Georgian Mathematical Journal Volume 8 (2001), Number 1, 33-59

WEIGHT INEQUALITIES FOR SINGULAR INTEGRALS DEFINED ON SPACES OF HOMOGENEOUS AND NONHOMOGENEOUS TYPE

D. E. EDMUNDS, V. KOKILASHVILI, AND A. MESKHI

Dedicated to Professor Hans Triebel on his 65th birthday

Abstract. Optimal sufficient conditions are found in weighted Lorentz spaces for weight functions which provide the boundedness of the Calderón– Zygmund singular integral operator defined on spaces of homogeneous and nonhomogeneous type.

2000 Mathematics Subject Classification: 42B20, 42B25.

Key words and phrases: Singular Integral, maximal function, Hardytype operator, space of homogeneous type, space of nonhomogeneous type, Lorentz space.

Introduction

In the present paper optimal sufficient conditions are found for the weight functions which provide the boundedness of Calderón–Zygmund singular integral operator defined on spaces of homogeneous as well as nonhomogeneous type in weighted Lorentz spaces. In the nonhomogeneous case the results are new even in Lebesgue spaces.

Two-weight strong type inequalities for singular integrals in Lebesgue spaces on spaces of homogeneous type (SHT) were established in [5] (see also [6], Chapter 9, and [14]), while a similar problem for singular integrals defined on homogeneous groups was considered in [16] in the case of Lorentz spaces.

Two-weight inequalities for Hilbert transforms with monotonic weights were studied in [17]. Analogous problems for singular integrals in Euclidean spaces were considered in [8] and generalized in [7] for singular integrals on Heisenberg groups. Optimal conditions for a pair of weights ensuring the validity of twoweight inequalities for Calderón–Zygmund singular integrals were obtained in [3] (and also in [19] for the case of Lorentz spaces). The latter result was generalized to the setting of homogeneous groups in [13], [15].

Finally, we would like to mention that singular integrals on spaces of nonhomogeneous type (measure spaces with a metric) were studied in [18].

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1. Preliminaries

In this section we give the notion of SHT and defined Lorentz space. The definition of a singular integral and several well-known results concerning the Hardy-type operators on SHT are also given.

Definition 1.1. A space of homogeneous type (SHT) (X, d, μ) is a topological space X endowed with a complete measure μ such that: (a) the space of compactly supported continuous functions is dense in L^{1}(X, μ), and (b) there exists a non-negative real-valued function (quasimetric) d : X × X → ℝ^{1} satisfying:

(i) d(x, x) = 0 for arbitrary x ∈ X;

(ii) d(x, y) > 0 for arbitrary x, y ∈ X, x ≠ y;

(iii) there exists a positive constant a_{0} such that the inequality

d(x, y) ≤ a_{0}d(y, x)

holds for all x, y ∈ X; (iv) there exists a positive constant a_{1} such that the inequality

d(x, y) ≤ a_{1}(d(x, z) + d(z, y))

holds for arbitrary x, y, z ∈ X;

(v) for every neighborhood V of any point x ∈ X there exists a number r > 0 such that the ball

B(x, r) = {y ∈ X : d(x, y) < r}

with center in x and radius r is contained in V;

(vi) the balls B(x, r) are measurable for all x ∈ X, r > 0 and, moreover, 0 < μB(x, r) < ∞;

(vii) there exists a positive constant b such that the inequality (doubling condition)

μB(x, 2r) ≤ bμB(x, r)

is true for all x ∈ X and for all positive r.

It will be supposed that for some x_{0} ∈ X

μ{x_{0}} = μ{x ∈ X : d(x_{0}, x) = a} = 0,

where

a ≡ sup {d(x_{0}, x) : x ∈ X}.

Note that the condition a = ∞ is equivalent to the condition μ(X) = ∞, and if a = ∞, then μ{x ∈ X : d(x_{0}, x) = a} = 0.

For the definitions, various examples and properties of SHT, we refer to [2],

[23] (also to [6]).

Definition 1.2. By a spaces of nonhomogeneous type we mean a measure space with a quasimetric (X, d, μ) satisfying conditions (i)–(v) of Definition 1.1, i.e., the doubling condition is not assumed and may fail.

In the sequel it will be assumed that there exists a point x_{0} ∈ X such that

B(x_{0}, R)\B(x_{0}, r) ≠ ∅  (1)

for all r and R provided 0 < r < R < a.

In the remaining part of this section (X, d, μ) is assumed to be a space of nonhomogeneous type.

Definition 1.3. An almost everywhere positive, locally integrable function w : X → ℝ^{1} is called a weight.

For weight functions w, we denote by L_{w}^{pq}(X) the Lorentz space with weight w which represents the class of all μ-measurable functions f : X → ℝ^{1} for which

∥f∥_{L_{w}^{pq}(X)} = (q ∫_{0}^{+∞}( ∫_{{x∈X: |f(x)|>λ}} w(x) dμ)^{q/p}λ^{q−1} dλ)^{1/q} < ∞

when 1 ≤ p ≤ ∞, 1 ≤ q < ∞, and

∥f∥_{L_{w}^{p∞}(X)} = sup_{λ>0} λ( ∫_{{x∈X: |f(x)|>λ}} w(x) dμ)^{1/p} < ∞

when 1 < p < ∞.

It is clear that if p = q, then L_{w}^{pq}(X) = L_{w}^{p}(X) is a Lebesgue space. If w ≡ 1, then we put

L_{w}^{p}(X) ≡ L^{p}(X).

The following lemma holds.

Lemma A. Let E ⊂ X be a μ-measurable set, 1 < p, q < ∞ and suppose that w is a weight function on X. Further, let f, f_{1} and f_{2} be μ-measurable functions on X. Then:

(i)

∥χ_{E}(·)∥_{L_{w}^{pq}(X)} = ( ∫_{E} w(x) dμ)^{1/p};

(ii)

∥f∥_{L_{w}^{pq_{1}}(X)} ≤ ∥f∥_{L_{w}^{pq_{2}}(X)}

for fixed p and q_{1} ≤ q_{2}; (iii) there exists a positive constant c such that

∥f_{1}f_{2}∥_{L_{w}^{pq}(X)} ≤ c∥f_{1}∥_{L_{w}^{p_{1}q_{1}}(X)}∥f_{2}∥_{L_{w}^{p_{2}q_{2}}(X)},

where \frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} , \frac{1}{q} = \frac{1}{q_{1}} + \frac{1}{q_{2}} .

The first part of this Lemma is proved in [22], Chpater V, Section 3 and the rest of the proof is given in [10].

Lemma B ([1], [19]). Let {E_{k}} be a sequence of μ-measurable subsets of X, such that ∑_{k} χ_{E_{k}} ≤ c_{0}χ_{∪E_{k}}. Then

(i) there exists a positive constant k_{1} = k_{1}(c_{0}), such that for arbitrary f, the following inequality holds:

∑_{k} ∥f(·)χ_{E_{k}(·)}∥_{L_{w}^{rs}(X)}^{λ} ≤ k_{1}∥f(·)χ_{∪E_{k}(·)}∥_{L_{w}^{rs}(X)}^{λ},

where max{r, s} ≤ λ;

(ii) there exists a positive constant k_{2} = k_{2}(c_{0}), such that for every f, the following inequality holds:

∥f(·)χ_{∪E_{k}(·)}∥_{L_{w}^{pq}(X)}^{γ} ≤ k_{2} ∑_{k} ∥f(·)χ_{E_{k}(·)}∥_{L_{w}^{pq}(X)}^{γ},

where 0 < γ ≤ min{p, q}.

Definition 1.4. If 1 < p < ∞, then A_{p}(X) is the set of all weights w such that

sup (\frac{1}{μ(B)} ∫_{B} w(x) dμ)(\frac{1}{μ(B)} ∫_{B} w^{1−p′}(x) dμ)^{p−1} < ∞, p′ = \frac{p}{p − 1},

where the supremum is taken over all balls B ⊂ X.

We recall that a ≡ sup{d(x_{0}, x) : x ∈ X} for some x_{0} ∈ X with μ{x_{0}} = 0.

Lemma C ([5]). Let a = ∞, 1 < p < ∞, ρ ∈ A_{p}(X) and 0 < c_{1} ≤ c_{2} < c_{3} < ∞. Then there exists a positive constant c such that the inequality

∫_{B(x_{0},c_{3}t)\B(x_{0},c_{2}t)} ρ(x) dμ ≤ c ∫_{B(x_{0},c_{1}t)} ρ(x) dμ

holds for arbtrary t > 0.

In the sequel we shall need the following results concerning the Hardy-type operators on the measure space (X, μ)

Hf(t) = ∫_{{x: d(x_{0},x)<t}} f(x) dμ(x), t ∈ (0, a),

and

H̃f(t) = ∫_{{x: d(x_{0},x)>t}} f(x) dμ(x), t ∈ (0, a).

The following result is from [5].

