CONDUCTOR SOBOLEV TYPE ESTIMATES AND ISOCAPACITARY INEQUALITIES

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1 CONDUCTOR SOBOLEV TYPE ESTIMATES AND ISOCAPACITARY INEQUALITIES JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE Abstract. In this paper we present an integral inequality connecting a function space (quasi-)norm of the gradient of a function to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the estimates 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 inequalities involving two measures, necessary and sufficient conditions for Sobolev isocapacitary type inequalities, and self-improvements for integrability of Lipschitz functions. 1. Introduction If Lip (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, cap(k, Ω) = inf f 1, f=1 on K f 2 2 (f Lip (Ω)), extended in the obvious way for any p 1 as the p-capacity cap p (K, Ω) = inf f 1, f=1 on K f p p was used in [M5] to obtain the Sobolev inequality (f Lip (Ω)), cap p ( M at, M t )d(t p ) c(a, p) f p p, (1) where M t is the level set {x Ω; f(x) > t} (t > ). This conductor inequality is a powerful tool with applications to Sobolev type imbedding theorems, which for p > 1 plays the same role as the co-area formula for p = 1. 2 Mathematics Subject Classification. Primary 46E3; Secondary 28A12. Key words and phrases. Convexity, lower estimates, Sobolev spaces, rearrangement invariant spaces, Sobolev-type inequalities. Partially supported by Grant MTM

2 2 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE With its variants, (1) has many applications to very different areas, such as Sobolev inequalities on domains of R n and on metric spaces, to linear and nonlinear partial differential equations, to calculus of variations, to Markov processes, etc. (See e.g. [AH], [AP], [Ci], [Da], [DKX], [Han], [K84], [MM], [MM1], [MM2], [M85], [M11], [M5], [M6], [MN], [MP], [Ra], [V99], and the references therein). An interesting extension based on the Lorentz space L p,q (Ω) (1 < p <, 1 q ) has been recently obtained in [CoMa], showing that cap p,q (M at, M t )d(t p ) c(a, p, q) f p L p,q (Ω) (1 q p) and where now cap p,q (M at, M t ) q/p d(t q ) c(a, p, q) f q L p,q (Ω) (p < q < ), cap p,q (K, Ω) = inf f 1, f=1 on K f p L p,q (Ω) (f Lip (Ω)). 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. Using the fact that the constant in the right hand side of the inequalities is one, they can be extended to an arbitrary set of disjoint functions, and L p,q satisfies lower estimates with constant one (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 certain particular kind of spaces because of the lower p-estimates with constant one, 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

3 CONDUCTOR SOBOLEV TYPE ESTIMATES 3 be applied to many examples, which include Lebesgue spaces, Lorentz spaces, classical Lorentz spaces, Orlicz spaces, 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 we present the most classical examples of spaces, not necessarily rearrangement invariant, satisfying this kind 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 < p <, we show the equivalence of the following properties: (i) For every compact set K on Ω, ϕ Y (µ(k)) Cap X (K). (ii) f Λ 1,p (Y ) f X (f Lip (Ω)). (iii) f Λ 1, (Y ) f X (f Lip (Ω)), where ϕ Y denotes the fundamental function of Y and Λ p,q (Y ) ( < q ) a Lorentz space, defined in Section 4. Moreover, under the appropriate conditions on Y, we show that f Λ 1, (Y ) f X f Λ 1,p (Y ) f X f Y f X. In the particular case when X = L p, p (1, n), and Y = L s with s = np we recover the well-known self-improvement of integrability of n p Lipschitz functions f L s,p = f Λ 1,p (L s ) f L p. 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], [M5] and [M6] 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 > (independent of all parameters involved) such that f cg, and f g means that f g f. 2. Preliminaries 2.1. Function spaces. Let (Ω, µ) be a measure space and L (Ω) the vector space of all (equivalence classes of) measurable real functions on

