GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 5, 1996, 423-446

CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES FOR FRACTIONAL MAXIMAL FUNCTIONS

A. GOGATISHVILI AND V. KOKILASHVILI

Abstract. A strong type two-weight problem is solved for fractional maximal functions defined in homogeneous type general spaces. A similar problem is also solved for one-sided fractional maximal functions.

1. Introduction

The solution of a two-weight problem for maximal functions in Euclidean spaces was obtained for the first time in [1]. This paper gives a new complete description of pairs of weight functions which provides the validity of strong type two-weighted inequalities for fractional maximal functions defined in homogeneous type general spaces. A similar result was obtained in [2] for homogeneous type spaces having group structure. It should be noted that condition (1.2) below, which has turned out to be a criterion of strong type two-weight estimates for fractional maximal functions, appeared for the first time in [3], [4].

In this paper we also solve a strong type two-weight problem for one-sided maximal functions of fractional order on the real axis.

The homogeneous type space (X, d, μ) is a topological space with a complete measure μ such that compactly supported functions are dense in the space L^{1}(X, μ). Moreover, it is assumed that there is a nonnegative realvalued function d : X × X → R^{1} satisfying the following conditions:

(i) d(x, x) = 0 for all x ∈ X;

(ii) d(x, y) > 0 for all x ≠ y in X;

(iii) there is a constant a_{0} such that d(x, y) ≤ a_{0}d(y, x) for all x, y in X; (iv) there is a constant a_{1} such that d(x, y) ≤ a_{1}(d(x, z) + d(z, y)) for all

x, y and z in X; 423

1991 Mathematics Subject Classification. 42B20, 42B25.

Key words and phrases. Homogeneous type space, fractional maximal function, strong type two-weighted inequality, one-sided fractional maximal functions.

1072-947X/96/0900-0423$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

(v) for each neighborhood V of x in X there is r > 0 such that the ball B(x, r) = {y ∈ X : d(x, y) < r} is contained in X.

The balls B(x, r) are measurable for all x and r > 0.

There is a constant b such that μB(x, 2r) ≤ bμB(x, r) for all nonempty B(x, r) (see [5]).

For a locally summable function f : X → R^{1} the fractional maximal function is defined as follows:

M_{γ}(f)(x) = sup(μB)^{γ−1} ∫_{B} |f(y)|dμ, 0 < γ < 1,  (1.0)

where the supremum is taken with respect to all balls B of positive measure containing the point x.

A measurable function w : X → R^{1}, which is positive almost everywhere and locally summable, is called a weight function (weight). For a μ-measurable set E we put

wE = ∫_{E} w(x)dμ.

Main Theorem. Let 1 < p < q < ∞, 0 < γ < 1, v and w be the weight functions. For a constant c > 0 to exist so that the inequality

( ∫_{X} (M_{γ}(f)(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c( ∫_{X} |f(x)|^{p}w(x)dμ)^{\frac{1}{p}}  (1.1)

would hold, it is necessary and sufficient that the condition

sup_{\begin{matrix}x∈X \\ r>0 \end{matrix}} (w^{1−p′}B(x, 2N_{0}r))^{\frac{1}{p′}} ×

×( ∫_{X\B(x,r)} v(y)(μB(x, d(x, y)))^{(γ−1)q}dμ)^{\frac{1}{q}} < ∞  (1.2)

be fulfilled.

This theorem was announced in [6].

2. Preliminary Results

We begin this section by generalizing the results of [1] for homogeneous type general spaces. After that we give the proof of the main theorem which is based on Theorem 2.1 to be proved below and our results obtained previously [7] (see also [8]) for fractional integrals defined in homogeneous type general spaces.

CRITERIA OF STRONG TYPE TWO-WEIGHTED INEQUALITIES 425

First we give a familiar covering lemma which is valid for arbitrary spaces with a quasi-metric.

