Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 22, 1997, 313–338

THINNESS IN NON-LINEAR POTENTIAL THEORY FOR NON-ISOTROPIC SOBOLEV SPACES

Tord Sjödin

University of Umeå, Department of Mathematics S-901 87 Umeå, Sweden; tord@abel.math.umu.se

Abstract. We consider non-isotropic Sobolev spaces H_{ρ,r,p} = H_{ρ,r,p}(R^{m} × R^{n}) of functions f on R^{d} = R^{m} × R^{n} of the form f = G_{ρ}^{m} ⊗ G_{r}^{n} ⋆ g. Here G_{ρ}^{m} and G_{r}^{n} are Bessel kernels or order ρ and r in R^{m} and R^{n} respectively and g belongs to L^{p}(R^{d}). We develop at least part of a potential theory for H_{ρ,r,p} that is analogous to the well-known non-linear L^{p} -potential theory for the Sobolev spaces H_{ρ,p}(R^{m}). We define thinness in H_{ρ,r,p}, prove the Choquet and Kellog properties and give a partial characterization of properties of certain non-linear potentials in terms of thinness.

0. Introduction

Let ρ ≥ 0, r ≥ 0, 1 < p < ∞ and let d = m+n, where m and n are positive integers. We define the non-isotropic Sobolev space H_{ρ,r,p} = H_{ρ,r,p}(R^{m} × R^{n}) as the linear space of functions f in R^{d} = R^{m} × R^{n} of the form

(0.1)  f = G_{ρ}^{m} ⊗ G_{r}^{n} ⋆ g.

Here G_{ρ}^{m} and G_{r}^{n} are Bessel kernels or order ρ and r in R^{m} and R^{n} respectively and g belongs to L^{p}(R^{d}) (see Section 1 for the exact definitions). Such spaces were recently used by P. Sjögren and P. Sjölin [SS] for p = 2 and n = 1 to study boundary values of time-dependent solutions of the Schrödinger equation.

It is our purpose to develop at least part of a potential theory for H_{ρ,r,p}. We try to do this in a way that is analogous to the well-known non-linear L^{p} -potential theory for the Sobolev spaces H_{ρ,p}(R^{d}) as presented for example by D.R. Adams [A], L.-I. Hedberg, Th. Wolff [HW], V.G. Maz’ya [Ma] and V.G. Maz’ya and T.O. Shaposhnikova [MS]. The current state of the non-linear potential theory is found in the new book by D.R. Adams and L.-I. Hedberg [AH].

Our starting point will be non-linear L^{p} -potential theory developed by N.G. Meyers [Me]. The first results in Section 2 are very general, but later on, in Sections 3, 4 and 6, we study more special situations.

We will consider kernels of the form

(0.2)  k(ξ, η) = k_{1}(x, y) · k_{2}(s, t),

1991 Mathematics Subject Classification: Primary 31C45; Secondary 46E35.

where ξ = (x, s) and η = (y, t) are points in R^{d} = R^{m} × R^{n} and k_{1} and k_{2} are kernels in R^{m} and R^{n} respectively. Taking as k the kernel G_{ρ}^{m} ⊗ G_{r}^{n} in (0.1) we get a non-linear potential theory for H_{ρ,r,p} that seems to be new.

We continue our study of the potential theory for H_{ρ,r,p} by introducing potentials W_{ρ,r,p}^{μ,δ} and 𝒲_{ρ,r,p}^{μ}, in analogy with the case of H_{ρ,p}(R^{m}) in [HW]. We define the corresponding capacities C_{ρ,r,p} and 𝒞_{ρ,r,p} and study their capacitary potentials and capacitary measures.

A set E ⊂ R^{d} is called C_{ρ,r,p} -thin at a point ξ_{0} ∈ \overline{E} if

∫_{0}^{δ} \frac{da}{a} ∫_{0}^{δ} \frac{db}{b}(\frac{C_{ρ,r,p}(E ∩ B(ξ_{0}; a, b))}{a^{m−ρp} · b^{n−rp}})^{p′−1} < ∞,

for some δ > 0. We can then prove that the Choquet and Kellog properties hold (Theorem 6.3), which generalizes the case of H_{ρ,p}(R^{m}) in [HW, Theorems 2 and 3].

The thinness of a set E at a point ξ_{0} ∈ \overline{E} is closely related to properties of suitable non-linear potentials. To study this problem we define another type of product capacity C_{ρ,r,p}^{⋆} (see Section 6 for the exact definition) and prove that if there exists a non-negative measure μ with compact support in B(ξ_{0}; δ, δ) such that

(0.3)  W_{ρ,r,p}^{μ,δ}(ξ_{0}) < lim inf_{ξ→ξ_{0}, ξ∈E} W_{ρ,r,p}^{μ,δ}(ξ),

for some δ > 0 and E and μ satisfy a cone condition at ξ_{0} , then E is thin at ξ_{0} relative to C_{ρ,r,p}^{⋆}, for 2 ≤ p < ∞ (Theorem 6.13).

The situation is a bit different when E is contained in one of the hyperplanes R^{m} × {s_{0}} or {x_{0}} × R^{n}. If for example E ⊂ R^{m} × {s_{0}} and 2 ≤ p < ∞ then (0.3) holds if and only if E is C_{ρ,p} -thin at x_{0} as a set in R^{m} (Theorem 6.4). Thus we here recover the case of H_{ρ,p}(R^{m}) in [HW]. For 1 < p < 2 we can only prove the if part.

The plan of this paper is as follows. Section 1 contains our definitions and notations. We begin Section 2 with reviewing the basic facts from non-linear L^{p} - potential theory and we study product kernels of the type (0.2). In Section 3 we apply these results to the non-isotropic Sobolev spaces H_{ρ,r,p}.

In Section 4 we look at the potential theory for H_{ρ,r,p} in another way. We define non-linear potentials W_{ρ,r,p}^{μ} and 𝒲_{ρ,r,p}^{μ} and show that they have equivalent energy integrals (Theorem 4.2). The rest of this section is devoted to a study of the potential theory for 𝒲_{ρ,r,p}^{μ}. Section 6 gives our treatment of thinness in H_{ρ,r,p}.

Thinness in non-linear potential theory for non-isotropic Sobolev spaces 315

1. Notation and definitions

We are going to use the notation and definitions from [Me] and [Sj]. Consider the d-dimensional Euclidean space R^{d} = R^{m} × R^{n} and denote points in R^{d} by ξ = (x, s), η = (y, t) and ζ = (z, u), where x, y, z ∈ R^{m} and s, t, u ∈ R^{n}. The Euclidean norm is written | · | and we let G and K denote open and compact sets respectively. A closed k -dimensional Euclidean ball is written B_{k}(w, r) and the k -dimensional Lebesgue measure of a set E is written |E|_{k}.

In R^{d} we define B(ξ; a, b) = B_{m}(x, a) × B_{n}(s, b) and we denote a rectangle by R = I × J, where I and J are cubes in R^{m} and R^{n} with their sides parallel to the coordinate axes. The side length of I is written l(I).

Measurability of sets and functions always refers to Lebesgue measure and we use standard notation for Lebesgue integrals. A function is called extended real valued if its values are real numbers or ±∞.

For 1 < p < ∞ we let L^{p}(R^{d}) be the linear space of extended real valued functions in R^{d} such that

∥f∥_{p} = (∫ |f(x, s)|^{p} dx ds)^{1/p}

is finite. The non-negative elements in L^{p}(R^{d}) are denoted by Lp_{+}(R^{d}).

A capacity in R^{d} is a non-negative set function C defined for all subsets of R^{d} such that: (i) C(ϕ) = 0, ϕ the empty set, and (ii) A_{1} ⊂ A_{2} implies that C(A_{1}) ≤ C(A_{2}).

Our terminology for measures and integrals is that in [Me]. A measure μ is the completion of an extended real valued and σ -additive set function defined on the Borel field, which is finite on compact sets. We say that μ is concentrated on the μ-measurable set A if μ(B) = 0 for all μ-measurable sets B in R^{d} \ A.

Let M be the space of Radon measures and let L_{1} be the Banach space of measures μ with finite total variation ∥μ∥_{1} . The non-negative elements are denoted by M_{+} and L1^{+} respectively and we take the usual weak topologies in M and L_{1} [Me, p. 258].

A kernel k = k(ξ, η) in R^{d} is a non-negative and lower semi-continuous function on R^{d} × R^{d}. When μ, ν belong to M we write

k(ν, μ) = ∫ k(ξ, η) dσ(ξ, η),

where σ is the tensor product σ = ν_{ξ} ⊗ μ_{η}. When ν = δ_{ξ} is Dirac measure at ξ we write k(ν, μ) = k(ξ, μ). Various constants are denoted by c or c(α, β, …) and a ∼ b means that a/b is bounded from above and below by positive finite constants.

2. Non-linear capacities and potentials

In this section we define the two set functions C_{k,p} and c_{k,p} and study their properties. Let 1 < p < ∞, k a kernel in R^{d} and let S denote the σ -algebra of sets which are measurable for all Borel measures in R^{d}. For any set A ⊂ R^{d} we define

C_{k,p}(A) = inf ∥f∥_{p}^{p},

where infimum is over all f ∈ Lp_{+}(R^{d}) such that k(ξ, f) ≥ 1, all ξ ∈ A. For any set A ∈ S we define

c_{k,p}(A) = sup ∥μ∥_{1},

where supremum is over all μ ∈ M_{+} which are concentrated on A and satisfies the inequality ∥k(μ, ·)∥_{p′} ≤ 1.

We will also use these capacities in other spaces than R^{d}, but that will be clear from the context. These capacities are studied in detail in [Me]. Among other things it is shown that all analytic sets are C_{k,p} -capacitable, i.e.

sup_{K⊂A} C_{k,p}(K) = C_{k,p}(A) = inf_{A⊂G} C_{k,p}(G),

and that C_{k,p}(A)^{1/p} = c_{k,p}(A), for all analytic sets A.