4 4 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE Ω. We shall say that X is a quasi-banach function space if it is a quasi-banach linear subspace of L (Ω) with the following properties: (1) (Lattice property) Given g X and f L (Ω) such that f g, then f X and f X g X. (2) (Fatou property) f n f a.e. f n X f X. For < 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 {f i } n X, ( n ) 1/p ( f i p n ) 1/p M f i p (2) or ( n ) 1/p ( f i p n ) 1/p M f i p, (3) 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 convenience, let us present some examples of spaces satisfying this kind of properties. As usual, if f L (Ω), f will denote the non-increasing rearrangement of f defined by f (t) = inf {λ > ; µ {x Ω; f(x) > λ} t}, and f (t) := t 1 t 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 (Ω) and g f imply g X and g X f X. A function F : (, ) (, ) is called quasi-increasing (resp. quasi-decreasing) if F (s) F (t) (resp. F (t) F (s)) for any < s < t. Moreover, F is said quasi-superadditive if there exists a constant d > such that F (x) + F (y) df (x + y) for all < x, y <, and it is said superadditive when d = 1. Example 2.1. The space X is said to be p-concave (resp. p-convex) if (3) (resp. (2)) holds for arbitrary functions. Since n g i p = n g i p when {g i } n X are disjointly supported, if X is p concave, then X satisfies a lower p-estimate. Example 2.2 (Lorentz spaces). Suppose < p < and let w be a weight on (, ) satisfying the 2 -condition 2t w(s)ds t w(s)ds,

5 CONDUCTOR SOBOLEV TYPE ESTIMATES 5 so that the classical Lorentz space Λ p (w) = { f L (Ω); f Λ p (w) := ( 1/p } f (x) p w(x) dx) <, is a quasi-banach space (see e.g. [CRS, Section 2.2]). It is well known that Λ p (w) is p-convex with constant one when w is decreasing, and p-concave with constant one when w is increasing (see [KM]). If w is decreasing, by [KP, Theorem 1], for r > p and p/r + 1/s = 1, the r-concavity constant of Λ p (w) is [ ( 1 t sup t> t ws ) 1/s ] 1/p. 1 t t w Moreover, if < x w(t)dt < and x t p w(t)dt <, then, for p r <, Λ p (w) satisfies a lower r estimate if and only if is quasi-increasing. t t p/r w(s)ds These spaces generalize many known spaces in the literature. For instance, if w(t) := t p/q 1 (1 + log(t)) λp, then we obtain the Lorentz- Zygmund space, that is, Λ p (w) = L q,p (LogL) λ (Ω) (see e.g. [BR]). More generally, a positive function b is said to be slowly varying on (1, ) (in the sense of Karamata), if for each ε >, t ε b(t) is quasiincreasing and t ε b(t) is quasi-decreasing. For example b(t) = exp( log t) and b(t) = (e + log t) α (log(e + log t)) β, with α, β R, are slowly varying. If w(t) = t q/p 1 b(1/t) q on ( < t < 1) with b slowly varying, then Λ q (w) is the Lorentz-Karamata space L p,q,b (Ω) (see e.g. [Nev]). Example 2.3 (Γ p (w)). Suppose that the weight w satisfies the nondegeneracy conditions 1 s p w(s) ds = w(s) ds =. 1 If t p t w(s)ds s p w(s) ds and 1 < p r <, then t { ) 1/p } Γ p (w) = f L (Ω); f Γ p (w) := f (x) p w(x) dx < satisfies a lower r estimate if and only if t p(1 1/r) s p w(s) ds is t quasi-increasing.