Lemma A ([5]). Let E be a bounded set in X and, for each point x ∈ E let a ball B_{x} = B(x, r_{x}) be given such that sup_{x∈X} rad B_{x} < ∞. Then from the family {B_{x}}_{x∈E} we can choose a (finite or infinite) sequence of pairwise disjoint balls (B_{j})_{j} for which E ⊂ ∪_{j≥1}N_{0}B_{j}, N_{0} = a_{1}(1 + 2a_{0}), and for each B_{x} ∈ {B_{x}}_{x∈E} there exists a ball B_{j_{0}} such that x ∈ N_{0}B_{j_{0}} and rad B_{x} ≤ 2 rad B_{j_{0}}.

We have

Theorem 2.1. Let 0 ≤ γ < 1, 1 < p ≤ q < ∞. Then the following two conditions are equivalent:

(i) there exists a constant c_{1} > 0 such that

( ∫_{X} (M_{γ}(f)(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c_{1}( ∫_{X} |f(x)|^{p}w(x)dμ)^{\frac{1}{p}}  (2.1)

for any f ∈ L_{w}^{p}(X, μ); (ii) there exists a constant c_{2} > 0 such that

( ∫_{B} (M_{γ}(χ_{B}w^{1−p′})(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c_{2}( ∫_{B} w^{1−p′}(x)dμ)^{\frac{1}{p}} (2.2)

for any ball B ⊂ X.

Proof. On substituting f = χ_{B}w^{1−p′} in (2.1), we obtain (2.2). Therefore the implication (i) ⇒ (ii) is fulfilled. We will prove (ii) ⇒ (i). Let N = a_{1}(1 + 2a_{1}(1 + a_{0})) and N_{1} = a_{1}(1 + a_{1}^{2}(1 + a_{0})(N + a_{0})). Assume b > 1 to be a constant such that

μ(N_{1}B) ≤ bμB

for an arbitrary ball B.

Let further B_{0} be an arbitrarily fixed ball and f be an arbitrary integrable function which is positive almost everywhere and satisfies the condition supp f ⊂ B_{0}.

We set

Ω_{k} = {x ∈ X : M_{γ}(f)(x) > b^{k}}, k ∈ ℤ.

Obviously, for each x ∈ Ω_{k} there exists a ball B(y, r) ∋ x such that

\frac{1}{(μB(y, r))^{1−γ}} ∫_{B(y,r)} f(z)dμ > b^{k}.

The set of radius lengths of such balls will be bounded by virtue of the fact that supp f ⊂ B_{0}.

Consider the values

R_{x}^{k} = sup \{r : ∃B(y, r) ∋ x, \frac{1}{(μB(y, r))^{1−γ}} ∫_{B(y,r)} f(z)dμ > b^{k}\}.

Obviously, for arbitrary x ∈ Ω_{k} there exist y_{x} ∈ X and r_{x} > \frac{R_{x}^{k}}{2} such that

\frac{1}{(μB(y_{x}, r_{x}))^{1−γ}} ∫_{B(y_{x},r_{x})} f(z)dμ > b^{k}.

Along with this, for each ball B′ which contains the point x and for which rad B′ ≥ 2r_{x} we have

\frac{1}{(μB′)^{1−γ}} ∫_{B′} f(z)dμ ≤ b^{k}.  (2.3)

By Lemma A for each k there exists a sequence (B_{j}^{k})_{j≥1} of nonintersecting balls such that ∪_{j≥1}NB_{j}^{k} ⊃ Ω_{k} and

\frac{1}{(μB_{j}^{k})^{1−γ}} ∫_{B_{j}^{k}} f(z)dμ > b^{k}.

Let us show that if B_{j}^{k} ∩ B_{i}^{n} ≠ ∅ and n > k, then

NB_{i}^{n} ⊂ NB_{j}^{k}.  (2.4)

To this end it will first be shown that if r_{i} = rad B_{i}, r_{j} = rad B_{j}, and n > k, then

r_{i}^{n} < \frac{r_{j}^{k}}{a_{1}(a_{0} + N)} .