Proposition A. Let 1 < p ≤ q < ∞, μ{x_{0}} = 0. Then the inequality

( ∫_{0}^{a} v(t)|Hf(t)|^{q} dt)^{1/q} ≤ c( ∫_{X} |f(x)|^{p}w(x) dμ)^{1/p},  (1.1)

holds with some c > 0 independent of f, f ∈ L_{w}^{p}(X), if and only if

D = sup_{0<t<a} ( ∫_{t}^{a} v(τ) dτ )^{1/q}( ∫_{{x: d(x_{0},x)<t}} w^{1−p′}(x) dμ)^{1/p′} < ∞.

Moreover, if c is the best constant in (1.2), then D ≤ c ≤ 4D.

Proposition B ([5]). Let 1 < p ≤ q < ∞, μ{x : d(x_{0}, x) = a} = 0. Then the inequality

( ∫_{0}^{a} v(t)|H̃f(t)|^{q} dt)^{1/q} ≤ c( ∫_{X} |f(x)|^{p}w(x) dμ)^{1/p}  (1.2)

holds with a constant c independent of f, f ∈ L_{w}^{p}(X), if and only if

D̃ = sup_{0<t<a} ( ∫_{0}^{t} v(τ) dτ )^{1/q}( ∫_{{x: d(x_{0},x)>t}} w^{1−p′}(x) dμ)^{1/p′} < ∞.

Moreover, if c is the best constant in (1.3), then D̃ ≤ c ≤ 4D̃_{1}.

Let f be a μ-measurable function defined on X and suppose that

H_{φψ}f(x) = φ(x) ∫_{B(x_{0},d(x_{0},x))} f(y)ψ(y)w(y) dμ(y)

is a Hardy-type operator on X, where φ and ψ are positive μ-measurable functions on X.

The following proposition is true (see Theorem 2.4 of [4] for a = ∞. The case a < ∞ can be obtained similarly).

Proposition C. Let r = s = 1 or r ∈ (0, ∞] and s ∈ [1, ∞], p = q = 1 or p ∈ (1, ∞), q ∈ [1, ∞], max{r, s} ≤ min{p, q}. The operator H_{φψ} is bounded from L_{w}^{rs}(X) to L_{v}^{pq}(X) if and only if

A ≡ sup_{0<t<a} ∥φ(·)χ_{{d(x_{0},y)>t}}(·)∥_{L_{v}^{pq}(X)}∥ψ(·)χ_{{d(x_{0},y)≤t}}(·)∥_{L_{w}^{r′s′}(X)} < ∞

with ∥H_{φψ}∥ ≈ A.

Let

H′_{φψ}f(x) = φ(x) ∫_{{y: d(x_{0},y)>d(x_{0},x)}} f(y)ψ(y) dμ(y).

By the dual arguments we deduce the following result:

Proposition D. Let r, s, p and q satisfy the conditions of Proposition C. The operator H′_{φ,ψ} is bounded from L_{w}^{rs}(X) to L_{v}^{pq}(X) if and only if:

A′ = sup_{t>0} ∥φ(·)χ_{{d(x_{0},y)≤t}}(·)∥_{L_{v}^{pq}(X)}∥ψ(·)χ_{{d(x_{0},y)>t}}(·)∥_{L_{w}^{r′s′}(X)} < ∞

with ∥H′_{φψ}∥ ≈ A′.

2. Singular integrals on SHT

In this section the existence of singular integrals is investigated and optimal sufficient conditions for the boundedness of singular integral operators in weighted Lorentz spaces defined on SHT (X, d, μ) are established.

Let k : X × X\{(x, x) : x ∈ X} → ℝ^{1} be a measurable function satisfying the condition

|k(x, y)| ≤ \frac{c}{μB(x, d(x, y))} ,

for all x, y ∈ X, x ≠ y, and

|k(x_{1}, y) − k(x_{2}, y)| + |k(y, x_{1}) − k(y, x_{2})| ≤ cω(\frac{d(x_{2}, x_{1})}{d(x_{2}, y)})\frac{1}{μ(B(x_{2}, d(x_{2}, y)))} ,

for all x_{1}, x_{2}, y ∈ X provided that d(x_{2}, y) > bd(x_{1}, x_{2}), where ω is a positive non-decreasing function on (0, ∞) satisfying the Δ_{2}-condition (i.e., ω(2t) ≤ cω(t) for all t ∈ (0, ∞)) and the Dini condition

∫_{0}^{1} \frac{ω(t)}{t} dt < ∞.

We shall also suppose that for some p_{0}, 1 < p_{0} < ∞, and all f ∈ L^{p_{0}}(X), the limit

Kf(x) = lim_{ε→0+} ∫_{X\B(x,ε)} k(x, y)f(y) dμ

exists a.e. on X and that the operatorK is bounded in L_{w}^{p}(X).

For the definition of singular integrals and for other related remarks see, e.g., [21], Chapter I, pp. 29–36, also [6], p. 295, and [5].

Theorem A ([9]). Let 1 < p < ∞. If w ∈ A_{p}(X), then the operator K is bounded in L_{w}^{p}(X).

Theorem B ([12], p. 207). Let 1 < p, q < ∞. If w ∈ A_{p}(X), then the operator K is bounded in L_{w}^{pq}(X). If the Hilbert transform H,

Hf(x) = p.v. ∫_{−∞}^{+∞}f(t)(x − t)^{−1} dt,

is bounded in L_{w}^{pq}(ℝ), then w ∈ A_{p}(ℝ).

The following Lemma holds.

Lemma 2.1. Let 1 < s ≤ p < ∞, ρ ∈ A_{p}(X). Suppose that for weight functions w and w_{1} the following conditions hold:

(1) there exist a positive increasing function σ defined on (0, 4a_{1}a) and a positive constant b such that

σ(d(x_{0}, x))ρ(x) ≤ bw(x)w_{1}(x)

for almost all x ∈ X; (2) for all t, 0 < t < a, we have

∥\frac{1}{w(·)w_{1}(·)} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then Kg(x) exists a.e. on X for any g satisfying the condition

∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)} < ∞.

Proof. Let us fix some positive α satisfying 0 < α < \frac{α}{α_{1}} and put

S_{α} = \{x ∈ X : d(x_{0}, x) ≥ \frac{α}{2}\}.

Assume that ∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)} < ∞. Introduce g in the following way:

g(x) = g_{1}(x) + g_{2}(x),

where g_{1}(x) = g(x) · χ_{Sa}(x), g_{2}(x) = g(x) − g_{1}(x).

For g_{1} we have

∫_{X} |g_{1}(x)|^{p}ρ(x) dμ = \frac{σ(α/2)}{σ(α/2)} ∫_{S_{a}} |g(x)|^{p}ρ(x) dμ

≤ \frac{1}{σ(α/2)} ∫_{S_{a}} |g(x)|^{p}ρ(x)σ(2a_{1}d(x_{0}, x)) dμ ≤ \frac{b}{σ(α/2)} ∫_{S_{a}} |g(x)|^{p}w_{1}^{p}(X)w(x) dμ

= \frac{b_{1}}{σ(α/2)} ∥g(·)w_{1}(·)∥_{L_{w}^{p}(X)}^{p} ≤ \frac{b}{σ(α/2)} ∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)}^{p}.

(In the latter inequality we have used Lemma B, part (ii).) Hence g_{1} ∈ L_{ρ}^{p}(X) and, according to Theorem A, Kg ∈ L_{ρ}^{p}(X) and consequently Kg(x) exists a.e. on X.

Now, let d(x_{0}, x) > αa_{1}, and let d(x_{0}, y) < α/2. We have

d(x_{0}, x) ≤ a_{1}(d(x_{0}, y) + d(y, x)) ≤ a_{1}(d(x_{0}, y) + a_{0}d(x, y)).

Hence

d(x, y) ≥ \frac{d(x_{0}, x)}{a_{1}a_{0}} − \frac{1}{a_{0}} d(x_{0}, y) ≥ \frac{α}{a_{0}} − \frac{α}{2a_{0}} = \frac{α}{2a_{0}} .

Moreover,

B(x_{0}, d(x, y)) ⊂ B(x, a_{1}(1 + a_{1}(1 + a_{0}^{2}))d(x, y))

and consequently we obtain the inequality

μB(x_{0}, α/2) ≤ c_{1}μB(x, d(x, y)),

where c_{1} is independent of x. So, for Kg_{2}, using the latter estimate and the Hölder’s inequality, we have

|Kg_{2}(x)| = | ∫_{X} g(y)k(x, y) dμ| ≤ c_{2} ∫_{B(x_{0},α/2)} \frac{|g(y)|}{μB(x, d(x, y))} dμ

≤ \frac{c_{3}}{μB(x_{0}, \frac{α}{2c_{0}} )} ∫_{B(x_{0},α/2)} |g(y)| dμ

≤ \frac{c_{3}}{μB(x_{0}, \frac{α}{2c_{0}} )} ∫_{B(x_{0},α/2)} |g(y)| \frac{1}{w(y)w_{1}(y)}w_{1}(y)w(y) dμ

≤ \frac{c_{3}}{μB(x_{0}, \frac{α}{2c_{0}} )} ∥χ_{B(x_{0}, \frac{α}{2} )}(·) \frac{1}{w(·)w_{1}(·)}∥_{L_{w}^{p′s′}(X)}∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)} < ∞.