6 6 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE For < p 1 and r 1, or for 1 < p < r <, Γ p (w) is r concave if and only if t p(1 1/r) ε s p w(s) ds is quasi-increasing for some ε >. For details see [KM1]. t Example 2.4 (Orlicz spaces). Let φ be a Young function and consider the Luxemburg norm defined by { µ(ω) } f φ := inf ε > ; φ( f(t) /ε)dt 1. For < q <, L φ (Ω) satisfies a lower q estimate if and only if φ(λu) λ q φ(u) for all λ 1 and all u. Suppose µ(ω) < and 1 < p 2 q < r <. If φ(λu) λ q φ(u) for all λ 1 and u u, then L φ (Ω) is r-concave. If µ(ω) =, then the above inequalities need to be satisfied for all u. For details see [K1] and [K2]. Function spaces that are not rearrangement invariant may also be considered: Example 2.5 (Mixed norm L p spaces). The space L q (Ω 2 )[L p (Ω 1 )] for 1 p, q, defined by the condition f := ( ( f(x, y) p dµ 1 (x)) q/p dµ2 (y)) 1/q <, satisfies a lower pq estimate with constant one. Indeed, if f and g are two disjointly supported functions, it follows from [BP, Theorem 1] that f + g pq f pq + g pq. Similarly, in the case L pn (µ n )[... [L p 1 (µ 1 )]] of n parameters we have a lower p 1 p n estimate with constant one. Example 2.6 (Mixed norm weighted Lorentz spaces). Suppose 1 p, q < and, for a measurable function f on Ω = Ω 1 Ω 1, denote fy (x, t) 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 u(t) dt is quasi-superadditive. Then the space Λ q (v)[λ p (u)] defined by the condition [ (s) ] q/pv(s)ds ) 1/q f Λ q (v)[λ p (u)] := (fy (, t)) u(t)dt) p <

7 CONDUCTOR SOBOLEV TYPE ESTIMATES 7 also satisfies a lower pq estimate. Let a 1. An application of Hölder s inequality gives ( x p + y p ) 1/p a 1 1 p x + (1 a) 1 1 p y (1 p < ). So 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 ( ) 1/p M (p) (Λ p (u)) f + g Λ p (u) f p Λ p (u) + g p Λ p (u) a 1 1/p f Λ p (u) + (1 a) 1 1/p g Λ p (u), if a 1. Let now f, g Λ q (v)[λ p (u)] be disjointly supported. Then, M (p) (Λ p (u)) f + g Λ q (v)[λ p (u)] [ ) (s) ] q/pv(s)ds ) 1/q = M (p) (Λ p (u)) (f + g) y(, t) p u(t)dt ) 1/q [a 1 1/p f y (s) q Λ p (u) + (1 a)1 1/p g y (s) q Λ p (u) ]v(s)ds a 1 1 pq f Λ q (v)[λ p (u)] + (1 a) 1 1 pq g Λ q (v)[λ p (u)]. Finally, choosing it follows that a = M (p) (Λ p (u)) f + g Λ q (v)[λ p (u)] f pq Λ q (v)[λ p (u)] f pq Λ q (v)[λ p (u)] + g pq Λ q (v)[λ p (u)] ( ) 1 f pq Λ q (v)[λ p (u)] + g pq pq Λ q (v)[λ p (u)]. Observe that, if U is superadditive, then M (p) (Λ p (u)) = 1. Remark 2.7. In Example 2.2, if w is decreasing (resp. increasing), then for all q > p, Λ p (w) is q-concave with constant one (resp. q- convex with constant one for all < q < p) if and only if Λ p (w) is isometric to L p. See [KP, Corollary 4]. For a Lorentz-Karamata space, we have (see [EP]) [ ] qdt ) 1/q f p,q,b = t 1/p 1/q f (t)b(t) = 1/q f (t) q t q/p 1 b(t) dt) q = f Λ q (w), where w is the weight defined as w(s) := s q/p 1 b(s) q, s >. For q > p, W (x) = x w(s)ds is quasi-superadditive and, by [CS, Lemma 3.2],,