Assume the opposite, i.e., r_{j}^{k} ≤ a_{1}(a_{0} + N)r_{i}^{n}. Then for y ∈ B_{j}^{k} and x ∈ B_{j}^{k} ∩ B_{i}^{n} we will have

d(x_{i}^{n}, y) ≤ a_{1}(d(x_{i}^{n}, x) + d(x, y)) ≤

≤ a_{1}(r_{i}^{n} + a_{1}(a_{0}d(x_{j}^{k}, x) + d(x_{j}^{k}, y))) < a_{1}(r_{i}^{n} + a_{1}(1 + a_{0})r_{j}^{k}) ≤

≤ a_{1}(1 + a_{1}^{2}(1 + a_{0})(N + a_{0}))r_{i}^{n} = N_{1}r_{i}^{n}.

where x_{j}^{k} are the centers of B_{j}^{k}. Therefore B_{j}^{k} ⊂ N_{1}B_{i}^{n}. Along with this, 2r_{j}^{k} < N_{1}r_{i}^{n}. Therefore by virtue of (2.3) we obtain

b^{n} < \frac{1}{(μB_{i}^{n})^{1−γ}} ∫_{B_{i}^{n}} f(z)dμ ≤ \frac{b}{μ(N_{1}B_{i}^{n})^{1−γ}} ∫_{N_{1}B_{i}^{n}} f(z)dμ ≤ b^{k+1}.

Thus n ≤ k, which leads to a contradiction. Therefore

r_{j}^{k} > a_{1}(a_{0} + N)r_{i}^{n}.

Now for x ∈ NB_{i}^{n} and y ∈ B_{i}^{n} ∩ B_{j}^{k} we derive

d(x_{j}^{k}, x) ≤ a_{1}(d(x_{j}^{k}, y) + a_{1}(a_{0}d(x_{i}^{n}, y) + d(x_{i}^{n}, x))) ≤
≤ a_{1}(r_{j}^{k} + a_{1}(a_{0} + N)r_{i}^{n}) ≤ 2a_{1}r_{j}^{k} < Nr_{j}^{k}.

Thus NB_{i}^{n} ⊂ NB_{j}^{k} provided that B_{j}^{k} ∩ B_{i}^{n} ≠ ∅ and n > k.

We introduce the sets

E_{j}^{k} = (NB_{j}^{k}\ ⋃_{i=1}^{j−1} NB_{i}^{k}) ∩ (Ω_{k}\Ω_{k+1}), k ∈ ℤ, j ∈ ℕ.

As is easy to verify,

⋃_{j=1}^{∞} E_{j}^{k} = Ω_{k}\Ω_{k+1} and E_{j}^{k} ∩ E_{i}^{n} = ∅ for k ≠ n or i ≠ j.

Therefore we have

∫_{X} (M_{γ}(f)(x))^{q}v(x)dμ ≤ ∑_{k=−∞}^{∞} ∑_{j=1}^{∞} b^{(k+1)q}vE_{j}^{k} ≤

≤ b^{q} ∑_{j,k} (\frac{1}{(μB_{j}^{k})^{1−γ}} ∫_{B_{j}^{k}} f(z)dμ)^{q}vE_{j}^{k} =

= c ∑_{j,k} (\frac{σ(NB_{j}^{k})}{(μB_{j}^{k})^{1−γ}} )^{q}vE_{j}^{k}(\frac{1}{σ(NB_{j}^{k})} ∫_{B_{j}^{k}} \frac{f(z)}{σ(z)}σ(z)dμ)^{q},  (2.5)

where σ = w^{1−p′}.

Let now ν be a discrete measure given on ℤ^{2} by

ν\{(j, k)\} = (\frac{σ(NB_{j}^{k})}{(μB_{j}^{k})^{1−γ}} )^{q}vE_{j}^{k}.

Define the operator T acting from L_{loc}^{1}(X, dμ) into the set of ν-summable functions defined on ℤ^{2} as follows:

T (g)(j, k) = \frac{1}{σ(NB_{j}^{k})} ∫_{B_{j}^{k}} |g(z)|σ(z)dμ.