Now let K be a compact set with C_{k,p}(K) < ∞, then the following statements (i)–(vii) hold:

(i) There exists a unique f_{K} ∈ Lp_{+}(R^{d}) such that ∥f_{K}∥_{p}^{p} = C_{k,p}(K), (ii) There exists μ_{K} ∈ M_{+}(K) which satisfies ∥k(μ_{K}, ·)∥_{p′} ≤ 1, and

∥μ_{K}∥_{1}^{p} = C_{k,p}(K),

(iii) f_{K} and μ_{K} are related by

f_{K}(η) = ∥μ_{K}∥_{1} · k(μ, η)^{p′−1},

(iv) μ_{K} is supported on the set {ξ; V_{k,p}^{μ_{K}} (ξ) = 1}, where V_{k,p}^{μ_{K}} (ξ) = k(ξ, f_{K}),

(v) V_{k,p}^{μ_{K}} (ξ) ≤ 1, all ξ in the support of μ_{K},

(vi) V_{k,p}^{μ_{K}} (ξ) ≤ 1, C_{k,p} -q.e. ξ ∈ K,

(vii) C_{k,p}(K) = sup μ(K), where supremum is over all μ ∈ M_{+} , supported by K and satisfying V_{k,p}^{μ}(ξ) ≤ 1, in the support of μ.

The properties (i)–(vi) are proved in [Me], while (vii) can be proved as in [HW, Theorem 1]. Any measure μ_{K} in (ii) and the function V_{k,p}^{μ_{K}} in (iv) is called the C_{k,p} -capacitary measure and the C_{k,p} -capacitary potential for K respectively. For the rest of this paper we will consider a special type of kernels, called product kernels, and defined as follows.

Definition 2.1. Let k_{1} and k_{2} be kernels in R^{m} and R^{n} respectively. Then k(ξ, η) = k_{1}(x, y)·k_{2}(s, t) is called a product kernel in R^{d}, where as usual ξ = (x, s) and η = (y, t).

It is obvious that product kernels are kernels in our sense so that the general theory above applies. These kernels are in a natural way adopted to measure product sets.

Theorem 2.2. Let 1 < p < ∞ and let k = k_{1} · k_{2} be a product kernel in R^{d}. Let A_{1} and A_{2} be analytic sets in R^{m} and R^{n} respectively such that C_{k_{1},p}(A_{1}) < ∞ and C_{k_{2},p}(A_{2}) < ∞. Then

C_{k,p}(A_{1} × A_{2}) = C_{k_{1},p}(A_{1}) · C_{k_{2},p}(A_{2}).

The theorem follows in a straight manner from the definitions of the capacities and the relation between C_{k,p} and c_{k,p}. We omit the proof.

The C_{k,p} -capacitary potential has a particularily simple form for product sets.

Theorem 2.3. Let 1 < p < ∞ and let k = k_{1} · k_{2} be a product kernel in R^{d}. Let K_{1} ⊂ R^{m} and K_{2} ⊂ R^{n} be compact sets with C_{k_{1},p}(K_{1}) < ∞ and C_{k_{2},p}(K_{2}) < ∞. Further let μ_{1} be a c_{k_{1},p} -capacitary measure for K_{1} and let μ_{2} be a c_{k_{2},p} -capacitary measure for K_{2} . Then μ = μ_{1} ⊗ μ_{2} is a c_{k,p} -capacitary measure for K_{1} × K_{2} and

V_{k,p}^{μ}(ξ) = V_{k_{1},p}^{μ_{1}}(x) · V_{k_{2},p}^{μ_{2}}(s)

is a C_{k,p} -capacitary potential for K_{1} × K_{2} .

The first part of the proof follows from Theorem 2.2 and properties of the capacities. A simple calculation gives the formula for V_{k,p}^{μ}(ξ).

Remark. Replacing L^{p}(R^{d}) by the mixed norm Lebesgue space L^{p,q}(R^{d}), where 1 < p, q < ∞, defined by the norm

∥f∥_{p,q} = (∫_{R^{n}}( ∫_{R^{m}} |f(x, s)|^{p} dx)^{q/p} ds)^{1/q},

gives the analogous capacities C_{k,p,q} and c_{k,p,q}, see [Sj]. When p = q we recover the present case. The statements (i)–(viii) above, as well as Theorems 2.2 and 2.3, have natural counterparts in this more general situation, but will not be treated in this paper.

3. Sobolev spaces of mixed norm

In this section we define the Sobolev spaces of mixed norm and apply the results from Section 2. Recall that for any positive integer k and any real number α the Bessel kernel G_{α}^{k} in R^{k} can be defined by its Fourier transform

Ĝ_{α}^{k}(ζ) = (1 + |ζ|^{2})^{−α/2}

cf. [St, p. 130]. The usual Sobolev spaces (Bessel potential spaces) H_{α,p}(R^{k}), α > 0 and 1 ≤ p < ∞, are defined as the linear space of functions f = G_{α}^{k} ⋆ g, where g ∈ L^{p}(R^{k}), with norm ∥f∥_{α,p} = ∥g∥_{p}.

Definition 3.1. Let ρ ≥ 0, r ≥ 0 and 1 ≤ p ≤ ∞. Then H_{ρ,r,p}(R^{d}) is the linear space of all functions f = G_{ρ}^{m} ⊗ G_{r}^{n} ⋆ g, where g ∈ L^{p}(R^{d}), with norm ∥f∥_{ρ,r,p} = ∥g∥_{p}.

If ρ > 0 and r > 0 every function f in H_{ρ,r,p} has a representation

f(ξ) = ∫ G_{ρ}^{m}(x − y) · G_{r}^{n}(s − t) · g(y, t) dy dt,

for an essentially unique g ∈ L^{p}(R^{d}). The function k(ξ, η) = G_{ρ}^{m}(x−y)·G_{r}^{n}(s−t) is a product kernel in R^{d} and we denote the corresponding capacities C_{k,p} and c_{k,p} by C_{ρ,r,p} and c_{ρ,r,p} respectively. For the Sobolev spaces H_{l,p}(R^{k}) we denote their capacities by B_{l,p}^{k} and b_{l,p}^{k}, see [Me, p. 280].

We can now apply Theorem 2.2 to the present situation. In particular we have the following result when restricted to products of balls.

Theorem 3.2. Let ρ > 0, r > 0 and 1 < p < ∞. Then for any a > 0 and b > 0 we have

C_{ρ,r,p}(B_{d}(ξ; a, b)) = B_{ρ,p}^{m}(B_{m}(x, a)) · B_{r,p}^{n}(B_{n}(s, b)).

If we insert the values of B_{ρ,p}^{m} and B_{r,p}^{n} on balls into the formula in Theorem 3.2 we get

(3.1)  C_{ρ,r,p}(B_{d}(ξ; a, b)) ∼ a^{m−ρp} · b^{n−rp},  0 < a, b ≤ 1,

when 0 < ρ < m/p and 0 < r < n/p. If ρ = m/p or r = n/p we replace a^{m−ρp} and b^{n−rp} by (log(2/a))^{1−p} and (log(2/b))^{1−p} respectively.

Remark. If C_{ρ,r,p,q} denotes the capacity in L^{p,q}(R^{d}) relative to the kernel G_{ρ}^{m} ⊗ G_{r}^{n} formula (3.1) takes the form

C_{ρ,r,p,q}(B_{d}(ξ; a, b))^{1/p} ∼ a^{m/ρ−p} · b^{n/r−q}, 0 < a, b ≤ 1,

and if B_{α,p,q}^{d} is the capacity in L^{p,q}(R^{d}) for the ordinary Bessel kernel G_{α}^{d} we get

B_{α,p,q}^{d}(B_{d}(ξ, a)) ∼ \{ \begin{matrix}a^{p(m/p+n/q−α)}, & if 0 < α < m/p + n/q, \\ (log 2/a)^{−p/q′}, & if α = m/p + n/q, \\ 1, & if α > m/p + n/q, \end{matrix}

for 0 < a ≤ 1.

It is well known that when k is a non-negative integer and 1 < p < ∞, H_{k,p}(R^{m}) can be identified with the space of distributions f ∈ L^{p}(R^{m}) with norm |f|_{k,p} = ∑_{|α|≤k} ∥D^{α}f∥_{p}, see [St, Ch. 5, Theorem 3]. We prove the analogous result for H_{ρ,r,p}(R^{d}).

Let k and l be non-negative integers and 1 < p < ∞. Define the mixed norm Sobolev space W_{k,l}^{p}(R^{d}) as the linear space of functions f ∈ L^{p}(R^{d}) such that D_{x}^{α}D_{s}^{β}f(x, s) ∈ L^{p}(R^{d}), for all |α| ≤ k and |β| ≤ l, with norm

|f|_{k,l,p} = ∑_{|α|≤k, |β|≤l} ∥D_{x}^{α}D_{s}^{β}f∥_{p}.

We then have the following relation between H_{ρ,r,p}(R^{d}) and W_{k,l}^{p}(R^{d}).

Theorem 3.3. Let k and l be non-negative integers and 1 < p < ∞. Then H_{k,l,p}(R^{d}) = W_{k,l}^{p}(R^{d}), with equivalence of norms.

Proof. It is easily seen that the Schwartz class 𝒮 (R^{d}) is dense in both spaces. Let g ∈ 𝒮 (R^{d}) and define f = G_{k}^{m} ⊗ G_{l}^{n} ⋆ g. Then also f ∈ 𝒮 (R^{d}) and

D_{x}^{α}D_{s}^{β}f(x, s) = D_{x}^{α}(∫ G_{k}^{m}(x − y)D_{s}^{β}(∫ G_{l}^{n}(s − t)g(y, t) dt) dy).

First assume that k > 0 and l > 0. For any fixed s ∈ R^{n} and |β| ≤ l

∑_{|α|≤k} ∫ |D_{x}^{α}D_{s}^{β}f(x, s)|^{p} dx ∼ ∫ |D_{x}^{α}(∫ G_{l}^{n}(s − t)g(y, t) dt)|^{p} dy,

with constants independent of s ∈ R^{n}. Summing over all |β| ≤ l, integrating w.r.t. s over R^{n} and changing the order of integration and summation gives

\begin{aligned}∑_{|α|≤k, |β|≤l} ∫ |D_{x}^{α}D_{s}^{β}f(x, s)|^{p} dx ds & ∼∫ dy ∑_{|β|≤l} ∫ |D_{s}^{β}(∫ G_{l}^{n}(s − t)g(y, t) dt)|^{p}ds \\ & ∼∫ dy ∫ |g(y, t)|^{p} dt = ∥g∥_{p}^{p}, \end{aligned}

with constants independent of f and g. The cases when k = 0 or l = 0 are treated similarly. Theorem 3.3 is proved.