Thus Kg_{2}(x) converges absolutely for all x with d(x_{0}, x) > αa_{1}. Since we can choose α arbitrarily small and μ{x_{0}} = 0, we conclude that Kg(x) exists μ- a.e. on X.

From Lemma 2.1 it is easy to derive

Lemma 2.2. Let 1 < s ≤ p < ∞. If u and u_{1}are positive increasing functions on (0, 4a_{1}a) and

∥\frac{1}{u(d(x_{0}, ·))u_{1}(d(x_{0}, ·))} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{u(d(x_{0},·))}^{p′s′}(X)} < ∞

for all t satisfying 0 < t < a, then for arbitrary g satisfying the condition

∥φ(·)u_{1}(d(x_{0}, ·))∥_{L_{u(d(x_{0},·))}^{ps}(X)} < ∞,

Kφ(x) exists a.e. on X.

The following lemma is proved in the same way as Lemma 2.1.

Lemma 2.3. Let a = ∞, 1 < s ≤ p < ∞, ρ ∈ A_{p}(X). If the weight functions w and w_{1} satisfy the conditions:

(1) there exists a decreasing positive function σ on (0, ∞) such that for almost all x ∈ X we have

σ(d(x_{0}, ·))ρ(x) ≤ bw(x)w_{1}^{p}(x),

where the positive constant b does not depend on x ∈ X; (2) if for every t > 0

∥\frac{μB(d(x_{0}, ·))^{−1}}{w(·)w_{1}(·)} χ_{\{d(x_{0},y)>t\}}(·)∥_{L_{w}^{p′s′}(X)} < ∞,

then Kg(x) exists a.e. on X for arbitrary g provided that ∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)} < ∞.

From Lemma 2.3 the next lemma follows easily:

Lemma 2.4. Let a = ∞, 1 < s ≤ p < ∞. Assume that u and u_{1} are positive decreasing functions on (0, ∞) and the condition

∥\frac{μB(d(x_{0}, ·))^{−1}}{u(d(x_{0}, ·))u_{1}(d(x_{0}, ·))} χ_{\{d(x_{0},y)>t\}}(·)∥_{L_{u(d(x_{0},·)}^{p′s′}(X)} < ∞

holds for any t > 0. Then Kg(x) exists a.e. on X for arbitrary g satisfying the condition g(·)u_{1}(d(x_{0}, ·)) ∈ L_{u(d(x_{0},·))}^{ps}(X).

Now we pass to the weight inequalities for the operator K.

Theorem 2.1. Let 1 < s ≤ p ≤ q < ∞, let w be a weight function on X, suppose that σ is a positive increasing continuous function on (0, 4a_{1}a), ρ ∈ A_{p}(X) and v(x) = σ(d(x_{0}, x))ρ(x). Suppose the following conditions are satisfied:

(a) there exists a positive constant b such that the inequality

σ(2a_{1}d(x_{0}, x))ρ(x) ≤ bw(x)

holds for almost every x ∈ X; (b)

B ≡ sup_{0<t<a} ∥μB(x_{0}, d(x_{0}, ·))^{−1}χ_{\{d(x_{0},y)>t\}} ∥_{L_{v}^{pq}(X)}∥\frac{1}{w(·)} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then there exists a positive constant c such that

∥Kf(·)∥_{L_{v}^{pq}(X)} ≤ c∥f(·)∥_{L_{w}^{ps}(X)}  (2.1)

for all f, f ∈ L_{w}^{ps}(X).

Proof. First, let us assume that σ is of the kind

σ(t) = σ(0+) + ∫_{0}^{t} φ(τ) dτ, φ ≥ 0.

Then we have

∥Kf(·)∥_{L_{v}^{pq}(X)} ≤ c_{1}(q ∫_{0}^{∞}λ^{q−1}( ∫_{{x∈X: |Kf(x)|>λ}} ρ(x)σ(0+) dμ)^{q/p} dλ)^{1/q}

+c_{1}(q ∫_{0}^{∞}λ^{q−1}( ∫_{{x∈X: |Kf(x)|>λ}} ρ(x)( ∫_{0}^{d(x_{0},x)}φ(t) dt) dμ)^{q/p} dλ)^{1/q} ≡ I_{1} + I_{2}.

In the case, where σ(0+) = 0, we have I_{1} = 0; otherwise, by Theorem B and Lemma A (part (ii)), we have

I_{1} = c_{1}σ(0+)^{1/p}∥Kf(·)∥_{L_{ρ}^{pq}(X)} ≤ c_{2}σ^{1/p}(0+)∥f(·)∥_{L_{ρ}^{pq}(X)}
≤ c_{2}σ^{1/p}(0+)∥f(·)∥_{L_{ρ}^{ps}(X)} ≤ c_{2}∥f(·)∥_{L_{w}^{ps}(X)}.

42 D. E. EDMUNDS, V. KOKILASHVILI, AND A. MESKHI

Now, let us estimate I_{2}. Set

f_{1t}(x) = f(x)χ_{\{d(x_{0},x)> \frac{t}{2a_{1}} \}}, f_{2t}(x) = f(x) − f_{1t}(x).

We have

I_{2} = c_{1}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{a} φ(t)( ∫_{{x: d(x_{0},x)>t,|Kf(x)|>λ}} ρ(x) dμ) dt)^{q/p} dλ)^{1/q}

≤ c_{3}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{a} φ(t)( ∫_{\{x: d(x_{0},x)>t\}} χ_{\{x: |Kf_{1t}(x)|> \frac{λ}{2} \}}ρ(x) dμ) dt)^{q/p} dλ)^{1/q}

+c_{3}(q ∫_{0}^{∞}λ^{q−1}( ∫_{0}^{a} φ(t)( ∫_{\{x: d(x_{0},x)>t\}} χ_{\{x: |Kf_{2t}(x)|> \frac{λ}{2} \}}ρ(x) dμ) dt)^{q/p} dλ)^{1/q}≡ I_{21} + I_{22}.

Applying the Minkowki’s inequality twice (\frac{q}{p} ≥ 1, \frac{p}{s} ≥ 1) and using Theorem B and Lemma A (part (ii)), we obtain

I_{21} ≤ c_{4}( ∫_{0}^{a} φ(t)( ∫_{0}^{∞}λ^{q−1}(  ∫_{\{x∈X: |Kf_{1t}(x)|> \frac{λ}{2} \}} ρ(x) dμ)^{q/p} dλ)^{p/q} dt)^{1/p}

≤ c_{5}( ∫_{0}^{a} φ(t)∥f_{1t}(·)∥_{L_{ρ}^{pq}(X)}^{p} dt)^{1/p} ≤ c_{5}( ∫_{0}^{a} φ(t)∥f_{1t}(·)∥_{L_{ρ}^{ps}(X)}^{p} dt)^{1/p}

≤ c_{5}( ∫_{0}^{∞}λ^{s−1}( ∫_{{x∈X: |f(x)|>λ}} ρ(x)( ∫_{0}^{2a_{1}d(x_{0},x)}φ(t) dt) dμ)^{s/p} dλ)^{1/s}

≤ c_{5}∥f(·)∥_{L_{w}^{ps}(X)} ≤ c_{6}∥f(·)∥_{L_{w}^{ps}(X)}. Next we shall estimate I_{22}. Note that if d(x_{0}, x) > t and d(x_{0}, y) < \frac{t}{2a_{1}} , then

d(x_{0}, x) ≤ a_{1}(d(x_{0}, y) + d(y, x)) ≤ a_{1}(d(x_{0}, y) + a_{0}d(x, y))

≤ a_{1}(\frac{t}{2a_{1}} + a_{0}d(x, y)) ≤ a_{1}(\frac{d(x_{0}, x)}{2a_{1}} + a_{0}d(x, y)).

Hence

\frac{d(x_{0}, x)}{2a_{1}a_{0}} ≤ d(x, y),

and we also have

μ(B(x, d(x_{0}, x))) ≤ bμ(B(x, \frac{d(x_{0}, x)}{2a_{1}a_{0}})) ≤ bμ(B(x, d(x, y))).

Moreover, it is easy to show that

μB(x_{0}, d(x_{0}, x)) ≤ b_{1}μB(x, d(x_{0}, x)).

Hence

μB(x_{0}, d(x_{0}, x)) ≤ b_{2}μB(x, d(x, y)).

Taking into account this inequality and using Proposition C we see that

I_{22} ≤ c_{7}∥\frac{1}{μB(x_{0}, d(x_{0}, ·))} ∫_{\{d(x_{0},y)<d(x_{0},·)\}} |f(y)| dy∥_{L_{v}^{pq}(X)} ≤ c_{9}∥f(·)∥_{L_{w}^{ps}(X)}.