Equality (2.5) can now be rewritten as

∫_{X} (M_{γ}(f)(x))^{q}ν(x)dμ ≤ c ∫_{ℤ^{2}} (T (\frac{f}{σ} )(z))^{q}dν.  (2.6)

Our aim is to prove that the operator T has a strong type (p, q). It is obvious that the operator T has a strong type (∞, ∞). Now let us show that T is the operator of a weak type (1, \frac{q}{p}), i.e.,

νΓ(λ) = ν{(j, k) ∈ ℤ^{2} : T (g)(j, k) > λ} ≤
cλ^{− \frac{q}{p}} ( ∫_{X} |g(z)|^{p}σ(z)dμ)^{\frac{q}{p}}  (2.7)

for any λ > 0 and g ∈ L^{1}(X, σdμ).

We fix some k_{0} ∈ ℤ and first show that

λ^{\frac{q}{p}} νΓ_{k_{0}}(λ) = ν\{(j, k) ∈ ℤ × ℤ_{k_{0}} : T (g)(j, k) > λ\} ≤
≤ ( ∫_{X} |g(z)|^{p}σ(z)dμ)^{\frac{1}{p}} ,  (2.8)

where ℤ_{k_{0}} = {k_{0}, k_{0} + 1, … }.

Now fix λ > 0 and consider the system of balls (B_{j}^{k})_{(j,k)∈Γ_{k_{0}} (λ)}. Choose from the latter system a subsystem of nonintersecting balls in the following manner: take all balls of “rank” k_{0}, i.e., all balls B_{j}^{k_{0}}, j = 1, 2, . . .. Pass to “rank” k_{0} + 1. If some B_{i}^{k_{0}+1} intersects with none of B_{j}^{k_{0}}, then include it in the subsystem and otherwise discard. Next compare the balls of rank k_{0} + 2 with the ones already chosen in the above-described manner and so on. We obtain thus the sequence of nonintersecting balls {B_{i}}_{i}. According to (2.4) if B_{j}^{k} ∈ {B_{i}}_{i}, then NB_{j}^{k} ⊂ NB_{i_{0}} for some i_{0} ≥ 1 and therefore

⋃_{i=1}^{∞} NB_{i} = ⋃_{(j,k)∈Γ_{k_{0}} (λ)} NB_{j}^{k}.

Hence we obtain

λ^{\frac{q}{p}} νΓ_{k_{0}}(λ)= λ^{\frac{q}{p}} ∑_{(j,k)∈Γ_{k_{0}} (λ)} (\frac{σ(NB_{j}^{k})}{(μB_{j}^{k})^{1−γ}} )^{q}vE_{j}^{k} ≤

≤ cλ^{\frac{q}{p}} ∑_{i=1}^{∞} ∑_{NB_{j}^{k}⊂NB_{i}} (\frac{σ(NB_{j}^{k})}{(μ(NB_{j}^{k}))^{1−γ}} )^{q}vE_{j}^{k} ≤

≤ cλ^{\frac{q}{p}} ∑_{i=1}^{∞} ∑_{NB_{j}^{k}⊂NB_{i}} ∫_{E_{j}^{k}} (M_{γ}(χ_{NB_{i}} σ)(z))^{q}v(z)dμ ≤

≤cλ^{\frac{q}{p}} ∑_{i=1}^{∞} ∫_{NB_{i}} (M_{γ}(χ_{NB_{i}} σ)(z))^{q}v(z)dμ≤cλ^{\frac{q}{p}} ∑_{i=1}^{∞} (σ(NB_{i}))^{\frac{q}{p}} ≤

≤ c ∑_{i=1}^{∞} (σ(NB_{i}))^{\frac{q}{p}} (\frac{1}{σ(NB_{i})} ∫_{B_{i}} |g(z)|σ(z)dμ)^{\frac{q}{p}} =

= c ∑_{i=1}^{∞} ( ∫_{B_{i}} |g(z)|σ(z)dμ)^{\frac{q}{p}} ≤ c( ∑_{i=1}^{∞} ∫_{B_{i}} |g(z)|σ(z)dμ)^{\frac{q}{p}} ≤

≤ c( ∫_{X} |g(z)|σ(z)dμ)^{\frac{q}{p}} .

Thus we have proved (2.8) where the constant does not depend on k_{0}. Making now k_{0} tend to −∞, we obtain (2.7). We have shown that the operator T has a weak type (1, \frac{q}{p}).