4. A non-linear potential theory for H_{ρ,r,p}(R^{m} × R^{n})

In this section we continue our study of a non-linear potential theory for H_{ρ,r,p}(R^{m} × R^{n}) in a different way. Hedberg and Wolff [HW] discovered in 1983 that the L^{p} -potential theory for the Sobolev space H_{ρ,p}(R^{m}) has an alternative formulation that parallels the classical potential theory and avoids some of the difficulties in earlier theories. It is our purpose here to carry out such a program also for the potential theory in H_{ρ,r,p}(R^{m} × R^{n}).

We are going to define these new non-linear potentials and capacities and establish their basic properties. Let us start with the following lower bound for the potential V_{ρ,r,p}^{μ}, cf. [HW, p. 164].

Lemma 4.1. Let ρ > 0, r > 0 and μ ∈ M_{+} . Then

V_{ρ,r,p}^{μ}(ξ) ≥ c(d, ρ, r, p)·∫_{0}^{∞}∫_{0}^{∞} μB(ξ; a, b)^{p′−1}·(G_{ρ}^{m}(4a)·G_{r}^{n}(4b))^{p′} ·a^{m−1}b^{n−1}da db.

For 0 < ρ < m, 0 < r < n, δ > 0 and μ ∈ M_{+} we define

W_{ρ,r,p}^{μ,δ}(ξ) = ∫_{0}^{δ} \frac{da}{a} ∫_{0}^{δ} \frac{db}{b}(\frac{μB(ξ; a, b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1}

and put W_{ρ,r,p}^{μ,δ} = W_{ρ,r,p}^{μ}, when δ = 1. It follows easily from Lemma 4.4 and properties of the Bessel kernel that V_{ρ,r,p}^{μ}(ξ) ≥ c(d, ρ, r, p)·W_{ρ,r,p}^{μ}(ξ), for all ξ ∈ R^{d}. The following partial converse turns out to be the key step in what follows.

Theorem 4.2. Let 0 < ρ ≤ m/p, 0 < r ≤ n/p, 1 < p < ∞ and μ ∈ M_{+} . Then

(4.1) ∫ V_{ρ,r,p}^{μ}(ξ) dμ(ξ) = ∥G_{ρ}^{m} ⊗ G_{r}^{n} ⋆ μ∥_{p′}^{p_{′}} ≤ c(d, ρ, r, p) · ∫ W_{ρ,r,p}^{μ}(ξ) dμ(ξ).

This was proved in [HW, Theorem 1] for the case of H_{ρ,p}(R^{m}). We postpone the proof of Theorem 4.2 to the next section in order to make our presentation easier to follow. We are now going to modify the potential W_{ρ,r,p}^{μ} in two more steps before we arrive at the potential 𝒲_{ρ,r,p}^{μ} that will be our main interest for the rest of this paper.

For each integer k we divide R^{m} into a net of non-intersecting congruent cubes with side length 2^{−k} by dividing every cube of side length 2^{−k} into 2^{m} cubes of side length 2^{−k−1} . Such cubes are called dyadic cubes in R^{m}. We divide R^{n} analogously and we call I × J a dyadic rectangle in R^{d}, where I and J are dyadic cubes in R^{m} and R^{n} respectively. For 0 < ρ ≤ m/p, 0 < r ≤ n/p, 1 < p < ∞ and μ ∈ M_{+} we define

W̃ _{ρ,r,p}^{μ}(ξ) = ∑_{l(I)≤1, l(J)≤1}(\frac{μ(I × J)}{l(I)^{m−ρ} · l(J)^{n−r}} )^{p′−1} · χ_{I×J}(ξ).

For a dyadic rectangle I × J we let ϕ_{I×J} be a C^{∞} -function supported in 3I × 3J such that 0 ≤ ϕ_{I×J}(ξ) ≤ 1, for all ξ ∈ R^{d} and ϕ_{I×J}(ξ) = 1 in I × J. We finally define

𝒲_{ρ,r,p}^{μ}(ξ) = ∑_{l(I)≤1, l(J)≤1}(\frac{μ(ϕ_{I×J})}{l(I)^{m−ρ} · l(J)^{n−r}} )^{p′−1} · ϕ_{I×J}(ξ),

where μ(ϕ_{I×J}) = ∫ ϕ_{I×J}(ξ) dμ(ξ). We also put

𝒥 (μ) = 𝒥_{ρ,r,p}(μ) = ∫ 𝒲_{ρ,r,p}^{μ}(ξ) dμ(ξ)

and call 𝒥 (μ) the energy integral associated with μ. As in [HW, p. 175] it follows from Theorem 4.2 and geometrical arguments that the four integrals

(4.2) ∫ V_{ρ,r,p}^{μ}(ξ) dμ(ξ), ∫ W_{ρ,r,p}^{μ}(ξ) dμ(ξ), ∫ W̃ _{ρ,r,p}^{μ}(ξ) dμ(ξ), ∫ 𝒲_{ρ,r,p}^{μ}(ξ) dμ(ξ)

are all equivalent with constants only depending on d, ρ, r and p.

The rest of this section is devoted to a formulation of a non-linear potential theory for 𝒲_{ρ,r,p}^{μ}. We let 0 < ρ < m, 0 < r < n and 1 < p < ∞. For any compact set K in R^{d} we define

𝒞_{ρ,r,p}(K)^{1/p} = sup{μ(K) ; μ ∈ M_{+}(K) and 𝒥 (μ) ≤ 1},

and we extend the definition of 𝒞_{ρ,r,p} in the usual way to an outer capacity on all sets. It follows from (4.2) that the capacities 𝒞_{ρ,r,p} and C_{ρ,r,p} are equivalent. Any measure μ ∈ M_{+}(K) such that 𝒥 (μ) ≤ 1 and μ(K) = 𝒞_{ρ,r,p}(K)^{1/p} is called a 𝒞_{ρ,r,p} -capacitary measure for K and 𝒲_{ρ,r,p}^{μ} is called a 𝒞_{ρ,r,p} -capacitary potential for K.

In the following we collect the properties of the 𝒞_{ρ,r,p} -capacity in a series of lemmas analogous to [HW, Propositions 1–9]. We first prove that 𝒞_{ρ,r,p} -capacitary measures and potentials exist for compact sets and have their usual equilibrium properties.

Lemma 4.3. Let K be a compact set. Then there exists γ ∈ M_{+}(K), γ(K) = 1 such that

(i) 𝒥 (γ) = 𝒞_{ρ,r,p}(K)^{1−p′},

(ii) 𝒲_{ρ,r,p}^{γ}(ξ) ≥ 𝒥 (γ), (ρ, r, p)-q.e. on K,

(iii) 𝒲_{ρ,r,p}^{γ}(ξ) ≤ 𝒥 (γ), everywhere on the support of γ.

The existence of such a γ satisfying (i) follows from the observation that

𝒞_{ρ,r,p}(K)^{−1} = inf{𝒥 (μ) ; μ ∈ M_{+}(K) and μ(K) = 1}

and a standard weak compactness argument. The properties (ii) and (iii) are proved as in [HW, Propositions 1 and 2].

We will also consider signed measures μ = μ_{+} −μ_{−}, where μ_{+} and μ_{−} belong to M_{+} and 𝒥 (μ_{+} + μ_{−}) < ∞. We define

𝒲_{ρ,r,p}^{μ}(ξ) = ∑_{l(I)≤1, l(J)≤1}(l(I)^{m−ρ} · l(J)^{n−r})^{1−p′} · |μ(ϕ_{I×J})|^{p′−2} · μ(ϕ_{I×J}) · ϕ_{I×J}(ξ),

and then 𝒥 (μ) = ∑_{l(I)≤1, l(J)≤1}(l(I)^{m−ρ} · l(J)^{n−r})^{1−p′} · |μ(ϕ_{I×J})|^{p′}. For λ > 0 we put E_{λ} = {ξ; 𝒲_{ρ,r,p}^{μ}(ξ) > λ or 𝒲_{ρ,r,p}^{μ_{+}+μ_{−}}(ξ) = ∞} then it follows that

𝒞_{ρ,r,p}(E_{λ}) ≤ \frac{1}{λ^{p}} · 𝒥 (μ),

cf. [HW, Proposition 3]. It is now a consequence of Lemma 4.3 that, at least for compact sets, the C_{ρ,r,p} -capacity can be defined in terms of the potential 𝒲_{ρ,r,p}^{μ}. For a proof see [HW, Propositions 4 and 5].

Lemma 4.4. Let K be a compact set then

(i) 𝒞_{ρ,r,p}(K) = inf{𝒥 (μ) ; μ ∈ M_{+} and 𝒲_{ρ,r,p}^{μ}(ξ) ≥ 1, (ρ, r, p)-q.e., ξ ∈ K}, (ii) 𝒞_{ρ,r,p}(K) = sup{μ(K) ; μ ∈ M_{+}(K) and 𝒲_{ρ,r,p}^{μ}(ξ) ≤ 1, ξ ∈ supp μ}.

The non-linear potentials 𝒲_{ρ,r,p}^{μ}, where μ ∈ M_{+} and 𝒥 (μ) < ∞, are C_{ρ,r,p} - quasi continuous in the following sense: For every ε > 0 there is an open set G such that 𝒞_{ρ,r,p}(G) < ε and the restriction of 𝒲_{ρ,r,p}^{μ}(ξ) to the closed set R^{d} \ G is continuous on R^{d} \ G. Cf. [HW, Proposition 6].

We conclude this section with an equilibrium theorem for sets of finite 𝒞_{ρ,r,p} - capacity.

Theorem 4.5. Let E be any set with 0 < 𝒞_{ρ,r,p}(E) < ∞. Then there is γ ∈ M_{+}(E) such that γ(E) = 1,

𝒲_{ρ,r,p}^{γ}(ξ) ≥ 𝒥 (γ) = 𝒞_{ρ,r,p}(E)^{1−p′}, (ρ, r, p)-q.e. on E

and

𝒲_{ρ,r,p}^{γ}(ξ) ≤ 𝒥 (γ), ξ ∈ supp γ.

The proof of Theorem 4.5 follows that of [HW, Propostions 7 and 8] almost word by word and is omitted.

5. Proof of Theorem 4.2

This section is devoted to a proof of Theorem 4.2. Since the proof follows [HW, Theorem 1] we will omit some of the details. However, for the readers’ convenience, we will carry out the crucial parts of the proof in a rather detailed manner.

