach function space, the set-function is a submeasure. Moreover, using that X satisfies a lower p-estimate, we conclude that if A 1 , . . . , A 1 are disjoint, then and we claim that ψ is a supermeasure satisfying an upper min(p, 1/p)-estimate. Indeed, given any ε > 0 and two disjoint sets A and B, choose finite partitions and ψ is a supermeasure. Let r = min(p, 1/p). Recall that ψ satisfies an upper r -estimate if ψ r is a submeasure.
Suppose first p ≥ 1, that is, r = 1/p, and let A, B be disjoint sets. If (C 1 , . . . , C n ) is a partition of A ∪ B, then, since φ is a submeasure, Therefore, taking the supremum over all partitions, we obtain that and ψ 1/p is a submeasure. If p < 1 and (C 1 , . . . , C n ) is a partition of A∪B, then, since φ is a submeasure, using that (x + y) p ≤ x p + y p (x, y ≥ 0), we have that Therefore, taking the supremum over all partitions, we obtain that and ψ p is a submeasure. We normalize ψ and define

Consider now
Since f ∈ Lip 0 (Ω), an easy computation shows that and the proof of (3.1) with c := c(a, p, M (p) (X)) = M (p) (X) p K r log a (a − 1) p ends by inserting the last estimate in the left-hand side of (3.7). If p = 1, then X satisfies a lower 1-estimate, and it follows from [LZ,Proposition 1.f.7] that X is q-concave for all q > 1. Therefore, X can be equivalently renormed so that, with the new norm, it satisfies a lower q-estimate with constant 1. Hence, the result follows with similar arguments to those in [CoMa].
The capacitary inequality (3.2) follows from using (3.1) with a = 2 and Cap X ( × M at ) ≤ Cap X ( × M at , M t ). In this case 2 p c = 2 p c(2, p, M (p) (X)) = M (p) (X) p K r 2 p log 2. ❐ Theorem 3.1 can be extended to the setting of quasi-Banach spaces by using Aoki-Rolewicz's Theorem (see, e.g., [BL,Section 3.10]): Theorem 3.2. Suppose 0 < p < ∞, and let a > 1 be a constant. If X is a quasi-Banach function space which satisfies a lower p-estimate, then where the constant c 1 depends on a, p, M (p) (X) and on the quasi-subadditivity constant c of the quasi-norm in X.
In particular, with c 1 depending on p, M (p) (X) and c.
Proof. The proof of Theorem 3.1 can be adapted to this case as follows.
By Aoki-Rolewicz's Theorem, if ̺ is defined as (2c) ̺ = 2, there is a 1seminorm · * such that, for all f ∈ X, Endowed with this 1-seminorm, X satisfies a lower p/̺-estimate, since if f 1 , . . . , f n are disjointly supported functions in X, then

Now consider
With the same arguments as in Theorem 3.1, it can be shown that ψ is a supermeasure that satisfies an upper r -estimate, and the proof ends in the same way, now with c 1 := c 1 (a, p, c, M (p) (X)) = 2 2p/̺ K r log aM (p) (X) p (a − 1) p , for ̺ such that (2c) ̺ = 2 and r = min(p/̺, ̺/p).

ISOCAPACITARY INEQUALITIES AND SOBOLEV-TYPE ESTIMATES
Let µ be a Borel measure on Ω and X be a quasi-Banach r.i. space on Ω. Recall that the distribution function of f is defined as and the fundamental function of X (see [BS] and [BrK]) is defined by Given 0 < p < ∞ and 0 < q ≤ ∞, the Lorentz space Λ p,q (X) associated to X is defined as with the usual changes when q = ∞. When p = q, we write Λ p (X) instead of Notice that if X = L 1 , then Λ p,q (L 1 ) = L p,q . It is well known that for 0 < q 0 ≤ q 1 ≤ ∞, Moreover, if X is a Banach space, then In fact, the spaces Λ 1 (X) and Λ 1,∞ (X) are respectively the smallest and largest r.i. spaces with fundamental function equal to ϕ X . Let X be an r.i. space on R n . Maz'ya's classical method shows that if and only if, for every Borel set A, where m + n is Minkowski's perimeter (see [M11] or [EG]). As shown in [MM2], the following self-improvement property follows for f ∈ Lip 0 (R n ): This Sobolev self-improvement obtained in the case q = 1 is also extended to the case q > 1 as In particular, if X is q-convex, then the space In summary, in terms of the X (q) scale of spaces, on Lipschitz functions we have the following equivalences (see [MMP]): In this section we shall extend this result to the setting of quasi-Banach r.i. spaces. As an application of Theorem 3.2, we characterize Sobolev-type estimates in terms of isocapacitary inequalities.
From now on, Ω will be a domain in R n , X a quasi-Banach function space on Ω, µ a Borel measure on Ω, and Y an r.i. space on (Ω, µ). The notation g ⋐ G means that g is an open set whose closure is a compact subset of the open set G.
An isocapacitary inequality is an inequality of the form Cap X (K) ≥ J(µ(K)), where J is a nonnegative function and K is any compact set in Ω.
Proposition 4.1. If the supremum being taken over all sets g, G such that g ⋐ G ⋐ R n , then for every compact subset K in Ω, ϕ Y (µ(K)) ≲ Cap X (K).
Let us consider some examples: It is well known that the Gagliardo-Nirenberg inequality allows us to see that if p ∈ (1, n), s = np/(n − p) and α = (n − 1)s/n, since But f Λ 1,∞ (L s ) = f L s,∞ ≲ ∇f L p , and then, since L p satisfies a lower p-estimate, from Theorem 4.2, we conclude that f L s,p = f Λ 1,p (L s ) ≲ ∇f L p (f ∈ Lip 0 (Ω)), and we have obtained a self-improvement.
If p = n, then we start from the Trudinger inequality, which gives the estimate ϕ(µ(K)) = 1 + log 1 and then f Λ 1,n (ϕ) ≲ ∇f L n . If r ≤ s < p, then L s,r satisfies an upper p-estimate and ϕ L s,r (t)/t 1/p is quasi-increasing, so that, since f L s, and then f Λ 1,p (L s,r ) ≲ ∇f L p . Therefore, if q ≤ p, then we obtain the selfimprovement f L s,p ≃ f Λ 1,p (L s,r ) ≲ ∇f L p,q (f ∈ Lip 0 (Ω)).