Since the operator T has a weak type (1, \frac{q}{p}) and a strong type (∞, ∞), by the Marcinkiewicz interpolation theorem we conclude that T has a strong type (p, q). Then (3.6) implies

∫_{X} (M_{γ}(f)(x))^{q}v(x)dμ ≤ c ∫_{ℤ^{2}} (T (\frac{f}{σ} )(x))^{q}dν ≤

≤ c_{1}( ∫_{X} (\frac{f}{σ} )^{p}(x)σ(x)dμ)^{\frac{q}{p}} = c_{1}( ∫_{X} f^{p}(x)w(x)dμ)^{\frac{q}{p}} .

Let now f be an arbitrary function from L^{p}(X, wdμ). By virtue of the foregoing arguments, for an arbitrary ball B_{0} we will have

( ∫_{X} (M_{γ}(χ_{B_{0}} f))^{q}v(x)dμ)^{\frac{1}{q}} ≤

≤ c( ∫_{B_{0}} f^{p}(x)w(x)dμ)^{\frac{1}{p}} ≤ c( ∫_{X} f^{p}(x)w(x)dμ)^{\frac{1}{p}} .

Making rad B_{0} tend to infinity, by the Fatou lemma we obtain (2.1).

Next we will give two theorems which are proved in [7]. They concern two-weight estimates of integral transforms with a positive kernel, in particular, analogs of fractional integrals defined in spaces with a quasi-metric and measure.

Consider the integral operators

K(f)(x) = ∫_{X} k(x, y)f(y)dy

K^{∗}(f)(x) = ∫_{X} k(y, x)f(y)dy.

Theorem A. Let 1 < p < q < ∞, k : X × X → R^{1} be an arbitrary positive measurable kernel; v and μ be arbitrary finite measures on X so that μ{x} = 0 for any x ∈ X. If the condition

c_{0} = sup_{\begin{matrix}x∈X \\ r>0 \end{matrix}} (vB(x, 2N_{0}r))^{\frac{1}{q}} ( ∫_{X\B(x,r)} k^{p′}(x, y)w^{1−p′}(y)dμ)^{\frac{1}{p′}} < ∞,

where N_{0} = a_{1}(1 + 2a_{0}) and the constants a_{0} and a_{1} are from the definition of a quasi-metric, is fulfilled, then there exists the constant c > 0 such that the inequality

v\{x ∈ X : K(f)(x) > λ\} ≤ cλ^{−q}( ∫_{X} |f(x)|^{p}dμ)^{\frac{q}{p}}

holds for any μ-measurable nonnegative function f : X → R^{1} and arbitrary λ > 0.

Definition 2.1. A positive measurable kernel k : X × X → R^{1} will be said to satisfy condition (V) (k ∈ V) if there exists a constant c > 0 such that

k(x, y) < ck(x′, y)

for arbitrary x, y, and x′ from X which satisfy the condition d(x, x′) < Nd(x, y), where N = 2N_{0}.

Theorem B. Let 1 < p < q < ∞, μ be an arbitrary locally finite measure, w be a weight, and k ∈ V. Then the following conditiona are equivalent:

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

w^{1−p′}\{x ∈ X : K(f)(x) > λ\} ≤ c_{1}λ^{−q}( ∫_{X} |f(x)|^{p}v^{\frac{1}{1−q}} (x)dμ)^{\frac{p′}{q′}}

holds for arbitrary λ > 0 and nonnegative f ∈ L^{p}(X, wdμ); (ii) there exists a constant c_{2} > 0 such that

( ∫_{X} (K^{∗}(χ_{B}w^{1−p′})(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c_{2}( ∫_{B} w^{1−p′}(y)dμ)^{\frac{1}{p}}

for an arbitrary ball B ⊂ X; (iii)

sup_{\begin{matrix}x∈X \\ r>0 \end{matrix}} (w^{1−p′}B(x, 2N_{0}r))^{\frac{1}{p′}} ( ∫_{X\B(x,r)} k^{q}(x, y)v(y)dμ)^{\frac{1}{q}} < ∞.

3. Proof of the Main Theorem

Using the results of the preceding sections we will prove the main theorem of this paper.