Proof of Theorem 4.2. We begin by reducing the proof to the case when the kernel is supported in a neighbourhood of the origin and the measure μ is supported in a unit cube. Define

R̃_{ρ}^{m}(x) = \{ \begin{matrix}|x|^{ρ−m}, & |x| < 1, \\ 0, & |x| ≥ 1, \end{matrix}

and analogously for R̃_{r}^{n}(s). We claim that it suffices to prove (4.1) with G_{ρ}^{m} ⊗ G_{r}^{n} replaced by R̃_{ρ}^{m} ⊗R̃_{r}^{n}. By properties of the Bessel kernel we have that

G_{ρ}^{m} ⊗ G_{r}^{n} ⋆ μ(ξ) ≤ c · (R̃_{ρ}^{m} ⊗R̃_{r}^{n} ⋆ μ(ξ)
+ ∫_{|t−s|>1, |y−x|≤1} |x − y|^{ρ−m}e^{−c|s−t|} dμ(y, t)
+ ∫_{|y−x|>1, |t−s|≤1} |s − t|^{r−n}e^{−c|x−y|} dμ(y, t)
+ ∫_{|y−x|>1, |t−s|>1} e^{−c|x−y|}e^{−c|s−t|} dμ(y, t))
= c · (R̃_{ρ}^{m} ⊗R̃_{r}^{n} ⋆ μ(ξ) + A(ξ) + B(ξ) + C(ξ)).

The second term is majorized by

\begin{aligned}A(ξ) & ≤ c · ∑_{l(J)=2^{−l}} e^{−c·dist(s,J)} · ∫ |x − y|^{ρ−m} dμ_{J}(y) \\ & ≤ c · ( ∑_{l(J)=2^{−l}}(∫ |x − y|^{ρ−m} dμ_{J}(y))^{p′} · e^{−c·dist(s,J)})^{1/p′}, \end{aligned}

and integration with respect to s over R^{n} gives

∫ A(ξ)^{p′} ds ≤ c · ∑_{l(J)=2^{−l}}(∫ |x − y|^{ρ−m} dμ_{J}(y))^{p′},

where μ_{J} is defined by μ_{J}(E) = μ(E × J), for all Borel sets E ⊂ R^{m}. Now by the R^{m} -case in [HW]

∫ A(ξ)^{p′} dx ds ≤ c · ∑_{l(J)=2^{−l}} ∫ dμ_{J}(y) ∫_{0}^{1}(\frac{μ_{J}B_{m}(y, a)}{a^{m−ρp}})^{p′−1} \frac{da}{a}

≤ c · ∑_{l(J)=2^{−l}} ∫_{R^{m}×J} dμ(y, t) ∫_{0}^{1}(\frac{μB(η; a, \frac{1}{2})}{a^{m−ρp}})^{p′−1} \frac{da}{a}

≤ c · ∫ 𝒲_{ρ,r,p}^{μ}(η) dμ(η),

provided l is chosen so large that t ∈ J implies that J ⊂ B_{n}(t; \frac{1}{2}). The terms B(ξ) and C(ξ) are handled analogously, proving our claim. It is also easy to see that we can assume μ is supported in a unit cube in R^{d}. We omit the details.

From now on we assume that μ has support in a unit cube Q_{0} = I_{0} × J_{0} in R^{d}. We first notice the pointwise estimate

W_{ρ,r,p}^{μ}(ξ) ≥ c · ∑_{l(I)≤2^{−γ}, l(J)≤2^{−γ}}(μ(I × J) · l(I)^{ρp−m} · l(J)^{rp−n})^{p′−1} · χ_{I}(x) · χ_{J}(s),

for some integer γ, only depending on m and n. Integrating w.r.t. μ and using the geometry of dyadic cubes gives

(5.1) ∫ W_{ρ,r,p}^{μ}(ξ) dμ(ξ) ≥ c · ∑_{l(I)≤1, l(J)≤1}(μ(I × J) · l(I)^{ρ−m} · l(J)^{r−n})^{p′} · |I| · |J|,

which will be our lower estimate for the right hand side of (4.1). Similarly we get

(5.2) R̃_{ρ}^{m} ⊗R̃_{r}^{n} ⋆ μ(ξ) ≤ c · ∑_{l(I)≤1, l(J)≤1} μ(Ĩ× J̃) · l(I)^{ρ−m} · l(J)^{r−n} · χ_{I}(x) · χ_{J}(s).

The proof will now be completed by repeated application of the estimates in [HW, Theorem 1]. We first have

\begin{matrix}∫ (R̃_{ρ}^{m} ⊗R̃_{r}^{n} ⋆ μ(ξ))^{p′} dμ(ξ) ≤ c · ∫_{Q_{0}}(R̃_{ρ}^{m} ⊗R̃_{r}^{n} ⋆ μ(ξ))^{p′} dμ(ξ) \\ ≤ c · ∫_{I_{0}} dx ∫_{J_{0}} ds( ∑_{l(J)≤1} χ_{J}(s) · l(J)^{r−n} × ( ∑_{l(I)≤1} l(I)^{ρ−m} · μ(Ĩ× J̃) · χ_{I}(x)))^{p′} \end{matrix}

by (5.2). For any fixed x ∈ I_{0} we define a Borel measure ν_{x} on R^{n} by

ν_{x}(E) = ∑_{l(I)≤1} l(I)^{ρ−m} · μ(Ĩ× E) · χ_{I}(x).

Then ν_{x} is supported in I_{0} and has finite mass by (5.1). Two applications of the relation (⋆) in [HW, p. 170] together with (5.1) gives

∫ (R̃_{ρ}^{m} ⊗R̃_{r}^{n} ⋆ μ(ξ))^{p′} dμ(ξ) ≤ c · ∫_{I_{0}} dx ∫_{J_{0}} ds( ∑_{l(J)≤1} l(J)^{r−n} · ν_{x}(J̃) · χ_{J}(s))^{p′}

≤ c · ∫_{I_{0}} dx ∑_{l(J)≤1}(l(J)^{r−n} · ν_{x}(J))^{p′} · |J|

= c · ∑_{l(J)≤1} l(J)^{(r−n)p′} · |J| · ∫_{I_{0}} dx( ∑_{l(I)≤1} l(I)^{ρ−m} · μ(Ĩ× J) · χ_{I}(x))^{p′}

≤ c · ∑_{l(J)≤1} l(J)^{(r−n)p′} · |J| · ∑_{l(I)≤1}(l(I)^{ρ−m} · μ(I × J))^{p′} · |I|

≤ c · ∫ W_{ρ,r,p}^{μ}(ξ) dμ(ξ),

which finally proves (4.1). Theorem 4.2 is thereby proved.

6. Thin sets in H_{ρ,r,p}(R^{m} × R^{n})

In this section we continue our study of a non-linear potential theory for 𝒲_{ρ,r,p}^{μ}. We are going to define the concept of a thin set in H_{ρ,r,p}(R^{m} × R^{n}) and study its basic properties. In particular we show that the Kellog and Choquet properties hold (Theorem 6.3). We also give a partial description of thinness in terms of potentials W_{ρ,r,p}^{μ,δ} (Theorems 6.4 and 6.13).

Definition 6.1. Let 1 < p < ∞, 0 < ρ ≤ m/p and 0 < r ≤ n/p. A set E ⊂ R^{d} is called C_{ρ,r,p} -thin at ξ_{0} ∈ R^{d}, if either ξ_{0} ∉ \overline{E} or ξ_{0} ∈ \overline{E} and

(6.1)  ∫_{0}^{δ} \frac{da}{a} ∫_{0}^{δ} \frac{db}{b}(\frac{C_{ρ,r,p}(E ∩ B(ξ_{0}; a, b))}{a^{m−ρp} · b^{n−rp}})^{p′−1} < ∞,

for some δ > 0. We put e_{ρ,r,p}(E) = {ξ ∈ R^{d}; E is C_{ρ,r,p} -thin at ξ}.

This concept of thinness has the following properties.

Lemma 6.2. (i) A set E is C_{ρ,r,p} -thin at ξ_{0} if and only if E ∩ B(ξ_{0}, λ, λ) is C_{ρ,r,p} -thin at ξ_{0} for some/all λ > 0.

(ii) If E = ⋃_{1}^{N} E_{i} are sets in R^{d} and ξ_{0} ∈ R^{d} then E is C_{ρ,r,p} -thin at ξ_{0} if and only if each set E_{i}, 1 ≤ i ≤ N, is C_{ρ,r,p} -thin at ξ_{0} .

The easy proof is left to the reader.

The first result in this section is a proof of the Choquet property for the potential theory in H_{ρ,r,p}(R^{m} × R^{n}).

Theorem 6.3. Let 1 < p < ∞, 0 < ρ ≤ m/p and 0 < r ≤ n/p. Then for any set E in R^{d} and any ε > 0 there is an open set G such that

e_{ρ,r,p}(E) ⊂ G  and  𝒞_{ρ,r,p}(E ∩ G) < ε.

It is an easy consequence of Theorem 6.3 that also the so called Kellog property holds: For every set E ∈ R^{d} we have

𝒞_{ρ,r,p}(e_{ρ,r,p}(E) ∩ E) = 0.

Proof of Theorem 6.3. We follow the method of Choquet [C, Theorem 1] as it is used in [HW, Theorem 3]. Let {O_{j}}_{1}^{∞} be an enumeration of the rational balls in R^{d} that intersect E and let 𝒲_{j} be the capacitary potential for E ∩ O_{j} whenever 𝒞_{ρ,r,p}(E ∩ O_{j}) > 0. Define A_{j} = \{ξ ∈ \overline{E} ∩ O_{j} ; 𝒲_{j}(ξ) < 1\} in this case and A_{j} = \overline{E} ∩ O_{j}, if 𝒞_{ρ,r,p}(\overline{E} ∩ O_{j}) = 0. Then e_{ρ,r,p}(E) ⊂ (\overline{E})^{c} ∪ (⋃_{j=1}^{∞} A_{j}) by the analogue of [HW, Proposition 10] in the present case.

Let ε > 0 be arbitrary and choose open sets G_{j} such that 𝒞_{ρ,r,p}(G_{j}) < ε·2^{−j}, the restriction of 𝒲_{j} to G_{j}^{c} is continuous on G_{j}^{c} and 𝒲_{j}(ξ) ≥ 1 on E ∩ O_{j} ∩ G_{j}^{c}. Let G_{j} = O_{j}, if 𝒞_{ρ,r,p}(E ∩ O_{j}) = 0.