8 8 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE Λ p (w) satisfies a lower q-estimate. For q p, by [KM, Theorem 6], Λ p (w) is not q-concave. A function φ is said to satisfy (RC) if φ(au) φ(u) + φ((1 a)v) φ(v) 1 for all u, v > and < a < 1. Assume that φ is an Orlicz function, that is, φ is strictly increasing and continuous with lim u φ(u) =, φ() = and φ(1) = 1. When µ(ω) <, if φ(u 1/p ) satisfies (RC), by [HKT, Corollary 3.3], L φ (Ω) satisfies a lower p-estimate with constant one. If µ(ω) =, then φ(u 1/p ) satisfies (RC) if and only if L φ (Ω) satisfies a lower p-estimate with constant one Capacities. Let Ω be a domain of R n endowed with the Lebesgue measure m n, Lip(Ω) the class of Lipschitz functions on Ω, and Lip (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 W (K, G) := {u Lip (G); u = 1 on a neighbourhood of K, u 1}. 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], [M5] and [Co]) we have: If K 1 K 2 are compact sets in G, Cap X (K 1, G) Cap X (K 2, G). If Ω 1 Ω 2 are open and K is a compact subset of Ω 1, then Cap X (K, Ω 2 ) Cap X (K, Ω 1 ). If {K i } is a decreasing sequence of compact subsets of G with K := K i, then Cap X (K, G) = lim i Cap X (K i, G). If {Ω i } is an increasing sequence of open subsets of Ω with Ω := Ω i and K is a compact subset of Ω 1, then Cap X (K, Ω) = lim i Cap X (K, Ω i ). We will write Cap X ( ) = Cap X (, Ω) if Ω has been fixed.

9 CONDUCTOR SOBOLEV TYPE ESTIMATES 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( ) = and φ(a) φ(b) whenever A B, and that φ is normalized when φ(ω) = 1. A monotone set-function φ is a submeasure if φ(a B) φ(a) + φ(b) whenever A, B A are disjoint, and φ is a supermeasure if φ(a B) φ(a) + φ(b) whenever A, B A are disjoint. For any < 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 < 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 K p = 2 1. (2 p 1/p 1) For a more complete treatment, see [KMo] and the references quoted therein. 3. Sobolev capacitary inequalities In this section we will present our extensions of [CoMa, Theorem 4.2] to an arbitrary parameter p, < p <, and X a Banach or quasi-banach function space which satisfies a lower p-estimate. Theorem 3.1. Suppose < p < and let a > 1 be a constant. If X is a Banach function space that satisfies a lower p estimate, then ) p t p dt Cap X ({ f > at}, { f > t} t c f p X (f Lip (Ω)), (4) where c is a constant that depends on a, p and M (p) (X). In particular, t p Cap X ({ f t}) p dt t 2p c f p X (f Lip (Ω)), (5) where c depends on p and M (p) (X). Proof. Without loss of generality, assume that f X <, and that f, since f f.

10 1 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE Since X is a Banach function space, the set-function φ(a) := f χ A X f X (A B(Ω)) is a submeasure. Moreover, using that X satisfies a lower p estimate, we conclude that, if A 1,, A 1 are disjoint, then φ(a 1 A n ) 1 M (p) (X) (φp (A 1 ) + + φ p (A n )) 1/p. (6) Let us consider the set-function ψ, defined by { n } ψ(a) := sup φ p (A i ), (7) the supremum being taken over all finite partitions (A 1,, A n ) of A. It follows from (6) and (7) that ψ ( M(p) (X) ) p φ p ψ, (8) and we claim that ψ is a supermeasure that satisfies an upper min(p, 1/p) estimate. Indeed, given any ε > and two disjoint sets A and B, choose finite partitions A = n a A i, B = n b j=1 B j such that n a n b ψ(a)(1 ε) φ p (A i ) and ψ(b)(1 ε) φ p (B j ). Then {D k } na+n b k=1 = {A k } na k=1 {B k} n b k=1 satisfies j=1 is a partition of A B which n a n b ψ(a)(1 ε) + ψ(b)(1 ε) φ p (A i ) + φ p (B j ) n a+n b k=1 j=1 φ p (D k ) ψ(a B) 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.