Proof. Our aim is to show that the implication (1.1) ⇔ (1.2) is valid. First we will prove the implication (1.2) ⇒ (1.1). Consider an operator given on L^{q′}(X, v^{\frac{1}{1−q}}) in the form

T_{γ}(f)(x) = ∫_{X} \frac{|f(y)|}{(μB(x, d(x, y)))^{1−γ}} dμ.

The latter operator is an analog of the Riesz potential for homogeneous type spaces.

Using Theorem A, from condition (1.2) we conclude that the weak type inequality

w^{1−p′}\{x ∈ X : T_{γ}(f)(x) > λ\} ≤ c_{4}λ^{−p′}( ∫_{X} |f(x)|^{q′}v^{\frac{1}{1−q}} (x)dμ)^{\frac{p′}{q′}}(3.1)

with the constant not depending on λ > 0 and f is valid.

Further, by virtue of Theorem B the latter inequality implies that there exists a constant c_{2} > 0 such that for any ball B ⊂ X we have

( ∫_{X} (T_{γ}^{∗}(χ_{B}w^{1−p′})(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c_{2}( ∫_{B} w^{1−p′}(x)dμ)^{\frac{1}{p′}} , (3.2)

where

T_{γ}^{∗}(φ)(x) = ∫_{X} \frac{|φ(y)|}{(μB(y, d(x, y)))^{1−γ}} dμ.

On the other hand, there exist constants c_{3} > 0 and c_{4} > 0 such that

c_{3}μB(y, d(x, y)) ≤ μB(x, d(x, y)) ≤ c_{4}μB(y, d(x, y)).  (3.3)

The latter follows from the fact that B(x, d(x, y)) ⊂ a_{1}(a_{0} + 1)d(x, y). Indeed, let d(x, z) ≤ d(x, y). Then

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

By virtue of the doubling property for measure we obtain

μB(y, a_{1}(a_{0} + 1)d(x, y)) ≤ c_{5}μB(y, d(x, y)).

Hence we conclude that (3.3) is valid. Next, from (3.3) and (3.2) we derive

( ∫_{X} (T_{γ}(χ_{B}w^{1−p′})(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c_{6}( ∫_{B} w^{1−p′}(x)dμ)^{\frac{1}{p}} . (3.4)

Now we will show that for an arbitrary nonnegative measurable function φ we have

M_{γ}(φ)(x) ≤ c_{7}T_{γ}(φ)(x),  (3.5)

where the constant c_{7} does not depend on φ and x.

First we will show that for any x there exists a ball B_{x} = B(x, r_{x}) such that

M_{γ}(φ)(x) ≤ \frac{c_{8}}{(μB_{x})^{1−γ}} ∫_{B} φ(z)dμ,  (3.6)

where the positive constant c_{8} does not depend on φ and B_{x}. Indeed, there exists a ball B(y, r) such that x ∈ B(y, r) and

M_{γ}(φ)(x) ≤ \frac{2}{(μB(y, r))^{1−γ}} ∫_{B(y,r)} φ(z)dμ.  (3.7)

Assuming now that z ∈ B(y, r), we obtain

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

Therefore B(y, r)⊂B(x, a_{1}(1+a_{0})r). On the other hand, we have B(x, a_{1}(1+ a_{0})r) ⊂ B(y, a_{1}(1 + a_{1}(1 + a_{0}))r) since

d(y, z) ≤ a_{1}(d(y, x) + d(x, z)) ≤ a_{1}(r + a_{1}(1 + a_{0}r)) = a_{1}(1 + a_{1}(1 + a_{0}))r

for any z ∈ B(x, a_{1}(1 + a_{0})r).

Now applying the doubling condition, from (3.7) we find that

M_{γ}(φ)(x) ≤ \frac{2}{(μB(y, r))^{1−γ}} ∫_{B(x,a_{1}(1+a_{0})r)} φ(z)dμ ≤

≤ \frac{c_{8}}{(μB(x, a_{1}(1 + a_{0})r))^{1−γ}} ∫_{B(x,a_{1}(1+a_{0})r)} φ(z)dμ.