Now define F = E ∩ (⋂_{j=1}^{∞} G_{j}^{c}) and G = (\overline{F})^{c}. Then by our construction e_{ρ,r,p}(E) ⊂ G and

𝒞_{ρ,r,p}(E ∩ G) ≤ ∑_{1}^{∞} 𝒞_{ρ,r,p}(G_{j}) < ε.

This proves the theorem.

In classical potential theory, as well as in L^{p} -potential theory, thinness can be characterized by properties of potentials of measures, cf. [HW] and the references found there. In the present setting it turns out that the situation is a bit more complicated, mainly because the kernel G_{ρ}^{m} ⊗G_{r}^{n} is singular not only at the origin but on the two hyperplanes R^{m} × {0} and {0} × R^{n}.

Let E ⊂ R^{d}, ξ_{0} = (x_{0}, s_{0}) ∈ E and consider the following property of the set E at the point ξ_{0} .

Property P_{ρ,r,p}: There exists a measure μ ∈ M_{+}(R^{d}) such that, for some δ > 0, μ has its compact support in the open ball B(ξ_{0}; δ, δ) and

(6.2)  W_{ρ,r,p}^{μ,δ}(ξ_{0}) < lim inf_{ξ→ξ_{0}, ξ∈E} W_{ρ,r,p}^{μ,δ}(ξ).

The existence of μ ∈ M_{+}(R^{d}) such that (6.2) holds is closely related to the thinness of E at ξ_{0} in a sense to be made precise below. We have to consider two separate cases, whether E is a subset of one of the hyperplanes R^{m} × {s_{0}} and {x_{0}} × R^{n} or if E is away from these hyperplanes. We start with the first case.

Theorem 6.4. Let 1 < p < ∞, 0 < ρ ≤ m/p and 0 < r ≤ n/p. Let ξ_{0} = (x_{0}, s_{0}) ∈ \overline{E}, where E ⊂ R^{m} × \{s_{0}\}.

(i) If E is C_{ρ,p} -thin at x_{0} as a subset of R^{m}, then E has property P_{ρ,r,p} at ξ_{0} in R^{d}.

(ii) Assume that 2 ≤ p < ∞. If E has property P_{ρ,r,p} at ξ_{0} in R^{d}, then E is C_{ρ,p} -thin at x_{0} as a subset of R^{m}.

Theorem 6.4 has an obvious analogue for subsets of {x_{0}} × R^{n}, which we leave to the reader. We extract the technical part of the proof in the following lemma, where we use an idea from the proof of [HW, Proposition 11]. The lemma will be used once more in the proof of Theorem 6.13 below.

Lemma 6.5. Let 2 ≤ p < ∞, 0 < ρ ≤ m/p, 0 < r ≤ n/p, δ > 0 and μ ∈ M_{+}(R^{d}). Then if

E_{λ} = \{x; ∫_{0}^{k} \frac{da}{a} ∫_{0}^{δ} \frac{db}{b}(\frac{μB(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1} > λ\}

it holds that for 0 < k < δ

(6.3) C_{ρ,p}(E_{λ} ∩ B_{m}(0, k)) ≤ c · λ^{1−p} · (∫_{0}^{δ} \frac{db}{b}(\frac{μB(0, 0; 2k, b)}{b^{n−rp}})^{p′−1})^{p−1},

where c = c(m, p, ρ, δ).

Proof. Throughout this proof c denotes constants depending on m, p, ρ and δ. Let K ⊂ E_{λ} ∩ B_{m}(0, k) be a compact set with 𝒞_{ρ,p} -capacitary measure τ, τ(K) = 𝒞_{ρ,p}(K). Define

g(x, a) = (∫_{0}^{δ} \frac{db}{b}(\frac{μB(x, 0; a, b)}{b^{n−rp}})^{p′−1})^{p−1}

and

M_{τ} g(x) = sup_{0<a≤5k} \frac{g(x, a/5)}{τB_{m}(x, a)}.

Then for any x in the support of τ

\begin{aligned}λ & < ∫_{0}^{k} \frac{da}{a}(\frac{g(x, a)}{a^{m−ρp}})^{p′−1} = c · ∫_{0}^{5k} \frac{da}{a}(\frac{τB_{m}(x, a)}{a^{m−ρp}})^{p′−1} · (\frac{g(x, a/5)}{τB_{m}(x, a)})^{p′−1} \\ & ≤ c · V_{ρ,p}^{τ}(x) · M_{τ} g(x)^{p′−1}· ≤ c · M_{τ} g(x)^{p′−1}, \end{aligned}

and hence M_{τ} g(x) > c · λ^{p−1} . For every such x we choose 0 < r_{x} ≤ 5k such that

\frac{g(x, r_{x}/5)}{τB_{m}(x, r_{x})} > c · λ^{p−1}.

By a well-known covering lemma we can cover the support of τ by a union of balls {B_{i}} = {B(x_{i}, r_{i})}_{1}^{∞} such that {B(x_{i}, r_{i}/5)}_{1}^{∞} are disjoint and

\frac{g(x_{i}, r_{i}/5)}{τB_{m}(x_{i}, r_{i})} > c · λ^{p−1}.

It then follows that

C_{ρ,p}(K) = τ(K) ≤ ∑_{1}^{∞} τB(x_{i}, r_{i}) ≤ c · λ^{1−p} · ∑_{1}^{∞} g(x_{i}, r_{i}/5)

≤ c · λ^{1−p} · (∫_{0}^{δ} \frac{db}{b}(∑_{1}^{∞} \frac{μB(x_{i}, r_{i}/5) × B(0, b)}{b^{n−rp}})^{p′−1})^{p−1}

≤ c · λ^{1−p} · (∫_{0}^{δ} \frac{db}{b}(\frac{μB(0, 0; 2k, b)}{b^{n−rp}})^{p′−1})^{p−1},

by the reversed Hölder inequality. Taking the supremum over all such K gives (6.3) by the inner regularity of the capacity.

Proof of Theorem 6.4. The proof of (i) is straightforward. By [HW, Theorem 4] there exists ν ∈ M_{+}(R^{m}) such that

W_{ρ,p}^{ν}(x_{0}) < lim inf_{x→x_{0}, x∈E} W_{ρ,p}^{ν}(x).

Let τ be any smooth measure in M_{+}(R^{n}) with compact support and W_{r,p}^{τ}(s_{0}) = 1. Then μ = ν ⊗ τ is easily seen to satisfy (6.2).

Conversely, assume that μ satisfies (6.2) with ξ_{0} = (0, 0). Let 0 < h < δ and let μ_{h} denote the restriction of μ to the set {(y, t) ; |y| < h or |t| < h}. If then v denotes the difference between the right and left hand sides of (6.2) we put γ = v^{1−p′} · μ_{h}. Then, given any ε > 0, it holds

W_{ρ,r,p}^{γ,δ}(0, 0) < ε  and  lim inf_{x→0, x∈E} W_{ρ,r,p}^{γ,δ}(x, 0) ≥ 1,

provided l is small enough. Then, if ε is sufficiently small, we can find k_{0} , depending on ε, such that if 0 < k < k_{0} then

∫_{0}^{k} \frac{da}{a} ∫_{0}^{δ} \frac{db}{b}(\frac{γB(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1} ≥ \frac{1}{2},

for all x ∈ E ∩ B_{m}(0, k). Lemma 6.5 now gives

∫_{0}^{k_{0}} \frac{dk}{k} (\frac{C_{ρ,p}(E ∩ B_{m}(0, k)}{k^{m−ρp}})^{p′−1} ≤ c(m, p, ρ, δ) · W_{ρ,r,p}^{γ,δ}(0, 0) < ∞,

and so by definition [HW, p. 165] E is C_{ρ,p} -thin at the origin in R^{m}.

Remark. If Lemma 6.5 holds for 1 < p < 2 then the same is true for the statement (ii) in Theorem 6.4.

The property P_{ρ,r,p} of a set E at ξ_{0} ∈ \overline{E} behaves in many aspects like C_{ρ,r,p} - thinness. For example the two properties in Lemma 6.2 carry over almost word by word. We restrict ourselves to the case (ii) in Lemma 6.2.

Lemma 6.6. Let E = ⋃_{1}^{N} E_{i} be sets in R^{d} and assume ξ_{0} ∈ \overline{E}_{i}, 1 ≤ i ≤ N. Then E has property P_{ρ,r,p} at ξ_{0} if and only if each of the sets E_{i}, 1 ≤ i ≤ N, have property P_{ρ,r,p} at ξ_{0} .

Proof. We only need to prove necessity. Let μ_{i} ∈ M_{+}(R^{d}) and δ_{i} > 0 be such that (6.2) holds for E_{i}, 1 ≤ i ≤ N. By the first part of the proof of (ii) in Theorem 6.4 there are γ_{i} ∈ M_{+}(R^{d}), 1 ≤ i ≤ N, such that

W_{ρ,r,p}^{γ_{i},δ_{i}}(ξ_{0}) < ε  and  lim inf_{ξ→ξ_{0}, ξ∈E_{i}} W_{ρ,r,p}^{γ_{i},δ_{i}}(ξ) ≥ 1,

Then μ = ∑_{1}^{N} γ_{i} satisfies (6.2) with δ = max δ_{i}, if ε > 0 is small enough.

For any set E in R^{d} and ξ_{0} ∈ \overline{E} we express E as a (except for ξ_{0} ) disjoint union E = E_{0} ∪E_{1} ∪E_{2} , where E_{1} = E ∩(R^{m} ×{s_{0}}) and E_{2} = E ∩({x_{0}}×R^{n}). Then by Lemma 6.6 the set E has property P_{ρ,r,p} at ξ_{0} if and only if each set E_{0} , E_{1} and E_{2} has property P_{ρ,r,p} at ξ_{0} .

In classical and L^{p} -potential theory for any set E that is thin at a point ξ_{0} we can find an open set containing E\{ξ_{0}} that is also thin at ξ_{0} [HW, p. 184]. In the following theorem we give a characterization of this property for C_{ρ,r,p} -thinness.

Theorem 6.7. Assume that E ⊂ R^{d} and ξ_{0} = (x_{0}, s_{0}) ∈ \overline{E}. Then there is an open set G containing E \ {ξ_{0}} that is C_{ρ,r,p} -thin at ξ_{0} if and only if E is C_{ρ,r,p} -thin at ξ_{0} and ξ_{0} is not a limit point of E ∩(R^{m} ×{s_{0}}) or E ∩({x_{0}}×R^{n}).