Now, let σ be a positive continuous, but not absolutely continuous, increasing function on (0, 4a_{1}a); then there exists a sequence of absolutely continuous functions σ_{n} such that σ_{n}(t) ≤ σ(t) and lim_{n→∞} σ_{n}(t) = σ(t) for arbitrary t ∈

(0, 4a_{1}a). For these functions, we can take σ_{n}(t) = σ(0+) + n ∫_{0}^{t}[σ(τ) − σ(τ −

\frac{1}{n})] dτ. Denote v_{n}(x) = ρ(x)σ_{n}(d(x_{0}, x)); then B_{n} < B, where

B_{n} ≡ sup_{t>0} ∥χ_{\{d(x_{0},y)>t\}}(·)μ(B(x_{0}, d(x_{0}, ·)))^{−1}∥_{L_{vn}^{pq}(X)}∥\frac{1}{w(·)} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{w}^{p′s′}(X)}.

By virtue of what has been proved, if B < ∞, then the following inequality holds:

∥Kf(·)∥_{L_{vn}^{pq}(X)} ≤ c∥f(·)∥_{L_{w}^{ps}(X)},

where the constant c > 0 does not depend on n.

By passing to the limit as n → ∞, we obtain inequality (2.1).

Using the representation σ(t) = σ(+∞)+∫_{t}^{∞}ψ(τ) dτ, where σ(+∞) = lim_{t→∞} σ(t)

and ψ ≥ 0 on (0, ∞) and Proposition D, we obtain the following result.

Theorem 2.2. Let a = ∞, 1 < s ≤ p ≤ q < ∞, and suppose that σ is a positive decreasing continuous function on (0, ∞). Assume that ρ ∈ A_{p}(X) and v(X) = σ(d(x_{0}, x))ρ(x). Suppose the following conditions are satisfied:

(a) there exists a positive constant b such that the inequality

ρ(x)σ(\frac{d(x_{0}, x)}{2a_{1}}) ≤ bw(x)

is true for almost every x ∈ X; (b)

B′ = sup_{t>0} ∥χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{v}^{pq}(X)}∥\frac{(μB(x_{0}, d(x_{0}, x)))^{−1}}{w(·)} χ_{\{d(x_{0},y)>t\}}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then inequality (2.1) holds.

Now let us consider the particular cases of Theorems 2.1 and 2.2.

Theorem 2.3. Let a = ∞, 1 < p ≤ q < ∞, suppose σ_{1} and σ_{2} are positive, increasing functions on (0, ∞), let σ_{1} be a continuous function and suppose that ρ ∈ A_{p}(X). Put v(x) = σ_{2}(d(x_{0}, x))ρ(x), w(x) = σ_{1}(d(x_{0}, x))ρ(x). If

sup_{t>0} ∥(μB(x_{0}, d(x_{0}, ·)))^{−1}χ_{\{d(x_{0},y)>t\}}(·)∥_{L_{v}^{pq}(X)}∥\frac{1}{w(·)} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{w}^{p′}(X)} < ∞,

then there exists a positive constant c such that

∥Kf(·)∥_{L_{v}^{pq}(X)} ≤ c∥f(·)∥_{L_{w}^{p}(X)}

for all f ∈ L_{w}^{p}(X).

Proof. By Theorem 2.1 it is sufficient to show that there exists a positive constant b such that σ_{2}(2a_{1}t) ≤ bσ_{1}(t) for all t ∈ (0, ∞).

From the doubling condition (see (vii) in Definition 1.1) and the condition (1.1) it follows that the measure μ satisfies the reverse doubling condition at the point x_{0} (see, e.g., [20] and [23], Lemma 20). In other words, there exist constants η_{1} > 1 and η_{2} > 1 such that

μ(B(x_{0}, η_{1}r)) ≥ η_{2}μ(B(x_{0}, r))

for all r > 0.

Applying the Hölder’s inequality and using Lemma C, the reverse doubling condition and the fact that ρ^{1−p′} ∈ A_{p′}(X), we obtain

\frac{σ_{2}(2a_{1}t)}{σ_{1}(t)} ≤ (μ(B(x_{0}, 2a_{1}η_{1}t)\B(x_{0}, 2a_{1}t)))^{−p}( ∫_{B(x_{0},2a_{1}η_{1}t)\B(x_{0},2a_{1}t)} ρ(x) dμ)

×( ∫_{B(x_{0},2a_{1}η_{1}t)\B(x_{0},2a_{1}t)} ρ^{1−p′}(x) dμ)^{p−1} \frac{σ_{2}(2a_{1}t)}{σ_{1}(t)}

≤ ((1 − \frac{1}{η_{2}} )μB(x_{0}, 2a_{1}ηt))^{−p}( ∫_{B(x_{0},2a_{1}η_{1}t)\B(x_{0},2a_{1}t)} ρ(x) dμ)

×( ∫_{B(x_{0},t)} ρ^{1−p′}(x) dμ)^{p−1} \frac{σ_{2}(2a_{1}t)}{σ_{1}(t)}

≤ c_{1}(μB(x_{0}, 2a_{1}ηt))^{−p}∥χ_{(B(x_{0},2a_{1}η_{1}t)\B(x_{0},2a_{1}t))}(·)∥_{L_{v}^{pq}(X)}^{p}∥\frac{1}{w(·)} χ_{B(x_{0},t)}(·)∥_{L_{w}^{p′}(X)}^{p}

≤ c_{2}∥(μB(x_{0}, d(x_{0}, ·)))^{−1}χ_{\{d(x_{0},y)>t\}}(·)∥_{L_{v}^{pq}(X)}^{p}∥\frac{1}{w(·)} χ_{B(x_{0},t)}(·)∥_{L_{w}^{p′}(X)}^{p} ≤ c.

An analogous theorem dealing with singular integrals on homogeneous groups was proved in [16], and for singular integrals on SHT in the Lebesgue space in [5].

Theorem 2.4. Let a = ∞, 1 < s ≤ p ≤ q < ∞; suppose σ_{1}, σ_{2}, u_{1} and u_{2} are weight functions defined on X. Let ρ ∈ A_{p}(X), and suppose v = σ_{2}ρ, w = σ_{1}ρ. Assume that the following conditions are fulfilled:

(1) there exists a positive constant b such that for all t > 0

sup_{F_{t}} σ_{2}^{1/p}(x) sup_{F_{t}} u_{2}(x) ≤ b inf_{F_{t}} σ_{1}^{1/p}(x) inf_{F_{t}} u_{1}(x)

holds, where F_{t} = \{x ∈ X : \frac{t}{a_{1}} ≤ d(x_{0}, x) < 8a_{1}t\};

WEIGHT INEQUALITIES 45

(2)

sup_{t>0} ∥u_{2}(·)(μ(B(x_{0}, d(x_{0}, ·))))^{−1}χ_{{d(x_{0},t)>t}}(·)∥_{L_{v}^{pq}(X)}

×∥\frac{1}{u_{1}(·)w(·)} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{w}^{p′s′}(X)} < ∞;

(3)

sup_{t>0} ∥u_{2}(·)χ_{\{d(x_{0},t)≤t\}}(·)∥_{L_{v}^{pq}(X)}∥\frac{(μB(x_{0}, d(x_{0}, ·)))^{−1}}{u_{1}(·)w(·)} χ_{\{d(x_{0},y)>t\}}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then

∥u_{2}(·)Kf(·)∥_{L_{v}^{pq}(X)} ≤ c∥u_{1}(·)f(·)∥_{L_{w}^{ps}(X)},  (2.2)

where the positive constant c does not depend on f. Proof. Let

E_{k} ≡ B(x_{0}, 2^{k+1})\B(x_{0}, 2^{k}), G_{k,1} ≡ B(x_{0}, 2^{k−1}/a_{1}), G_{k,2} ≡ B(x_{0}, a_{1}2^{k+2})\B(x_{0}, 2^{k−1}/a_{1}), G_{k,3} ≡ X\B(x_{0}, a_{1}2^{k+2}).

We obtain

∥u_{2}(·)Kf(·)∥_{L_{v}^{pq}(X)}^{p} ≤ c_{1}∥ ∑_{k∈Z} u_{2}(·)K(fχ_{G_{k,1}})(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(X)}^{p}

+c_{1}∥ ∑_{k∈Z} u_{2}(·)K(fχ_{G_{k,2}})(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(X)}^{p}

+c_{1}∥ ∑_{k∈Z} u_{2}(·)K(fχ_{G_{k,3}})(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(X)}^{p} ≡ c_{1}(S_{1}^{p} + S_{2}^{p} + S_{3}^{p}).

Now we estimate S_{1}. Note that

d(x_{0}, y) < \frac{2^{k−1}}{a_{1}} ≤ \frac{d(x_{0}, x)}{2a_{1}}

when x ∈ E_{k}, and y ∈ G_{k,1}. From the latter inequality we have

μB(x, d(x_{0}, x)) ≤ b_{1}μB(x, d(x, y)).

Indeed,

d(x_{0}, x) ≤ a_{1}(d(x_{0}, y) + d(y, x)) ≤ a_{1}(\frac{d(x_{0}, x)}{2a_{1}} + a_{0}d(x, y)).

Hence

\frac{1}{2a_{1}a_{0}} d(x_{0}, x) ≤ d(x, y).