11 CONDUCTOR SOBOLEV TYPE ESTIMATES 11 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, ( ) ( ) 1/p 1/p φ p (C i ) = φ p ((C i A) (C i B)) i ( ( φ(ci A) + φ(c i B) ) p i = {φ(c i A) + φ(c i B)} n l p = ) 1/p {φ(c i A)} n l p + {φ(c i B)} n l p ( ) 1/p ( 1/p φ p (C i A) + φ p (C i B)) ψ(a) 1/p + ψ(b) 1/p. Therefore, taking the supremum over all partitions we obtain that ψ(a B) = sup φ p (C i ) ( ψ(a) 1/p + ψ(b) 1/p) p 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 we have that ( ) p φ p (C i ) (x + y) p x p + y p (x, y ), ( ( φ(ci A) + φ(c i B) ) p i ) p ( ) p ( p φ p (C i A) + φ p (C i B)) ψ(a) p + ψ(b) p. Therefore, taking the supremum over all partitions we obtain that ψ(a B) = sup φ p (C i ) (ψ(a) p + ψ(b) p ) 1/p and ψ p is a submeasure. We normalize ψ and define ϕ(a) := ψ(a) ψ(ω), a normalized supermeasure which satisfies an upper r estimate. Thus, by [KMo, Theorem 2.2], there is a measure µ on Ω such that ϕ µ and µ(ω) K r. (9)

12 12 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE Now, if M t := { f > t} = {f > t}, the function γ(t) := µ(m t ) is decreasing on (, ) and the limits γ() and γ( ) exist, so that (γ(t) γ(at)) dt t = lim ε,n N as an improper integral. We have µ(m t ) = µ(m t \ M at ) + µ(m at ) and therefore µ(m t \ M at ) dt t = By (8) and (9) we obtain so that µ(m t \ M at ) dt t Consider now 1 ψ(ω) = 1 ψ(ω) f χmt\m at p X ε (γ(t) γ(at)) dt t (µ(m t ) µ(m at )) dt t = µ(m ) log a. ϕ(m t \ M at ) dt t = φ p (M t \ M at ) dt t f χmt\m at p X f p X ψ(m t \ M at ) dt ψ(ω) t dt t, dt t ψ(ω)µ(m ) log a f p X (1) K r M (p) (X) p log a f p X. { } ( f t)+ Λ t (f) = min (a 1)t, 1. Since f Lip (Ω), an easy computation shows that Λ t (f) = 1 (a 1)t f χ M t\m at and obviously f χmt\mat p = (a X 1)p t p Λ t (f) p X. Moreover, since Λ t (f) W (M at, M t ), f χmt\m at p (a X 1)p t p Cap X (M at, M t ) p, and the proof of (4) with c := c(a, p, M (p) (X)) = M (p)(x) p K r log a (a 1) p

13 CONDUCTOR SOBOLEV TYPE ESTIMATES 13 ends by inserting the last estimate in the left hand side of (1). 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 one. Hence, the result follows with similar arguments to those in [CoMa]. The capacitary inequality (5) follows using (4) 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 using Aoki-Rolewicz s Theorem (see e.g. [BL, Section 3.1]): Theorem 3.2. Suppose < p < and let a > 1 be a constant. If X is a quasi-banach function space which satisfies a lower p estimate, then t p Cap X ({ f > at}, {f > t}) p dt t c 1 f p X (f Lip (Ω)), 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, t p Cap X ({ f t}) p dt t 2p c 1 f p X (f Lip (Ω)), 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 ϱ is defined as (2c) ϱ = 2, there is a 1-seminorm such that, for all f X, f f ϱ X 2 f. Endowed with this 1-seminorm X satisfies a lower p/ϱ estimate, since, if f 1,, f n are disjointly supported functions in X, then ( n ) ϱ/p ( n ) ϱ/p ( f i ) p/ϱ f i p X M(p) (X) ϱ n f i Now consider 2M (p) (X) ϱ n f i. { n } ψ(a) = sup φ p/ϱ (A i ) ϱ X