Replacing r_{x} by the number a_{1}(1 + a_{0})r, we obtain (3.6).

Now we will prove (3.5). We have

T_{γ}(φ)(x) ≥ ∫_{B(x,r_{x})} \frac{φ(y)}{(μB(x, d(x, y)))^{1−γ}} dμ ≥

≥ \frac{1}{(μB(x, r_{x}))^{1−γ}} ∫_{B(x,r_{x})} φ(z)dμ ≥ \frac{1}{c_{8}} M_{γ}(φ)(x).

From (3.4) and (3.5) we obtain

( ∫_{X} (M_{γ}(χ_{B}w^{1−p′})(x))^{q}v(x)dμ)^{\frac{1}{q}} ≤ c_{9}( ∫_{B} w^{1−p′}(x)dμ)^{\frac{1}{p}} .

By Theorem 2.1 we conclude that inequality (1.1) is valid.

Thus we have proved the implication (1.2) ⇒ (1.1).

Let us show the validity of the inverse implication (1.1) ⇒ (1.2). Fix an arbitrary ball B(x, r) and assume

f(y) = χ_{NB(x,r)}w^{1−p′}(y),

where N is an arbitrary number greater than unity. Obviously, by virtue of the doubling condition we have

M_{γ}(f)(y) ≥ \frac{1}{μB(x, Nd(x, y))}  ∫_{B(x,Nd(x,y))∩NB(x,r)} w^{1−p′}(z)dz ≥

≥ \frac{c_{10}}{μB(x, d(x, y))} ∫_{NB(x,r)} w^{1−p′}(z)dμ

for any y ∈ X\B(x, r).

Therefore (1.1) implies

( ∫_{NB(x,r)} w^{1−p′}(z)dμ)( ∫_{X\B(x,r)} \frac{v(y)}{(μB(x, d(x, y))^{(1−γ)q}} dμ)^{\frac{1}{q}} ≤

≤ c( ∫_{NB(x,r)} w^{1−p′}(z)dμ)^{\frac{1}{p}} .

From the latter inequality we obtain the validity of (1.2).

Definition 3.1. Measure ν satisfies the reverse doubling condition (ν ∈ (RD)) if there exist numbers δ and ε from (0, 1) such that

νB(x, r_{1}) ≤ ενB(x, r_{2})

for μB(x, r_{1}) ≤ δμB(x, r_{2}), 0 < r_{1} < r_{2}.

In the particular case where measure w^{1−p′} satisfies the reverse doubling condition, (1.2) in the main theorem can be replaced by a simpler condition.

Theorem 3.1. Let 1 < p < q < ∞, 0 < γ < 1 and measure w^{1−p′} satisfy the reverse doubling condition. Then (1.1) holds iff

sup_{\begin{matrix}x∈X \\ r>0 \end{matrix}} (μB(x, r))^{γ−1}(w^{1−p′}B(x, r))^{\frac{1}{p′}} (vB(x, r))^{\frac{1}{q}} < ∞.  (3.8)

Proof. The implication (1.1) ⇒ (3.8) is obtained immediately by substituting the function f(y) = χ_{B}w^{1−p′}(y) into (1.1).

By virtue of the main theorem to prove the implication (3.8) ⇒ (1.1) it is sufficient to show that if w^{1−p′} ∈ (RD), then (3.8) ⇒ (1.2). Let x ∈ X and r > 0 be fixed. Choose numbers r_{k} (k = 0, 1, …) as follows:

r_{0} = 2N_{0}r and r_{k} = inf {r : μB(x, r_{k−1}) < δμB(x, r)}.

Obviously,

μB(x, r_{k−1}) < δμB(x, r_{k}) < cμB(x, r_{k−1}).

Again applying the condition (RD), we obtain

w^{1−p′}B(x, r_{0}) ≤ ε^{k}w^{1−p′}B(x, r_{k}) k = 1, 2, ….