Correspondingly,

μB(x, d(x_{0}, x)) ≤ b_{2}μB(x, \frac{d(x_{0}, x)}{2a_{1}a_{0}}) ≤ b_{2}μB(x, d(x, y)).

46 D. E. EDMUNDS, V. KOKILASHVILI, AND A. MESKHI

It is easy to see that

μB(x_{0}, d(x_{0}, x)) ≤ b_{3}μB(x, d(x_{0}, x))

and, finally, we obtain

μB(x_{0}, d(x_{0}, x)) ≤ b_{4}μB(x, d(x, y)).

By considering the latter inequality we have

|K(fχ_{G_{k1}})(x)| ≤ b_{5} ∫_{X} \frac{|f(y)|χ_{G_{k1}}(y)}{μB(x, d(x, y))} dμ

≤ \frac{b_{6}}{μB(x_{0}, d(x_{0}, x))} ∫_{B(x_{0},d(x_{0},x))} |f(y)| dμ,

when x ∈ E_{k}, and by Proposition C we obtain

S_{1}^{p} ≤ c_{2}∥u_{2}(·)(μB(x_{0}, d(x_{0}, ·)))^{−1}  ∫_{B(x_{0},d(x_{0},x))} |f(y)| dμ∥_{L_{v}^{pq}(X)}^{p}
≤ c_{3}∥u_{1}(·)f(·)∥_{L_{w}^{ps}(X)}^{p}.

Now we shall estimate S_{3}^{p}. It is easy to check that if x ∈ E_{k} and y ∈ G_{k,3}, then d(x_{0}, y) ≤ d(x, y) and

μB(x_{0}, d(x_{0}, y)) ≤ b_{7}μB(x, d(x, y)).

By virtue of Proposition D we obtain

S_{3}^{p} ≤ c_{4}∥u_{2}(·) ∫_{\{d(x_{0},y)>d(x_{0},x)\}} \frac{|f(y)|}{μB(x_{0}, d(x_{0}, y))} dμ∥_{L_{v}^{pq}(X)}^{p} ≤ c_{5}∥u_{1}(·)f(·)∥_{L_{w}^{pq}(X)}^{p}.

Now let us estimate S_{2}^{p}. By Lemma B (part (ii)),

S_{2}^{p} ≤ ∑_{k∈u} ∥u_{2}(·)K(fχ_{G_{k,2}})(·)χ_{E_{k}} (·)∥_{L_{v}^{pq}(X)}^{p} ≡ ∑_{k∈u} S_{k,2}^{p}.

We shall use the following notation:

u_{2,k} ≡ sup_{x∈E_{k}} u_{2}(x), σ_{2,k} ≡ sup_{x∈E_{k}} σ_{2}(x), u_{1,k} ≡ inf_{x∈G_{k,2}} u_{1}(x), σ_{1,k} ≡ inf_{x∈G_{k,2}} σ_{1}(x).

By Theorem C and Lemma A we have

S_{k,2} ≤ u_{2,k}σ_{2,k}^{1/p}∥K(fχ_{G_{k,2}})(·)∥_{L_{ρ}^{pq}(X)} ≤ c_{6}u_{2,k}σ_{2,k}^{1/p}∥f(·)χ_{G_{k,2}}(·)∥_{L_{ρ}^{pq}(X)}
≤ c_{6}u_{2,k}σ_{2,k}^{1/p}∥f(·)χ_{G_{k,2}}(·)∥_{L_{ρ}^{ps}(X)} ≤ c_{7}u_{1,k}σ_{1,k}^{1/p}∥f(·)χ_{G_{k,2}}(·)∥_{L_{ρ}^{ps}(X)}^{p}
≤ c_{8}∥u_{1}(·)f(·)χ_{G_{k,2}}(·)∥_{L_{w}^{ps}(X)}.

By Lemma B (part (i)) we finally obtain S_{2}^{p} ≤ c_{9}∥u_{1}(·)f(·)∥_{L_{w}^{ps}(X)}.

Remark 2.1. It is easy to check that Theorem 2.4 is still valid if we replace condition (1) by the condition

(1′)

sup_{E_{x}} σ_{2}(y) sup_{E_{x}} u_{2}^{p}(y) ≤ \overline{b}σ_{1}(x)u_{1}^{p}(x),

where \overline{b} > 0 does not depend on x ∈ X and where

E_{x} = \{y : \frac{d(x_{0}, x)}{4a_{1}} ≤ d(x_{0}, y) < 4a_{1}d(x_{0}, x)\}.

Indeed, we have

S_{k,2} ≤ σ_{2,k}^{1/p}u_{2,k}∥T (fχ_{G_{k,2}})(·)∥_{L_{ρ}^{pq}(X)} ≤ σ_{2,k}^{1/p}u_{2,k}∥T (fχ_{G_{k,2}})(·)∥_{L_{ρ}^{p}(X)}

≤ b_{1}σ_{2,k}^{1/p}u_{2,k}∥f(·)χ_{G_{k,2}}(·)∥_{L_{ρ}^{p}(X)}

= b_{1} ∫_{G_{k,2}} ( sup_{2^{k}≤d(x_{0},y)<2^{k+1}} σ_{2}(y))( sup_{2^{k}≤d(x_{0},y)<2^{k+1}} u_{2}^{p}(y))|f(x)|^{p}ρ(x) dμ)^{1/p}

≤ b_{1}( ∫_{G_{k,2}} ( sup_{E_{x}} σ_{2}(y))( sup_{E_{x}} u_{2}^{p}(y))|f(x)|^{p}ρ(x) dμ)^{1/p}

≤ b_{2}( ∫_{G_{k,2}} σ_{1}(x)u_{1}^{p}(x)ρ(x)|f(x)|^{p} dμ)^{1/p}

= b_{2}∥f(·)u_{1}(·)χ_{G_{k,2}}(·)∥_{L_{w}^{p}(X)} ≤ b_{2}∥f(·)u_{1}(·)χ_{G_{k,2}}(·)∥_{L_{w}^{sp}(X)}.

In [19] it is proved that conditions (2) and (3) of Theorem 2.4 are also necessary for equality (2.2) to be fulfilled when K is the Hilbert transform.

Theorem 2.5. Let μ(X) = ∞, 1 < s ≤ p ≤ q and suppose φ_{1}, φ_{2}, and v are positive increasing functions on (0, ∞).

If the condition

sup_{t>0} B(t) ≡ sup_{t>0} ∥φ_{2}(d(x_{0}, ·))(μB(x_{0}, d(x_{0}, x)))^{−1}χ_{{d(x_{0},y)>t}}(·)∥_{L_{v(d(x_{0},·))}^{pq}(X)}

× ∥\frac{1}{φ_{1}(d(x_{0}, ·))} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L^{p′s′}(X)} < ∞

is fulfilled, then the following weighted inequality holds:

∥Kf(·)φ_{2}(d(x_{0}, ·))∥_{L_{v(d(x_{0},·))}^{pq}(X)} ≤ c∥f(·)φ_{1}(d(x_{0}, ·))∥_{L^{ps}(X)}.  (2.3)

Proof. First, let us prove the inequality

φ_{2}(8a_{1}t)v^{1/p}(8a_{1}t) ≤ b_{1}φ_{1}(\frac{t}{a_{1}} ),

where the positive constant b_{1} does not depend on t > 0. Indeed, by Lemma A (part (i)) and by the reverse doubling condition for μ we have

c ≥ B(t) ≥ ∥φ_{2}(d(x_{0}, ·))(μB(x_{0}, d(x_{0}, ·))^{−1}χ_{{t<d(x_{0},y)<η_{1}t}}(·)∥_{L_{v(d(x_{0},·))}^{ps}(X)}

× ∥\frac{1}{φ_{1}(d(x_{0}, ·))} χ_{\{d(x_{0},y)≤ \frac{t}{8a^{2}_{1}} \}}(·)∥_{L^{p′s′}(X)}

≥ c_{1}(μB(x_{0}, η_{1}t))^{−1}φ_{2}(t)( ∫_{{t<d(x_{0},y)<η_{1}t}} v(d(x_{0}, y)) dμ)^{1/p}

× φ_{1}^{−1}(\frac{t}{8a_{1}^{2}} )(μB(x_{0}, \frac{t}{8a_{1}^{2}} ))^{1/p}

≥ c_{2}φ_{2}(t)v^{1/p}(t)μB(x_{0}, t)^{−1}μB(x_{0}, t)^{1/p′}φ_{1}^{−1}(\frac{t}{8a_{1}^{2}} )μB(x_{0}, t)^{1/p′}

= c_{2} \frac{φ_{2}(t)v^{1/p}(t)}{φ_{1}(\frac{t}{8a_{1}^{2}} )} .

Now, we are to show that the following condition is fulfilled:

sup_{t>0} B_{1}(t) ≡ sup_{t>0} ∥φ_{2}(d(x_{0}, ·)) χ_{{d(x_{0},y)≤t}}(·)∥_{L_{v}^{pq}(X)}

× ∥\frac{μB(x_{0}, d(x_{0}, ·))^{−1}}{φ_{1}(d(x_{0}, ·))} χ_{\{d(x_{0},y)>t\}}(·)∥_{L^{p′s′}(X)} < ∞.