14 14 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE with φ(a) = f χ A f. 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)) = 22p/ϱ K r log am (p) (X) p (a 1) p, for ϱ such that (2c) ϱ = 2 and r = min(p/ϱ, ϱ/p). 4. 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 µ f (λ) := µ{x Ω : f(x) > λ}, (λ ), and the fundamental function of X (see [BS] and [BrK]) is defined by ϕ X (t) = χ A X, (t = µ(a)). Given < p < and < q, the Lorentz space Λ p,q (X) associated to X is defined as { f L (Ω); f Λ p,q (X) = pt q 1( ϕ X (µ f (t)) ) q ) 1 } q p dt < with the usual changes when q =. When p = q we write Λ p (X) instead of Λ p,p (X). Notice that if X = L 1, then Λ p,q (L 1 ) = L p,q. It is well known that for < q q 1 Λ p,q (X) Λ p,q 1 (X). Moreover, if X is a Banach space, then Λ 1 (X) X Λ 1, (X) 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 f X f L 1 (f Lip (R n )) if and only if, for every Borel set A, ϕ X (m n (A)) m + n (A), where m + n is Minkowski s perimeter (see [M11] or [EG]).

15 CONDUCTOR SOBOLEV TYPE ESTIMATES 15 As shown in [MM2], the following self-improvement property follows for f Lip (R n ) f Λ 1, (X) f L 1 f X f L 1 f Λ 1 (X) f L 1. This Sobolev self-improvement obtained in the case q = 1 is also extended to the case q > 1 as f Λ q, (X) f L q f Λ 1,q (X) f L q. In particular, if X is q convex, then the space is an r.i. space and X (q) = {f; f 1/q X}, f X(q) = f 1/q q X Λ 1,q (X) = Λ q (X (q) ) X (q) Λ 1, (X (q) ). In summary, in terms of the X (q) scale of spaces, on Lipschitz functions we have the following equivalences (see [MMP]) f Λ 1, (X (q) )) f L q f Λ q (X (q) ) f L q f X(q) f L q. 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 sup ϕ Y (µ(g)) Cap X (g, G) <, 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). Proof. Let K be a compact subset in Ω and d := d(k, Ω c ) >. Denote λ n = 1/n and consider the smallest n N, n, such that 1/n d. For each n n, let G(λ n ) := {x Ω; d(k, x) < λ n }, K(λ n ) := {x Ω; d(k, x) λ n }. Then n n G(λ n ) = K and n n K(λ n ) = K. Since ϕ Y (µ(g(λ k ))) Cap X (G(λ k )) (k n ),