Proof. To prove the sufficiency we may assume that E does not intersect R^{m} × {s_{0}} or {x_{0}} × R^{n}, by Lemma 6.2(i). Let ξ_{0} = 0 and define G as follows. For any integers k ≥ 0, l ≥ 0 there are open sets U_{k,l} such that

(a)  E ∩ B(0; 2^{−k}, 2^{−l}) ⊂ U_{k,l} ⊂ B(0; 2^{−k}, 2^{−l}),
(b)  U_{k_{1},l_{1}} ⊂ U_{k,l}, if k_{1} ≥ k and l_{1} ≥ l,
(c)  C_{ρ,r,p}(U_{k,l}) ≤ C_{ρ,r,p}(E ∩ B(0; 2^{−k}, 2^{−l})) + ε · 2^{−Θk} · 2^{−Θl},

where ε > 0 is arbitrary and Θ > 0 will be defined below. Now define

G = ⋃_{k,l}(U_{k,l} \ \overline{(B(0; 2^{−k−2}, 1) ∪ B(0; 1, 2^{−l−2}))} ∪ \overline{B(0; 1/2, 1/2)}^{c}.

It is easy to see that E ⊂ G and

G ∩ B(0; 2^{−k}, 2^{−l}) ⊂ U_{k−1,l−1},

from which it follows that G is C_{ρ,r,p} -thin at the origin, if Θ is large enough.

Conversely, assume that such an open set G exists. Then E is clearly C_{ρ,r,p} - thin at the origin. Assume for a moment that ξ_{0} = 0 is a limit point of E ∩(R^{m} × {0}). Let δ > 0 be arbitrary and take ξ_{1} = (x_{1}, 0) ∈ E such that 0 < |x_{1}| < δ. There is v > 0 such that B(ξ_{1}; v, v) ⊂ G and |x_{1}| + v < δ. If a ≥ |x_{1}| + v and 0 < b < v then B(ξ_{1}; v, b) ⊂ G ∩ B(0; a, b) and hence

C_{ρ,r,p}(G ∩ B(0; a, b)) ≥ c · v^{m−ρp} · b^{n−rp},

with appropriate changes when ρp = m or rp = n. It now follows easily that the integral (6.1), with E = G, diverges for all δ > 0. This contradiction completes the proof.

Before we proceed along our main line we define two other types of product capacities as follows. Let 1 < p < ∞, ρ > 0, r > 0 and E ⊂ R^{d} and define

\begin{aligned}C_{r,p} ⊗ C_{ρ,p}(E) & = ∫_{0}^{∞} C_{r,p}\{s; C_{ρ,p}(E_{s}) > u\} du, \\ C_{ρ,p} ⊗ C_{r,p}(E) & = ∫_{0}^{∞} C_{ρ,p}\{x; C_{r,p}(E_{x}) > u\} du, \end{aligned}

where E_{s} = {x; (x, s) ∈ E} and E_{x} = {s; (x, s) ∈ E} for fixed s ∈ R^{n} and x ∈ R^{m} respectively. Both these set functions are capacities in our sense, with additional regularity, see [Ce, Theorems 3.5 and 4.7]. We also define

\begin{aligned}C_{r,p} ⊙ C_{ρ,p}(E) & = sup_{u>0} u · C_{r,p}\{s; C_{ρ,p}(E_{s}) > u\}, \\ C_{ρ,p} ⊙ C_{r,p}(E) & = sup_{u>0} u · C_{ρ,p}\{x; C_{r,p}(E_{x}) > u\}. \end{aligned}

It is easy to see that C_{r,p} ⊙ C_{ρ,p}(E) = C_{r,p} ⊗ C_{ρ,p}(E) = C_{ρ,p}(E_{1}) · C_{r,p}(E_{2}) = C_{ρ,r,p}(E), for products E = E_{1} ×E_{2} of Borel sets. We get C_{r,p} ⊙C_{ρ,p}(E) ≤ C_{r,p} ⊗ C_{ρ,p}(E), for arbitrary sets, by the definitions. Here we can of course interchange the orders of C_{ρ,p} and C_{r,p}.

Lemma 6.9. Let 1 < p < ∞, ρ > 0, r > 0 and E ⊂ R^{d}, then

max(C_{r,p} ⊗ C_{ρ,p}(E), C_{ρ,p} ⊗ C_{r,p}(E)) ≤ c · C_{ρ,r,p}(E).

Proof. Let f ≥ 0 be as in the definition of C_{ρ,r,p}(E). Then for fixed s ∈ R^{n} and all x ∈ E_{s}

1 ≤ ∫ G_{ρ}(x − y)(∫ G_{r}(s − t) · f(y, t) dt) dy = ∫ G_{ρ}(x − y) · f_{s}(y) dy.

Hence C_{ρ,p}(E_{s})^{1/p} ≤ ∥f_{s}∥_{p} ≤ G_{r} ⋆ g(s), where g(t) = (∫ f(y, t)^{p} dy)^{1/p}, and so by the capacitary strong type inequality [AH, Theorem 7.1.1]

C_{r,p} ⊗ C_{ρ,p}(E) ≤ ∫_{0}^{∞} C_{r,p}{s; G_{r} ⋆ g(s) > u} d(u^{p}) ≤ c · ∥g∥_{p}^{p} = c · ∥f∥_{p}^{p}.

Taking infimum over all such f proves that C_{r,p} ⊗ C_{ρ,p}(E) ≤ c · C_{ρ,r,p}(E). The other inequality is proved in the same way.

Before we state and prove our general results about potentials and thinness we prove an analogue of [HW, Proposition 11]. We use the notation C_{ρ,r,p}^{⋆}(E) = max(C_{r,p} ⊙ C_{ρ,p}(E), C_{ρ,p} ⊙ C_{r,p}(E)).

Theorem 6.10. Let 2 ≤ p < ∞, 0 < ρ ≤ m/p, 0 < r ≤ n/p and μ ∈ M_{+} . Then for all λ > 0

(6.4)  C_{ρ,r,p}^{⋆}\{ξ; W_{ρ,r,p}^{μ,δ}(ξ) > λ\} ≤ \frac{c}{λ^{p−1}} · ∥μ∥_{1},

where c = c(d, ρ, r, p, δ).

Proof. In this proof c denotes constants depending on d, ρ, r, p and δ. We may and will assume that the support of μ is contained in B(0; 1/80, 1/80), λ > A · ∥μ∥_{1}^{p′−1} and 0 < δ < 1/80, where A depends on d, ρ, r, p and δ. Define G = {(x, s) ; W_{ρ,r,p}^{μ,δ}(x, s) > λ} then G is an open set and for fixed s ∈ R^{n} we put G_{s} = {x; (x, s) ∈ G}. If G_{s} is non-empty we choose a finite union K_{s} of closed cubes in G_{s} and a closed cube I_{s} ⊂ R^{n}, with side length l_{s}, containing s such that K_{s} × I_{s} ⊂ G. The proof is now in two steps. In the first step we estimate C_{ρ,p}(K_{s}) (and thereby C_{ρ,p}(G_{s})) and in the second step we put the pieces together to get an estimate of C_{r,p} ⊙ C_{ρ,p}(G). We start from the estimate

\begin{aligned}W_{ρ,r,p}^{μ,δ}(ξ) & ≤ c · ∫_{0}^{1/2} \frac{da}{a} ∫_{0}^{1/2} \frac{db}{b}(\frac{μB(x, s; a/5, b)}{a^{m−ρp} · b^{n−rp}})^{p′−1} \\ & = c · ∫_{0}^{1/2} \frac{da}{a}(\frac{g_{s}(x, a/5; μ)}{a^{m−ρp}})^{p′−1}, \end{aligned}

where

g_{s}(x, a; μ) = (∫_{0}^{1/2} \frac{db}{b}(\frac{μB(x, s; a, b)}{b^{n−rp}})^{p′−1})^{p−1},

by a change of variables. Let γ be the 𝒞_{ρ,r,p} -capacitary measure for K_{s} × I_{s}. Then for all (x, s) in the support of γ we have

\begin{aligned}λ & < W_{ρ,r,p}^{μ,δ}(x, s) ≤ c · ∫_{0}^{1/2} \frac{da}{a}(\frac{g_{s}(x, a; γ)}{a^{m−ρp}})^{p′−1} · (\frac{g_{s}(x, a/5; μ)}{g_{s}(x, a; γ)})^{p′−1} \\ & ≤ c · 𝒲_{ρ,r,p}^{γ}(x, s) · M_{γ}μ(x, s)^{p′−1} ≤ c · M_{γ}μ(x, s)^{p′−1}, \end{aligned}

where

M_{γ}μ(x, s) = sup_{0<a≤1/2} \frac{g_{s}(x, a/5; μ)}{g_{s}(x, a; γ)}

is the appropriate maximal function in R^{m}. For any such (x, s) there is 0 < a_{x} ≤ 1/2 with g_{s}(x, a_{x}/5; μ) > c · λ^{p−1} · g_{s}(x, a_{x}; γ). A standard covering theorem gives a disjoint class of balls {B_{m}(x_{j}, a_{j}/5)} such that {B_{m}(x_{j}, a_{j})} covers K_{s}. Then

(6.5) \begin{aligned}∑_{j} g_{s}(x_{j}, a_{j}; γ) & ≤ \frac{c}{λ^{p−1}} · ∑_{j} g_{s}(x_{j}, a_{j}/5; μ) \\ & ≤ \frac{c}{λ^{p−1}} · (∫_{0}^{1/2} \frac{db}{b}(\frac{∑_{j} μB(x_{j}, s; a_{j}/5, b)}{b^{n−rp}})^{p′−1})^{p−1} \\ & ≤ \frac{c}{λ^{p−1}} · (∫_{0}^{1/2} \frac{db}{b}(\frac{μB(0, s; 1, b)}{b^{n−rp}})^{p′−1})^{p−1} \\ & = W_{r,p}^{\overline{μ},1/2}(s), \end{aligned}

by the reverse triangle inequality. Here \overline{μ} is the measure in R^{n} defined by \overline{μ}(E) = μ(B_{m}(0, 1) × E). For the left hand side of (6.5) we get

\begin{aligned}∑_{j} g_{s}(x_{j}, a_{j}; γ) & ≥ c · ∑_{j} \frac{γB(x_{j}, s; a_{j}, l_{s}\sqrt{n})}{l_{s}^{n−rp}} ≥ c · \frac{γ(K_{s} × I_{s})}{l_{s}^{n−rp}} \\ & = c · \frac{𝒞_{ρ,r,p}(K_{s} × I_{s})}{l_{s}^{n−rp}} ≥ c · \frac{C_{ρ,r,p}(K_{s} × I_{s})}{l_{s}^{n−rp}} ≥ c · C_{ρ,p}(K_{s}) \end{aligned}

by equivalence of the capacities and Theorem 2.2, for r · p < n. When r · p = n we replace l_{s}^{n−rp} by (log(1/l_{s}))^{1−p} and the same estimate holds. Combining the last estimate with (6.5) and taking the supremum over all such sets K_{s} gives

(6.6)  C_{ρ,p}(G_{s}) ≤ \frac{c}{λ^{p−1}} · W_{r,p}^{\overline{μ},1/2}(s)^{p−1},

which completes the first step of the proof.