Indeed, by the monotonic property of the functions φ_{1}, φ_{2}, and v and by Lemma A (part (i)), we obtain:

B_{1}(t) ≤ c_{3}φ_{2}(t)(μB(x_{0}, t))^{1/p}v^{1/p}(t)φ_{1}^{−1}(\frac{t}{8a_{1}^{2}} )
× ∥(μB(x_{0}, d(x_{0}, ·))^{−1}χ_{{d(x_{0},y)>t}}(·)∥_{L^{p′s′}(X)}.

On the other hand, we have

∥(μB(x_{0}, d(x_{0}, ·))^{−1}χ_{{d(x_{0},y)>t}}(·)∥_{L^{p′s′}(X)}

= (s′ ∫_{0}^{∞}λ^{s′−1}(μ{x : (μB(x_{0}, d(x_{0}, x))^{−1} > λ} ∩ {d(x_{0}, x) ≥ t}))^{s′/p′} dλ)^{1/s′}

= (s′ ∫_{0}^{(μB(x_{0},t))^{−1}}λ^{s′−1}(μ{x : μB(x_{0}, d(x_{0}, x)) < λ^{−1}})^{s′/p′} dλ)^{1/s′}

≤ c_{4}( ∫_{0}^{(μB(x_{0},t))^{−1}}λ^{s′−1}λ^{−s′/p′} dλ)^{1/s′} = c_{5}(μB(x_{0}, t))^{−1/p}.

Here we have used the inequality

μ{x : μB(x_{0}, d(x_{0}, x)) < λ^{−1}} ≤ bλ^{−1},

where the positive constant b is from the doubling condition for μ. Thus we obtain

B_{1}(t) ≤ c_{6}φ_{2}(t)v^{1/p}(t)φ_{1}^{−1}(\frac{t}{8a_{1}^{2}} ) ≤ c_{7}

for arbitrary t > 0.

By Theorem 2.4 we conclude that inequality (2.3) holds.

Remark 2.2. All results of this section remain valid if we omit the continuity of σ, but require that μB(x_{0}, r) be continuous with respect to r, where x_{0} is the same fixed point as above. This follows from the fact that in this case the sequence σ_{n}(d(x_{0}, x)), where σ_{n} are absolutely continuous functions (see the proof of Theorem 2.1.), converges to σ(d(x_{0}, x)) a.e. on X. On the other hand, the continuity of μB(x_{0}, r) with respect to r is equivalent to the condition μ{x : d(x_{0}, x) = r} = 0 for all r > 0.

3. Singular Integrals on Spaces of Nonhomogeneous Type

In this section we present weighted inequalities for Calderón–Zygmund singular integrals defined on spaces of nonhomogeneous type.

Let (X, d, μ) be a spaces of nonhomogeneous type with metric d and measure μ (i.e., a_{0} = 1, a_{1} = 1, we have equality instead of the inequality in (iii) and (vii) need not be valid in Definition 1.1) satisfying the condition

μ\overline{B}(x, r) ≤ r^{α}, x ∈ X, r > 0,

for some α > 0, where \overline{B}(x, r) ≡ \{y : d(x, y) ≤ r\}.

Let the kernel k satisfy the following conditions:

(1)

|k(x, y)| ≤ c_{1}d(x, y)^{−α},

for all x, y ∈ X, x ≠ y; (2) there exist c_{2} > 0 and ε ∈ (0, 1] such that

max \{|k(x, y) − k(x_{1}, y)|, |k(y, x) − k(y, x_{1})|\} ≤ c_{2} \frac{d(x, x_{1})^{ε}}{d(x, y)^{α+ε}}

whenever d(x, x_{1}) ≤ 2^{−1}d(x, y), x ≠ y. Assume also that the integral operator

T f(x) = lim_{ε→0} ∫_{X\B(x_{0},ε)} f(y)k(x, y) dμ

is bounded in L^{2}(X).

The following theorem is valid.

Theorem C ([18]). The operator T is bounded in L^{p}(X) for 1 < p < ∞ and is of weak type (1, 1).

Recall that a ≡ sup{d(x_{0}, x) : x ∈ X}, where x_{0} is a given point of X and B(x_{0}, R)\B(x_{0}, r) ≠ ∅ whenever 0 < r < R < a.

We remark that the condition μ{x} = 0 is automatically satisfied for all x ∈ X.

Lemma 3.1. Let 1 < p < ∞, and let w be a weight function on X. Assume that the following two conditions are satisfied:

(i) there exists a positive increasing function v on (0, 4a) such that for almost all x ∈ X the inequality

v(2d(x_{0}, x)) ≤ b_{1}w(x)

holds, where the positive constant b_{1} does not depend on x; (ii)

I(t) ≡ ∫_{B(x_{0},t)} w^{1−p′}(x) dμ < ∞

for all t > 0. Then T φ(x) exists μ-a.e. for any φ ∈ L_{w}^{p}(X).

Proof. Let 0 < α < a and let us denote

S_{β} = {x ∈ X : d(x_{0}, x) ≥ β/2}.

Suppose φ ∈ L_{w}^{p}(X) and represent φ as follows:

φ(x) = φ_{1}(x) + φ_{2}(x),

where φ_{1} = φχ_{S_{β}} and φ_{2} = φ − φ_{1}. Due to condition (i) we see that

∫_{X} |φ_{1}(x)|^{p} dμ = \frac{v(\frac{β}{2} )}{v(\frac{β}{2} )} ∫_{S_{β}} |φ(x)|^{p} dμ

≤ \frac{1}{v(\frac{β}{2} )} ∫_{S_{β}} |φ(x)|^{p}v(2d(x_{0}, x)) dμ ≤ \frac{c_{1}}{v(\frac{β}{2} )} ∫_{S_{β}} |φ(x)|^{p}w(x) dμ < ∞

for arbitrary β, 0 < β < a. Consequently T φ_{1} ∈ L^{p}(X) and, according to Theorem C, T φ_{1}(x) exists μ-a.e. on X.

Now let x be such that d(x_{0}, x) > β. If y ∈ X and d(x_{0}, y) < \frac{β}{2} , then

d(x_{0}, x) ≤ d(x_{0}, y) + d(x, y) ≤ \frac{d(x_{0}, x)}{2} + d(x, y).

Hence β/2 < \frac{d(x_{0},x)}{2} ≤ d(x, y) and we obtain

|T φ_{2}(x)| = | ∫_{X} φ_{2}(y)k(x, y) dμ| ≤ c_{2} ∫_{B(x_{0}, \frac{β}{2} )} \frac{|φ(y)|}{d(x, y)^{α}} dμ

≤ c_{3} ∫_{B(x_{0}, \frac{β}{2} )} \frac{|φ(y)|}{d(x, y)^{α}} dμ ≤ c_{4}β^{−α} ∫_{B(x_{0}, \frac{β}{2} )} |φ(y)| dμ

≤ c_{4}β^{−α}( ∫_{B(x_{0}, \frac{β}{2} )} |φ(y)|^{p}w(y) dμ)^{1/p}( ∫_{B(x_{0}, \frac{β}{2} )} w^{1−p′}(y) dμ)^{1/p′} < ∞.

Thus T φ(x) is absolutely convergent for arbitrary x such that d(x_{0}, x) > β. As we can take β arbitrarily small and μ{x_{0}} = 0, we conclude that T φ(x) exists μ-a.e. on X.

Theorem 3.1. Let 1 < p < ∞. Assume that v is a positive increasing continuous function on (0, 4a). Suppose that w is a weight on X. Let the following two conditions hold:

(i) there exists a constant b_{1} > 0 such that the inequality

v(2d(x_{0}, x)) ≤ b_{1}w(x)

is fulfilled for μ-almost all x ∈ X; (ii)

sup_{0<t<a} ( ∫_{X\B(x_{0},t)} \frac{v(d(x_{0}, x))}{d(x_{0}, x)^{αp}} dμ)^{1/p}( ∫_{B(x_{0},t)} w^{1−p′}(x) dμ)^{1/p′} < ∞.

Then T is bounded from L_{w}^{p}(X) to L_{v(d(x_{0},·))}^{p}(X).

Proof. Without loss of generality we can suppose that v has the form

v(t) = v(0+) + ∫_{0}^{t} ϕ(τ) dτ, ϕ ≥ 0.

We have

∫_{X} |T f(x)|^{p}v(d(x_{0}, x)) dμ = v(0+) ∫_{X} |T f(x)|^{p} dμ

+ ∫_{X} |T f(x)|^{p}( ∫_{0}^{d(x_{0},x)}ϕ(t) dt) dμ ≡ I_{1} + I_{2}.

If v(0+) = 0, then I_{1} = 0. If v(0+) ≠ 0, by Theorem C we obtain

I_{1} ≤ c_{1}v(0+) ∫_{X} |f(x)|^{p} dμ ≤ c_{1} ∫_{X} |f(x)|^{p}v(d(x_{0}, x)) dμ ≤ c_{2} ∫_{X} |f(x)|^{p}w(x) dx.