16 16 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE by the properties of Cap X, ϕ Y (µ(k)) lim k ϕ Y (µ(g(λ k ))) lim k Cap X (G(λ k )) = Cap X (K), and the result follows. Theorem 4.2. Let < p < and assume that X satisfies a lower p estimate. Then the following properties are equivalent: (i) ϕ Y (µ(k)) Cap X (K) for every compact set K on Ω. (ii) f Λ 1,p (Y ) f X (f Lip (Ω)). (iii) f Λ 1, (Y ) f X (f Lip (Ω)). Moreover, for q p, if Y is q-convex or, if Y satisfies an upper q estimate and ϕ Y (t)/t 1/p is quasi-increasing, then, for every f Lip (Ω), f Λ 1, (Y ) f X f Λ 1,p (Y ) f X f Y f X. (11) Proof. (i) (ii) If a > 1, by Theorem 3.2 we have ) 1/p t p 1 ϕ X (µ(m t )) p dt f Λ 1,p (X) ) 1/p t p 1 Cap X (M t ) p dt 1/p a s p 1 Cap X (M as, M s ) ds) p f X. (ii) (iii) Observe that Λ 1,p (X) Λ 1, (X). (iii) (i) Trivial using Proposition 4.1. To prove (11), if Y is q-convex, then ϕ Y (t) q is quasi-decreasing and f Y = f q 1/q Y (q) f q 1/q Λ q (Y (q) ) = f q 1/q Λ 1,q (Y ). Then (11) follows. If Y satisfies a upper q estimate and ϕ Y (t)/t 1/p is quasi-increasing, then it also satisfies a upper p estimate and then, for every simple function s = i a iχ Ai with A i A j = if i j we obtain Y ( s Y = a i χ Ai = ) 1/p a p i χ Y A i i i ( ) 1/p ( ) 1/p. M (p) (X) a i χ Ai p Y = M (p) (X) a i p ϕ Y (µ(a i )) p i Since ϕ Y is also the fundamental function of Λ 1,p (Y ) and ϕ Y (t)/t 1/p is quasi-increasing, we know that Λ 1,p (Y ) satisfies a lower p estimate t i

17 CONDUCTOR SOBOLEV TYPE ESTIMATES 17 (see [KM, Theorem 8]). Hence ( ) 1/p ( a p i ϕ Y (µ(a i )) p = a i χ Ai p Λ 1,p (Y ) i i ) 1/p ( ) 1/p a p i χ Λ A i 1,p (Y ) = s Λ 1,p (Y ). Then, by the Fatou property, for every positive function f we have f Y f Λ 1,p (Y ). Therefore, if f Λ 1,p (Y ) f X, then f Y f X. Conversely, if f Y f X, then, since Y Λ 1, (Y ), it follows that f Λ 1, (Y ) f X, and we conclude that f Λ 1,p (Y ) f X. Remark 4.3. Theorem 4.2 is to be compared with the results in [MM2, Section 1]. Let us consider some examples: It is well known that the Gagliardo- Nirenberg inequality f L n/(n 1) f L 1 allows us to see that, if p (1, n), s = f s L s = f α n/(n 1) L n 1 n, then f s(n 1)/n L s i (f Lip (Ω)), np n p (n 1)s and α =, since α f α 1 f L 1 f s/p L s f L p, where p is the conjugate exponent of p. Hence f L s f L p. Therefore, since L s L s,, it follows that f L s, f L p. 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 (Ω)), and we have obtained a self-improvement. If p = n, then we start from the Trudinger inequality, ( t f ) (s) n n 1 n n 1 t(1 + log 1) f L n, t which gives the estimate ( 1 ) 1 n n ϕ(µ(k)) = 1 + log µ(k) and then f Λ 1,n (ϕ) f L n. Cap L n(k), n

18 18 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE But, ) 1/n ( 1 ( Λ 1,n (ϕ) = t n 1 (ϕ(µ f (t))) n f (s) ) n ds dt = (1 + log 1) s s ) 1/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, = f Λ 1, (L s ) f L p, f L s, f Λ 1, (L s,r ) f L p f L p,q (q p), and then f Λ 1,p (L s,r ) f L p. Therefore, if q p, then we obtain the self-improvement f L s,p f Λ 1,p (L s,r ) f L p,q (f Lip (Ω)). 5. Sobolev-Poincaré estimates for two measure spaces In [CoMa], characterizations for Sobolev-Lorentz type inequalities involving two measures are proved, extending results obtained in [M5] and [M6]. 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 < p <. Let X be a quasi-banach function space on Ω, Y 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 > such that f Λ 1,p (Y ) A( f X + f Λ 1,p (Z)) (ii) There exists a constant B > such that ϕ Y (µ(g)) B(Cap X (g, G) + ϕ Z (ν(g)) (f Lip (Ω)). (g G Ω). Proof. (i) (ii). Choose g G Ω and consider f W (g, G) arbitrary. Since g {f 1}, it follows that with ϕ Y (µ(g)) p 1 ϕ Y (µ({f 1})) p dt p 1 p f p Λ 1,p (Y ) f p X + f p Λ 1,p (Z), ϕ Y (µ({f > t})) p dt p 1 f p Λ 1,p (Z) t p 1 ϕ Z (ν(g)) p dt = 1 p ϕ Z(ν(G)) p, and (ii) follows by taking infimum over all functions f W (g, G).