In the second step we simply get

C_{r,p} ⊙ C_{ρ,p}(G) = sup_{u>0} u · C_{r,p}{s; C_{ρ,p}(G_{s}) > u}

≤ sup_{u>0} u · C_{r,p}\{s; W_{r,p}^{\overline{μ},1/2}(s) > c · u^{p′−1} · λ\}

≤ c · \frac{∥\overline{μ}∥_{1}}{λ^{p−1}} = c · \frac{∥μ∥_{1}}{λ^{p−1}} ,

by (6.6) and [HW, Proposition 11]. The corresponding inequality for C_{ρ,p}⊙C_{r,p}(G) is proved in the same way interchanging the roles of R^{m} and R^{n}.

Remark. The conclusion in Theorem 6.10 is false if C_{ρ,r,p}^{⋆} is replaced by C_{r,p} ⊗ C_{ρ,p} or C_{ρ,p} ⊗ C_{r,p} and hence by Lemma 6.9 also when C_{ρ,r,p}^{⋆} is replaced by C_{ρ,r,p}. The simplest example is μ = δ_{0} , the Dirac measure at the origin. If 0 < ρ < m/p and 0 < r < n/p then

W (x, s) = W_{ρ,r,p}^{μ,δ}(x, s) ≥ (c_{1} · |x|^{ρp−m} · |s|^{rp−n})^{p′−1} = \overline{W} (x, s),

for |x| < δ/2 and |s| < δ/2. Define \overline{W} (x, s) = 0, elsewhere. It is clearly sufficient to consider \overline{W} instead of W. Let λ^{p−1} > c_{1} · δ^{ρp−m+rp−n} · 2^{n−rp}, put G = \{(x, s) ; \overline{W} (x, s) > λ\} and for fixed |s| < δ/2 define G_{s} = \{x; (x, s) ∈ G\}. Then G_{s} = B_{m}(0, δ/2), for |s|^{n−rp} ≤ c_{1} · (δ/2)^{ρp−m} · λ^{1−p} = A^{n−rp} and

G_{s} = {x; |x|^{m−ρp} < c_{1} · |s|^{rp−n} · λ^{1−p}},

for A < |s| < δ/2. Note that A < δ/4 by our choice of λ. From this we get

C_{r,p} ⊗ C_{ρ,p}(G) ≥ ∫_{a}^{b} C_{r,p}{s; C_{ρ,p}(G_{s}) > u} du

≥ ∫_{a}^{b} C_{r,p}{s; |s|^{n−rp} < c · λ^{1−p} · 1/u} du ≥ c · λ^{1−p} · log(b/a)

≥ c · λ^{1−p} · log(c · λ^{p−1}),

where a = C_{ρ,p}{x; |x|^{m−ρp} < c_{1} · (δ/2)^{rp−n} · λ^{1−p}} and b = C_{ρ,p}{x; |x| < δ/2}, which proves that μ has the desired property. We can get similar examples when μ is Lebesgue measure restricted to |x| < h and |s| < k, for suitable small h, k and λ.

The method of proof in Theorem 6.10 also gives another result of the same kind for the other type of product capacity, when p > 2. We have no such result for p = 2.

Theorem 6.11. Let p, ρ, r, δ, E, μ be as in Theorem 6.10 and put G = {ξ; W_{ρ,r,p}^{μ,δ}(ξ) > λ}. Then

max(C_{r,p−1} ⊗ C_{ρ,p}(G), C_{ρ,p−1} ⊗ C_{r,p}(G)) ≤ \frac{c}{λ^{p−1}} · ∥μ∥_{1},

for p > 2.

Proof. From the definitions, (6.6) and [HW, p. 164] we get

C_{r,p−1} ⊗ C_{ρ,p}(G) ≤ \frac{c}{λ^{p−1}} · ∫_{0}^{∞} C_{r,p−1}\{s; G_{r}^{n} ⋆ (G_{r}^{n} ⋆ \overline{μ})^{p′−1}(s) > v\} · v^{p−2} dv,

by a change of variables. The capacitary strong type inequality [AH, Theorem 7.1.1] with exponent p − 1 then gives

C_{r,p−1} ⊗ C_{ρ,p}(G) ≤ \frac{c}{λ^{p−1}} · ∫ G_{r}^{n} ⋆ \overline{μ}(s) ds = \frac{c}{λ^{p−1}} · ∥\overline{μ}∥_{1} = \frac{c}{λ^{p−1}} · ∥μ∥_{1}.

Remark. When p = 2 we here get C_{2r} ⊙ C_{ρ,2}(G) ≤ (c/λ) · ∥μ∥_{1} , where we have defined

C_{2r}(E) = inf{ν(E) ; supp ν ⊂ E and G_{2r}^{n} ⋆ ν(s) ≥ 1, for all s ∈ E },

for Borel sets E ⊂ R^{n}. It is however easy to see from [HW, p. 163] that C_{2r} is equivalent to C_{r,2} and hence we only recover Theorem 6.10.

Definition 6.12. A set E ⊂ R^{d} satisfies a cone condition at ξ_{0} = (x_{0}, s_{0}) (relative to the hyperplanes R^{m} × {s_{0}} and {x_{0}} × R^{n}) if there are constants R > 0, M > 0 such that (6.7) E ∩ B(ξ_{0}; R, R) ⊂ {ξ = (x, s) ; |s − s_{0}| ≤ M · |x − x_{0}| and |x − x_{0}| ≤ M · |s − s_{0}|}.

Analogously, a measure μ ∈ M_{+} satisfies a cone condition at ξ_{0} if (6.7) holds with E equal to the support of μ.

Theorem 6.4 gives a partial characterization (exact if 2 ≤ p < ∞) of property P_{ρ,r,p} in terms of thinness for sets contained in one of the hyperplanes R^{m} × {s_{0}} or {x_{0}} × R^{n}. Our next theorem gives a partial answer to the same question for sets E and measures μ satisfying a cone condition. We say that a set E ⊂ R^{d} is C_{ρ,r,p}^{⋆} -thin at ξ ∈ \overline{E} if (6.1) holds with C_{ρ,r,p} replaced by C_{ρ,r,p}^{⋆}.

Theorem 6.13. Let 2 ≤ p < ∞, 0 < ρ ≤ m/p and 0 < r ≤ n/p. Assume that E ⊂ R^{d} and μ ∈ M_{+} both satisfy a cone condition (6.7) at ξ_{0} and

W_{ρ,r,p}^{μ,δ}(ξ_{0}) < lim inf_{ξ→ξ_{0}, ξ∈E} W_{ρ,r,p}^{μ,δ}(ξ).

Then E is C_{ρ,r,p}^{⋆} - thin at ξ_{0} .

Proof. Throughout this proof c denotes various constants that may depend on d, ρ, r, p, δ and M. Let ξ_{0} = 0, 0 < ε < 1 and assume that μ ∈ M_{+} satisfies

v = lim inf_{ξ→ξ_{0}, ξ∈E}(W_{ρ,r,p}^{μ,δ}(ξ) − W_{ρ,r,p}^{μ,δ}(ξ_{0})) > 0.

Define γ = v^{1−p′} · μ_{h}, where μ_{h} is the restriction of μ to the set {(y, t) ; |y| < h or |t| < h}. Then as in [HW, Theorem 4] we get

W_{ρ,r,p}^{γ}(0) < ε  and  lim inf_{ξ→0, ξ∈E} W_{ρ,r,p}^{γ}(ξ) ≥ 1,

if h, depending on ε, is small enough. From now on such a measure γ is fixed, for some arbitrarily small h, 0 < h < δ. Choose 0 < δ_{0} < h such that W_{ρ,r,p}^{γ,δ}(ξ) ≥ \frac{1}{2} on E ∩ B(0; δ_{0}, δ_{0}) and let 0 < k, l < δ_{0}/2. If h is small enough the support of γ is contained in B(0; δ/2, δ/2) by the cone condition (6.6). It is clearly sufficient to consider the following two cases.

Case I: 0 < l < k/2M. This is the principal case that motivates the cone condition (6.7). Here E ∩ B(0; k, l) = E ∩ B(0; Ml, l) and analogously for the support of μ. We define four sets

A = {(y, t) ; |y| < 2Ml and |t| < 2l},  B = {(y, t) ; |y| < 2Ml and |t| ≥ 2l},
C = {(y, t) ; |y| ≥ 2Ml and |t| < 2l},  D = {(y, t) ; |y| ≥ 2Ml and |t| ≥ 2l},

and note that E ∩ C is empty. Let γ_{A}, γ_{B} and γ_{D} be the restriction of γ to each of these sets. By the definition of W_{ρ,r,p}^{γ,δ} we have

(6.8)  W_{ρ,r,p}^{γ,δ}(ξ) ≤ c · (W_{ρ,r,p}^{γ_{A},δ}(ξ) + W_{ρ,r,p}^{γ_{B},δ}(ξ) + W_{ρ,r,p}^{γ_{D},δ}(ξ)).

We let ξ = (x, s), where |x| ≤ Ml, |s| ≤ l, and estimate each of the terms in the right hand side of (6.8). It is easy to see that

W_{ρ,r,p}^{γ_{D},δ}(ξ) ≤ ∫_{Ml}^{δ} \frac{da}{a} ∫_{l}^{δ} \frac{db}{b}(\frac{γB(0; 2a, 2b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1}

≤ c · ∫_{2Ml}^{2δ} \frac{da}{a} ∫_{2l}^{2δ} \frac{db}{b}(\frac{γB(0; a, b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1}

≤ c · W_{ρ,r,p}^{γ,δ}(0) ≤ c · ε,

since γ has support in B(0; δ/2, δ/2). Then by (6.8) the set E ∩ B(0; Ml, l) is contained in the union of the sets where each of the potentials W_{ρ,r,p}^{γ_{A},δ} and W_{ρ,r,p}^{γ_{B},δ} exceeds some positive constant λ = c(d, ρ, r, p), provided ε is small enough. First we have

(6.9)  C_{ρ,r,p}^{⋆}({ξ ∈ E; W_{ρ,r,p}^{γ_{A},δ}(ξ) > λ} ∩ B(0; Ml, l)) ≤ c · γB(0; k, 2l),

by Theorem 6.10. This is our estimate for W_{ρ,r,p}^{γ_{A},δ}.