SOBOLEV-POINCARÉ ESTIMATES FOR TWO MEASURE SPACES
In [CoMa], characterizations for Sobolev-Lorentz-type inequalities involving two measures are proved, extending results obtained in [M05] and [M06]. Here, we extend those results and derive, with similar methods, necessary and sufficient conditions for such Sobolev-type inequalities involving two rearrangement invariant spaces subjected to appropriate convexity conditions. From now on, µ and ν are two Borel measures on Ω, and 0 < p < ∞. Let X be a quasi-Banach function space on Ω, Y be an r.i. space on (Ω, µ), and Z be an r.i. space on (Ω, ν).
Theorem 5.1. Suppose that X satisfies a lower p-estimate. Then, the following properties are equivalent: (i) There is a constant A > 0 such that (ii) There exists a constant B > 0 such that Proof. (i) ⇒ (ii) Choose g ⋐ G ⋐ Ω, and consider f ∈ W (ḡ, G) arbitrary.

EXTENSION TO CAPACITARY FUNCTION SPACES
Let us now extend our results to the capacitary function spaces setting considered in [Ce], [CMS] and [CMS1]. By a capacity C on a measurable space (Ω, Σ) we mean a set function defined on Σ satisfying at least the following properties: With this notation, Theorem 3.2 states that if X satisfies a lower p-estimate, then f L 1,p (Cap X ) ≲ ∇f X (f ∈ Lip 0 (Ω)).
Let us denote by C (p) := C 1/p the p-convexification of C (see [Ce]). Theorem 6.1. Suppose 0 < p, s, q < ∞, and let C andC be two capacities on (Ω, Σ). If X satisfies a lower q-estimate, then the following properties are equivalent: (i) f L p,q (C) ≲ ∇f X + f L s,q (C) for every f ∈ Lip 0 (Ω).
(ii) ⇒ (i) Consider f ∈ Lip 0 (Ω), and take for a > 1 and t > 0 the open sets, g := M at and G := M t . By hypothesis we have C (p) (M at  ≤ ∇f X + f L s,q (C) .

❐
In a similar way, we obtain the following result. Theorem 6.2. Let 0 < p, q < ∞. Suppose that X satisfies a lower q-estimate, and let C be a capacity on (Ω, Σ). The following properties are equivalent: (i) f L p,q (C) ≲ ∇f X for every f ∈ Lip 0 (Ω).
(ii) C (p) (g) ≲ Cap X (ḡ, G) if g ⋐ G ⋐ Ω. We say that the capacity C is Fatou if C(A n ) → C(A) whenever A n ↑ A and that it is concave if The capacity C is said to be µ-invariant, where µ is a measure on (Ω, Σ), if C(A) = C(B) whenever µ(A) = µ(B), and it is said to be quasi-concave with respect to µ if there exists a constant γ ≥ 1 such that, whenever µ(A) ≤ µ(B), the following two conditions are satisfied: (
Indeed, since ϕ X is continuous except possibly at the origin, C is a Fatou capacity on (Ω, m n ) and L p (C) is complete. Moreover, C is m n -invariant and quasi-concave with respect to m n , and, by [CMS,Theorem 8], there exists a Fatou concave capacity C 1 which is equivalent to C. For such a capacity, L p (C 1 ) is a normed space.