19 CONDUCTOR SOBOLEV TYPE ESTIMATES 19 (ii) (i). From M t = { f > t} supp (f) and M at M t if a > 1, ϕ Y (µ(m at )) p Cap X (M at, M t ) p +ϕ Z (ν(m t )) p, and Theorem 3.2 yields ) 1/p f Λ 1,p (Y ) = a p 1 s p 1 ϕ Y (µ(m as )) p ads + { a ) 1/p s p 1 Cap X (M as, M s ) p ds ) 1/p } s p 1 ϕ Z (ν(m s )) p ds f X + f Λ 1,p (Z). 6. 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: (a) C( ) =, (b) C(A), (c) C(A) C(B) if A B, and (d) Quasi-subadditivity: C(A B) c(c(a) + C(B)), where c 1 is a constant. Then the Lorentz spaces L p,q (C) are defined by the condition 1/q f L p,q (C) := q t q 1 C{ f > t} dt) q/p <. 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 (Ω)). Let us denote by C (p) := C 1/p the p-convexification of C (see [Ce]). Theorem 6.1. Suppose < p, s, q <, and let C and C 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 (Ω). (ii) C (p) (g) Cap X (g, G) + C (s) (G) for all sets g and G such that g G Ω. Proof. (i) (ii) Choose g G Ω and any f W (g, G). Then f L p,q (C) f X + f L s,p ( C), so that ( 1 ) 1/q C (p) (g) C (p) ({f > t}) q dt q f X + f L s,p ( C),

20 2 JOAN CERDÀ, JOAQUIM MARTÍN, AND PILAR SILVESTRE and f L s,p ( C) = ( 1 C{ f > s} p/s ds p) 1/p C(s) (G). Taking the infimum over all f W (g, G) we conclude that C (p) (g) Cap X (g, G) + C (s) (G). (ii) (i) Consider f Lip (Ω) and take for a > 1 and t > the open sets, g := M at and G := M t. By hypothesis we have C (p) (M at ) Cap X (M at, M t ) + C (s) (M t ), and then, by Theorem 3.2, ) 1/q s q 1 Cap X (M as, M s ) q ds f L p,q (C) In a similar way, + ) 1/q s q 1 C(s) (M s ) q ds f X + f L s,q ( C). Theorem 6.2. Let < 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 (Ω). (ii) C (p) (g) Cap X (g, 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 C(A B) + C(A B) C(A) + C(B) (A, B Σ). 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: (a) C(A) γ C(B), and (b) C(B) γ C(A). µ(b) µ(a) Assume that X is an r.i. quasi-banach space. If we define C(A) := ϕ X (m n (A)), then L p (C) = Λ p (X) is a Banach space. 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. Acknowledgment: We thank Mario Milman for having informed us about the article [CoMa].

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23 CONDUCTOR SOBOLEV TYPE ESTIMATES 23 [Nev] Neves, J. S. Spaces of Bessel-potential type and embeddings: the superlimiting case, Math. Nachr. 265 (24), [Ra] Rao, M. Capacitary inequalities for energy, Israel. J. of Math. 61 (1988), no. 1, [V99] Verbitksky, I. E. Nonlinear potentials and trace inequalities, Operator Theory, Advances and Applications, Vol. 11, The Maz ya Anniversary Collection, vol. 2, Birkhäuser (1999), Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Barcelona, Spain address: jcerda@ub.edu, pilar.silvestre@ub.edu Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Spain address: jmartin@mat.uab.cat