52 D. E. EDMUNDS, V. KOKILASHVILI, AND A. MESKHI

Changing the order of integration in I_{2}, we have

I_{2} = ∫_{0}^{a} ϕ(t)( ∫_{{x: d(x_{0},x)>t}} |T f(x)|^{p} dμ) dt

≤ c_{3} ∫_{0}^{a} ϕ(t)(  ∫_{\{x: d(x_{0},x)>t\}} |  ∫_{\{y: d(x_{0},y)> \frac{t}{2} \}} f(y)k(x, y) dμ|^{p} dμ) dt
+c_{3} ∫_{0}^{a} ϕ(t)(  ∫_{\{x: d(x_{0},x)>t\}} |  ∫_{\{y: d(x_{0},y)≤ \frac{t}{2} \}} f(y)k(x, y) dμ|^{p} dμ) dt ≡ I_{21} + I_{22}.

Using again the boundedness of T in L^{p}(X) we obtain

I_{21} ≤ c_{4} ∫_{0}^{a} ϕ(t)( ∫_{\{y: d(x_{0},y)> \frac{t}{2} \}} |f(y)|^{p} dμ) dt = c_{4} ∫_{X} |f(y)|^{p}( ∫_{0}^{2d(x_{0},y)}ϕ(t) dt) dμ

≤ c_{4} ∫_{X} |f(y)|^{p}v(2a_{1}d(x_{0}, y)) dμ ≤ c_{5} ∫_{X} |f(y)|^{p}w(y) dμ.

Now let us estimate I_{22}. When d(x_{0}, x) > t and d(x_{0}, y) ≤ \frac{t}{2} we have

d(x_{0}, x) ≤ d(x_{0}, y) + d(y, x) = d(x_{0}, y) + d(x, y)

≤ \frac{t}{2} + d(x, y) ≤ \frac{d(x_{0}, x)}{2} + d(x, y).

Hence

\frac{d(x_{0}, x)}{2} ≤ d(x, y).

Consequently,

I_{22} ≤ c_{6} ∫_{0}^{a} ϕ(t)(  ∫_{\{x: d(x_{0},y)>t\}} (  ∫_{\{y: d(x_{0},y)≤ \frac{t}{2} \}} \frac{|f(y)|}{d(x, y)^{α}} dμ(y))^{p} dμ(x)) dt

≤ c_{7} ∫_{0}^{a} ϕ(t)( ∫_{\{x: d(x_{0},x)>t\}} \frac{1}{d(x_{0}, x)^{αp}} dμ)( ∫_{B(x_{0},t)} |f(y)| dμ)^{p} dt.

It is easy to see that for any s, 0 < s < a, we have

∫_{s}^{a} ϕ(t)( ∫_{\{x: d(x_{0},x)>t\}} \frac{1}{d(x_{0}, x)^{αp}} dμ) dt

≤  ∫_{\{x: d(x_{0},x)≥s\}} \frac{1}{d(x_{0}, x)^{αp}} ( ∫_{s}^{d(x_{0},x)}ϕ(t) dt) dμ ≤  ∫_{\{x: d(x_{0},x)≥s\}} \frac{v(d(x_{0}, x))}{d(x_{0}, x)^{αp}} dμ.

Finally, using Proposition A, we obtain

I_{22} ≤ c_{8} ∫_{X} |f(x)|^{p}w(x) dμ.

Lemma 3.2. Let a = ∞, 1 < p < ∞, and let w be a weight function on X. Suppose the following conditions are fulfilled:

(i) there exists a positive decreasing function v on (0, ∞) such that

v(\frac{d(x_{0}, x)}{2}) ≤ cw(x) a.e.;

(ii) for all t > 0

∫_{X\B(x_{0},t)} w^{1−p′}(x)(d(x_{0}, x))^{−αp′} dμ < ∞.

Then T φ(x) exists μ-a.e. for arbitrary φ ∈ L_{w}^{p}(X).

Proof. Fix arbitrarily β > 0 and let

S_{β} = {x : d(x_{0}, x) ≥ β}.

Represent φ as follows:

φ(x) = φ_{1}(x) + φ_{2}(x),

where φ_{1}(x) = φ(x)χ_{S_{β}} (x) and φ_{2}(x) = φ(x) − φ_{1}(x).

For φ_{2} we have

∫_{X} |φ_{2}(x)|^{p} dμ = \frac{v(β)}{v(β)} ∫_{B(x_{0},β)} |φ(x)|^{p} dμ

≤ \frac{1}{v(β)} ∫_{B(x_{0},β)} |φ(x)|^{p}v(d(x_{0}, x)) dμ ≤ \frac{c_{1}}{v(β)} ∫_{B(x,β)} |φ(x)|^{p}w(x) dμ < ∞.

Consequently, φ_{2} ∈ L^{p}(X) and by Theorem C we have T φ_{2} ∈ L^{p}(X). Hence T φ_{2}(x) exists a.e. on X.

Now let x ∈ X and let d(x_{0}, x) < β/2. If d(x_{0}, y) ≥ β, then

\frac{d(x_{0}, y)}{2} ≥ d(x, y).

Using this inequality, we obtain

|T φ_{1}(x)| ≤ c_{2} ∫_{S_{β}} \frac{|φ(y)|}{d(x, y)^{α}} dμ ≤ c_{3} ∫_{S_{β}} \frac{|φ(y)|}{d(x_{0}, y)^{α}} dμ

≤ c_{3}( ∫_{S_{β}} |φ(y)|^{p}w(y) dμ)^{1/p}( ∫_{S_{β}} w^{1−p′}(y)d(x_{0}, y)^{−αp′} dμ)^{1/p′} < ∞.

As we may take β arbitrarily large, we conclude that T φ(x) exists a.e.

Theorem 3.2. Let a = ∞, 1 < p < ∞ and let v be a positive continuous decreasing function (0, ∞). Suppose that w is a weight function on X and the following conditions are satisfied:

(i) for almost all x

v(d(x_{0}, x)/2) ≤ cw(x);

(ii)

sup_{t>0} ( ∫_{B(x_{0},t)} v(d(x_{0}, x)) dμ)^{1/p}( ∫_{X\B(x_{0},t)} w^{1−p′}(x)d(x_{0}, x)^{−αp′} dμ)^{1/p′} < ∞.

Then the operator T is bounded from L_{w}^{p}(X) to L_{v(d(x_{0},·))}^{p}(X).

Proof. Without loss of generality we can represent v as

v(t) = v(+∞) + ∫_{t}^{∞}ϕ(τ) dτ, ϕ ≥ 0.

Further,

∫_{X} |T f(x)|^{p}v(d(x_{0}, x)) dx

= v(+∞) ∫_{X} |T f(x)|^{p} dμ + ∫_{X} |T f(x)|^{p}( ∫_{d(x_{0},x)}^{∞} ϕ(t)dt) dμ ≡ I_{1} + I_{2}.

If v(+∞) = 0, then I_{1} = 0. But if v(+∞) ≠ 0, then by virtue of the boundedness of T in L^{p}(X) we have

I_{1} ≤ c_{1}v(+∞) ∫_{X} |f(x)|^{p} dμ ≤ c_{1} ∫_{X} |f(x)|^{p}v(d(x_{0}, x)) dμ ≤ c_{2} ∫_{X} |f(x)|^{p}w(x) dμ.

Now we pass to I_{2} :

I_{2} = ∫_{0}^{∞}ϕ(t)( ∫_{B(x_{0},t)} |T f(x)|^{p} dμ) dt ≤ c_{3} ∫_{0}^{∞}ϕ(t)( ∫_{B(x_{0},t)} |T f_{t}^{(1)}(x)|^{p} dμ) dt

+ c_{3} ∫_{0}^{∞}ϕ(t)( ∫_{B(x_{0},t)} |T f_{t}^{(2)}(x)|^{p} dμ) dt = I_{21} + I_{22},

where f_{t}^{(1)} = fχ_{B(x_{0},2t)} and f_{t}^{(2)} = f − f_{t}^{(1)}. Again using Theorem C we have

I_{21} ≤ c_{4} ∫_{0}^{∞}ϕ(t)( ∫_{B(x_{0},2t)} |f(y)|^{p} dμ) dt

= c_{4} ∫_{X} |f(y)|^{p}( ∫_{\frac{d(x_{0},x)}{2}}^{∞} ϕ(t) dt) dμ ≤ c_{5} ∫_{X} |f(y)|^{p}w(y) dμ.

It remains to estimate I_{22}. If x ∈ B(x_{0}, t) and y ∈ X\B(x_{0}, 2t), then

\frac{d(x_{0}, y)}{2} ≤ d(x, y).

Consequently,

I_{22} ≤ c_{5} ∫_{0}^{∞}ϕ(t)( ∫_{B(x_{0},t)} dμ)( ∫_{X\B(x_{0},2a_{1}t)} |f(y)|d(x_{0}, y)^{−α} dμ)^{p} dt.