We are thus left with the term W_{ρ,r,p}^{γ_{B},δ}, which has no counterpart in the R^{n} - case. By the definition of γ_{B} we have

\begin{aligned}W_{ρ,r,p}^{γ_{B},δ}(ξ) & ≤ ∫_{0}^{δ} \frac{da}{a} ∫_{l}^{δ} \frac{db}{b}(\frac{γ_{B}B(x, 0; a, 2b)}{a^{m−ρp} · b^{n−rp}})^{p′−1} \\ & ≤ c · ∫_{0}^{δ} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}})^{p′−1}, \end{aligned}

since again γ is supported in B(0; δ/2, δ/2). Next

W_{ρ,r,p}^{γ_{B},δ}(ξ) ≤ c · (∫_{0}^{Ml} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b} + ∫_{Ml}^{δ} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b})(\frac{γ_{B}B(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}})^{p′−1},

and the second iterated integral is at most

∫_{Ml}^{δ} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(0; 2a, b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1} ≤ c·∫_{2Ml}^{δ} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(0; a, b)}{a^{m−ρp} · b^{n−rp}} )^{p′−1} ≤ c·ε.

The key estimate is now

C_{ρ,r,p}^{⋆}({(x, s) ; W_{ρ,r,p}^{γ_{B},δ}(ξ) > λ} ∩ B(0; Ml, l))

≤ C_{ρ,r,p}^{⋆}(\{(x, s) ; ∫_{0}^{Ml} \frac{da}{a}∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}})^{p′−1} > \frac{λ}{c}\}∩B(0; Ml, l))

≤ c · l^{n−rp} · (∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(0; 2Ml, b)}{b^{n−rp}})^{p′−1})^{p−1},

if ε is small and rp < n, by Lemma 6.5. If rp = n we get an extra factor (log(1/l))^{1−p}. Observing that γ_{B}B(0; 2Ml, b) ≤ γ_{B}B(0; 2Ml, 2M^{2}l) by (6.7) gives the estimate

(6.10) \begin{aligned}∫{δ_{0} \atop 0} & \frac{dk}{k} ∫_{0}^{k/2M} \frac{dl}{l} (\frac{C_{ρ,r,p}^{⋆}(\{ξ ∈ E; W_{ρ,r,p}^{γ_{B},δ}(ξ) > λ\} ∩ B(0; 2Ml, l))}{k^{m−ρp} · l^{n−rp}})^{p′−1} \\ & ≤ c · ∫_{0}^{δ_{0}} \frac{dk}{k} ∫_{0}^{k/2M} \frac{dl}{l} · ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(0; k, 2M^{2}l)}{k^{m−ρp} · b^{n−rp}})^{p′−1} \\ & ≤ c · ∫_{0}^{δ_{0}} \frac{dk}{k} ∫_{0}^{k/2M} \frac{dl}{l} (\frac{γ_{B}B(0; k, 2M^{2}l)}{k^{m−ρp} · l^{n−rp}})^{p′−1} \\ & = c · ∫_{0}^{δ_{0}} \frac{dk}{k} ∫_{0}^{k/2M} \frac{dl}{l} (\frac{γ_{B}B(0; k, l)}{k^{m−ρp} · l^{n−rp}} )^{p′−1} \\ & ≤ c · W_{ρ,r,p}^{γ,δ}(0) ≤ c · ε, \end{aligned}

for rp < n and some δ_{0} > 0, by a change of variables. If rp = n we get an extra factor (log(1/l)) that cancels against the factor (log(1/l))^{1−p} above, raised to power p′ − 1.

Case II: k/2M ≤ l ≤ 2Mk. We define
A = {(y, t) ; |y| < 2k and |t| < 2l}, B = {(y, t) ; |y| < 2k and |t| ≥ 2l},
C = {(y, t) ; |y| ≥ 2k and |t| < 2l}, D = {(y, t) ; |y| ≥ 2k and |t| ≥ 2l},

and let γ_{A}, γ_{B}, γ_{C} and γ_{D} be the restriction of γ to each of these sets. Since we will follow the first case very closely, we do not repeat all arguments in detail. We handle W_{ρ,r,p}^{γ_{D},δ} as above and for W_{ρ,r,p}^{γ_{A},δ} we get

C_{ρ,r,p}^{⋆}({ξ ∈ E; W_{ρ,r,p}^{γ_{A},δ}(ξ) > λ} ∩ B(0; , k, l)) ≤ c · γB(0; 2k, 2l),

by Theorem 6.10. It therefore only remains to estimate W_{ρ,r,p}^{γ_{B},δ} and W_{ρ,r,p}^{γ_{C},δ} in E ∩ B(0; k, l). We start from

W_{ρ,r,p}^{γ_{B},δ}(ξ) ≤ ∫_{0}^{δ} \frac{da}{a} ∫_{l}^{δ} \frac{db}{b}(\frac{γ_{B}B(x, 0; a, 2b)}{a^{m−ρp} · b^{n−rp}})^{p′−1}

≤ c · ∫_{0}^{δ} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}})^{p′−1}

≤ c · ∫_{0}^{k} \frac{da}{a} ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(x, 0; a, b)}{a^{m−ρp} · b^{n−rp}})^{p′−1}

and hence

\begin{aligned}C_{ρ,r,p}^{⋆}(\{ξ ∈ E & ; W_{ρ,r,p}^{γ_{B},δ}(ξ) > λ\} ∩ B(0; , k, l)) \\ & ≤ c · l^{n−rp} · (∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(0; 2k, b)}{b^{n−rp}})^{p′−1})^{p−1} \end{aligned}

by Lemma 6.5, for rp < n. If rp = n we get an extra factor (log(1/l))^{1−p} that is to our advantage. It follows at once that

(6.11) \begin{aligned}∫{δ_{0} \atop 0} & \frac{dk}{k} ∫_{k/2M}^{2Mk} \frac{dl}{l} (\frac{C_{ρ,r,p}^{⋆}(\{ξ ∈ E; W_{ρ,r,p}^{γ_{B},δ}(ξ) > λ\} ∩ B(0; k, l))}{k^{m−ρp} · l^{n−rp}})^{p′−1} \\ & ≤ c · ∫_{0}^{δ_{0}} \frac{dk}{k} ∫_{k/2M}^{2Mk} \frac{dl}{l} ∫_{2l}^{δ} \frac{db}{b}(\frac{γ_{B}B(0; 2k, b)}{k^{m−ρp} · b^{n−rp}} )^{p′−1} ≤ c · W_{ρ,r,p}^{γ,δ}(0), \end{aligned}

for some δ_{0} > 0. Estimating W_{ρ,r,p}^{γ_{C},δ} in the same way, combining (6.9) with (6.10) and (6.11) and interchanging the roles of k and l finally proves the theorem.

Remark. Theorem 6.13 is not true for sets that are close to the hyperplanes R^{m} × {s_{0}} or {x_{0}} × R^{n}. To see this take E ⊂ R^{m} × {s_{0}} such that E is (ρ, p)- thin at ξ_{0} in R^{m} and ξ_{0} is a limit point of E. By Theorem 6.4(i) there exists μ ∈ M_{+} such that

W_{ρ,r,p}^{μ,δ}(ξ_{0}) < v < lim inf_{ξ→ξ_{0}, ξ∈E} W_{ρ,r,p}^{μ,δ}(ξ).

The open set G = {ξ; W_{ρ,r,p}^{μ,δ}(ξ) > v} is not C_{ρ,r,p}^{⋆} -thin at ξ_{0} by the last part of the proof of Theorem 6.7, since it contains E \ {ξ_{0}} near ξ_{0} , but clearly

W_{ρ,r,p}^{μ,δ}(ξ_{0}) < v ≤ lim inf_{ξ→ξ_{0}, ξ∈G} W_{ρ,r,p}^{μ,δ}(ξ).

Acknowledgement. The author thanks the referee for pointing out an error in the first version of Theorem 6.13 and for several valuable comments that have improved the presentation.

338 Tord Sjödin

References

[A] Adams, D.R.: Lectures on L^{p} -potential theory. - Report no. 2, Department of Mathematics, University of Umeå, 1981.

[AH] Adams, D.R., and L.-I. Hedberg: Function Spaces and Potential Theory. - Springer-Verlag, Berlin, 1995.

[Ce] Cegrell, U.: On product capacities with applications to complex analysis. - In: Lecture Notes in Math. 822, Springer-Verlag, 1980, 33–45.

[Ch] Choquet, G.: Sur les points d’effilement d’un ensemble application à l’étude de la capacité. - Ann. Inst. Fourier 9, 1959, 91–101.

[HW] Hedberg, L.-I., and Th. Wolff: Thin sets in non-linear potential theory. - Ann. Inst. Fourier 33, 1983, 161–187.

[Ma] Maz’ya, V.G.: Sobolev Spaces. - Springer-Verlag, Berlin–Heidelberg, 1985.

[MS] Maz’ya, V.G., and T.O. Shaposhnikova: Theory of Multipliers in Spaces of Differentiable Functions. - Pitman Publ. Co., New York, 1985.

[Me] Meyers, N.G.: A theory of capacities for potentials of functions in Lebesgue classes. - Math. Scand. 26, 1970, 255–292.

[Sj] Sjödin, T.: Non-linear potential theory in Lebesgue spaces with mixed norm. - In: Potential Theory, Proceedings of a Conference on Potential Theory, Prague, 1987, 325–331, edited by J. Král, I. Netuka, J. Lukeš, J. Veselý. Plenum Press, New York, 1988.

[SS] Sjögren, P., and P. Sjölin: Convergence properties for the time dependent Schrödinger equation. - Ann. Acad. Sci. Fenn. Ser. A I Math. 14, 1989, 13–25.

[St] Stein, E.M.: Singular Integrals and Differentiability of Functions. - Princeton Univ. Press, New Jersey, 1970.

Received 23 October 1995