Moreover,

∫_{0}^{s} ϕ(t)( ∫_{B(x_{0},t)} dμ) dt = ∫_{B(x_{0},s)} ( ∫_{d(x_{0},x)}^{s} ϕ(t) dt) dμ ≤ ∫_{B(x_{0},s)} v(d(x_{0}, x)) dμ

and due to Proposition B we finally obtain the boundedness of T.

Now we are going to establish weighted estimates for the operator T in Lorentz spaces defined on spaces of nonhomogeneous type. Theorem C and the interpolation theorem imply

Proposition E. Let 1 < p, q < ∞. Then T is bounded in L^{pq}(X).

The following lemmas are obtained in the same way as in the homogeneous case. Instead of Theorem A we need to use Theorem C.

Lemma 3.3. Let 1 < s ≤ p < ∞. Let the weight functions w and w_{1} satisfy the conditions:

(1) there exists an increasing function v on (0, 4a) such that the inequality

v(d(x_{0}, x)) ≤ bw(x)w_{1}(x)

holds for almost all x ∈ X; (2) for every t, 0 < t < a, the norm

∥\frac{1}{w(·)w_{1}(·)} χ_{\{d(x_{0},y)≤t\}}(·)∥_{L_{w}^{p′s′}(X)}

is finite.

Then T g(x) exists a.e. on X for any g satisfying the condition

∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)} < ∞.

Lemma 3.4. Let 1 < s ≤ p < ∞. Suppose also that u and u_{1} are positive increasing functions on (0, 4a_{1}a) and

∥\frac{1}{u(d(x_{0}, ·))u_{1}(d(x_{0}, ·))} χ_{\overline{B}(x_{0},t)}(·)∥_{L_{u(d(x_{0},·))}^{p′s′}(X)} < ∞

for all t satisfying the condition 0 < t < a. Then for arbitrary φ with

∥φ(·)u_{1}(d(x_{0}, ·))∥_{L_{u(d(x_{0},·))}^{ps}(X)} < ∞,

T φ(x) exists a.e. on X.

The following lemmas are also true.

Lemma 3.5. Let a = ∞, 1 < s ≤ p < ∞. Suppose that for weights w and w_{1} the following conditions are satisfied:

(1) there exists a decreasing positive function v on (0, ∞) such that

v(d(x_{0}, ·)) ≤ bw(x)w_{1}^{p}(x)

for almost all x ∈ X; (2) for every t > 0,

∥\frac{d(x_{0}, ·)^{−α}}{w(·)w_{1}(·)} χ_{X\\overline{B}(x_{0},t)}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then T g(x) exists a.e. on X for arbitrary g satisfying ∥g(·)w_{1}(·)∥_{L_{w}^{ps}(X)} < ∞.

From the previous lemmas easily follows

Lemma 3.6. Let a = ∞, 1 < s ≤ p < ∞. Suppose also that for the decreasing functions u and u_{1} on (0, ∞) the following condition is satisfied

∥\frac{d(x_{0}, ·)^{−α}}{u(d(x_{0}, ·))u_{1}(d(x_{0}, ·))} χ_{X\\overline{B}(x_{0},t)}(·)∥_{L_{u(d(x_{0},·))}^{p′s′}(X)} < ∞,

for all t > 0. Then T g(x) exists a.e. on X for g satisfying the condition

g(·)u_{1}(d(x_{0}, ·)) ∈ L_{u(d(x_{0},·))}^{ps}(X).

Using Propositions E and C, we obtain the following result in the same way as Theorem 2.1.

Theorem 3.3. Let 1 < s ≤ p ≤ q < ∞, and let w be a weight function on X. Assume that v is a positive increasing continuous function on (0, 4a). Suppose also that the following two conditions are satisfied:

(1) there exists a positive constant c such that the inequality

v(2a_{1}d(x_{0}, x)) ≤ cw(x)

holds for almost every x ∈ X; (2)

sup_{0<t<a} ∥(d(x_{0}, ·))^{−α}χ_{X\\overline{B}(x_{0},t)} ∥_{L_{v}^{pq}(X)}∥\frac{1}{w(·)} χ_{\overline{B}(x_{0},t)}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then the operator T is bounded from L_{w}^{ps}(X) to L_{v(d(x_{0},·)}^{pq}(X).

Theorem 3.4. Let a = ∞, and let 1 < s ≤ p ≤ q < ∞; suppose that v is a positive decreasing continuous function on (0, ∞). Assume also that:

(i) there exists a positive constant b such that the inequality

v(\frac{d(x_{0}, x)}{2a_{1}}) ≤ bw(x)

is true for a.e. x ∈ X; (ii)

sup_{t>0} ∥χ_{\overline{B}(x_{0},t)}(·)∥_{L_{v}^{pq}(X)}∥\frac{(d(x_{0}, ·))^{−α}}{w(·)} χ_{X\\overline{B}(x_{0},t)}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then T acts boundedly from L_{w}^{ps}(X) into L_{v(d(x_{0},·)}^{pq}(X).

Finally, we formulate the following result:

Theorem 3.5. Let a = ∞, 1 < s ≤ p ≤ q < ∞. Suppose that v, w, u_{1} and u_{2} are weights on X. Assume that the following conditions are fulfilled:

(1) there exists a positive constant b such that for all t > 0

sup_{F_{t}} v^{1/p}(x) sup_{F_{t}} u_{2}(x) ≤ b inf_{F_{t}} w^{1/p}(x) inf_{F_{t}} u_{1}(x)

holds, where F_{t} = \{x ∈ X : \frac{t}{a_{1}} ≤ d(x_{0}, x) < 8a_{1}t\}; (2)

sup_{t>0} ∥u_{2}(·)(d(x_{0}, ·))^{−α}χ_{X\\overline{B}(x_{0},t)}(·)∥_{L_{v}^{pq}(X)}∥\frac{1}{u_{1}(·)w(·)} χ_{\overline{B}(x_{0},t)}(·)∥_{L_{w}^{p′s′}(X)} < ∞;

(3)

sup_{t>0} ∥u_{2}(·)χ_{\overline{B}(x_{0},t)}(·)∥_{L_{v}^{pq}(X)}∥(d(x_{0}, ·))^{−α}(u_{1}(·)w(·))^{−1}χ_{X\\overline{B}(x_{0},t)}(·)∥_{L_{w}^{p′s′}(X)} < ∞.

Then the following inequality holds:

∥u_{2}(·)T f(·)∥_{L_{v}^{pq}(X)} ≤ c∥u_{1}(·)f(·)∥_{L_{w}^{ps}(X)},

where the positive constant c does not depend on f.

Remark 3.1. The results of this section remain valid if we do not require the continuity of v, but assume that μB(x_{0}, r) is continuous with respect to r.

4. Examples of weight functions

Let (X, dμ) be an SHT such that the condition μ(B(x, r)) ≈ r holds (if a < ∞, then we assume that this condition is fulfilled for 0 < r ≤ 1). It is known (see [5]) that if μ(X) < ∞, then

∫_{X} |Kf(x)|^{p}(d(x_{0}, x))^{p−1} dμ ≤ c ∫_{X} |f(x)|^{p}(d(x_{0}, x))^{p−1} log^{p} \frac{b}{d(x_{0}, x)} dμ,

where x_{0} ∈ X, 1 < p < ∞, and b = 8a_{1}ae^{p′} and the positive constant c does not depend on f.

Example 4.1. Let 1 < p < q < ∞, and let a < ∞; put v(t) = t^{p−1}, w(t) = t^{p−1} ln^{γ} \frac{b}{t} for t ∈ (0, 4a_{1}a), where b = 8a_{1}ae^{\frac{γ}{p−1}} , γ = \frac{p}{q} + p − 1, and a_{1} is the positive constant from Definition 1.1. Then from Theorem 2.1 it follows that

∥Kf(·)∥_{L_{v(d(x_{0},·))}^{pq}(X)} ≤ c∥f(·)∥_{L_{w(d(x_{0},·))}^{p}(X)},  (4.1)

where the positive constant c does not depend on f.

Example 4.2. Let 1 < p < q < ∞, a = ∞, v(t) = t^{p−1} when 0 < t ≤ 1

and v(t) = t^{α} when t > 1, and let w(t) = t^{p−1} ln^{γ} ^{\frac{2e^{\frac{γ}{p−1}}}{t}} when t ≤ 1, and w(t) = t^{β} ln^{β}(2e^{\frac{γ}{p−1}} ) when t > 1, where γ = \frac{p}{q} + p − 1, 0 < α ≤ β < p − 1. Then inequality (4.1) holds.

An appropriate example for the conjugate function

f̃(x) = \frac{1}{π} ∫_{−π}^{π} \frac{f(x + t)}{2 tg \frac{t}{2}} dt

is presented in [11].

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(Received 23.08.2000)

Authors’ Addresses:

D. E. Edmunds Centre for Mathematical Analysis and its Applications University of Sussex Brighton BN1 9QH Sussex United Kingdom E-mail: D.E.Edmunds@sussex.ac.uk

V. Kokilashvili and A. Meskhi A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: kokil@rmi.acnet.ge

meskhi@rmi.acnet.ge