The latter inequalities imply

(w^{1−p′}B(x, 2N_{0}r))^{\frac{1}{p′}} ( ∫_{X\B(x,r)} v(y)(μB(x, d(x, y)))^{(γ−1)q}dμ(y))^{\frac{1}{q}} ≤

≤(w^{1−p′}B(x, r_{0}))^{\frac{1}{p′}} ∑_{k=1}^{∞} ( ∫_{B(x,r_{k})\B(x,r_{k−1})} v(y)(μB(x, d(x, y)))^{(γ−1)q}dμ(y))^{\frac{1}{q}} +

+(w^{1−p′}B(x, r_{0}))^{\frac{1}{p′}} ( ∫_{B(x,r_{0})\B(x,r)} v(y)(μB(x, d(x, y)))^{(γ−1)q}dμ(y))^{\frac{1}{q}} ≤

≤ ∑_{k=1}^{∞} ε^{\frac{k}{p′}} (w^{1−p′}B(x, r_{k}))^{\frac{1}{p′}} (vB(x, r_{k}))^{\frac{1}{q}} (μB(x, r_{k−1}))^{γ−1} +

+(w^{1−p′}B(x, r_{0}))^{\frac{1}{p′}} (vB(x, r_{0}))^{\frac{1}{q}} (μB(x, r_{0}))^{γ−1} ≤

≤ c ∑_{k=0}^{∞} ε^{\frac{k}{p′}} (μB(x, r_{k}))^{γ−1}(w^{1−p′}B(x, r_{k}))^{\frac{1}{p′}} (vB(x, r_{k}))^{\frac{1}{q}} ≤

≤ c ∑_{k=0}^{∞} ε^{\frac{k}{p′}} < ∞.

Remark. A similar result was obtained in [2] under the assumption that the measure w^{1−p′} satisfies the doubling condition. Theorem 3.1 contains a stronger result, since condition (RD) is weaker than the doubling condition. (For instance, the function e^{|x|} ∈ (RD) but it does not satisfy the doubling condition.) Along with this, the proof in [2] essentially differs from the above.

The above proofs also remains valid for fractional maximal functions:

M_{γ}(fdσ)(x) = sup_{B∋x} \frac{1}{(μB)^{1−γ}} ∫_{B} |f(y)|dσ(y),  (3.9)

where 0 ≤ γ < 1 and σ is a Borel measure while the supremum is taken over all balls B ⊂ X with a positive measure containing x.

Theorem 3.2. Let X be an arbitrary homogeneous type space, 1 < p < q < ∞, 0 < γ < 1, ω and σ be positive Borel measures. For a constant c > 0 to exist such that the inequality

( ∫_{X} (M_{γ}(fdσ)(x))^{q}dω(x))^{\frac{1}{q}} ≤ c( ∫_{X} |f(x)|^{p}dω(x))^{\frac{1}{p}}  (3.10)

holds, it is necessary and sufficient that the condition

sup_{\begin{matrix}x∈X \\ r>0 \\ μB(x,r)≠0 \end{matrix}} (σB(2N_{0}r))^{\frac{1}{p′}} ×

×( ∫_{X\B(x,r)} (μB(x, d(x, y)))^{(γ−1)q}dω(y))^{\frac{1}{q}} < ∞  (3.11)

be fulfilled.

Theorem 3.3. Let X be an arbitrary homogeneous type space, 1 < p ≤ q < ∞, 0 ≤ γ < 1, and σ and ω be positive Borel measures. Then the following two conditions are equivalent:

(i) there exists a constant c_{1} > 0 such that

( ∫_{X} (M_{γ}(fdσ)(x))^{q}dω(x))^{\frac{1}{q}} ≤ c_{1}( ∫_{X} |f(x)|^{p}dσ(x))^{\frac{1}{p}} , (3.12)

for any f ∈ L^{p}(X, dσ); (ii) there exists a constant c_{2} > 0 such that

( ∫_{B} (M_{γ}(χ_{B}dσ)(x))^{q}dω(x))^{\frac{1}{q}} ≤ c_{2}(σB)^{\frac{1}{p}}  (3.13)

for any B ⊂ X.

Theorem 3.4. Let 1 < p < q < ∞, 0 < γ < 1, and the measure σ satisfy the reverse doubling condition (σ ∈ (RD)). Then (3.10) holds iff

sup_{\begin{matrix}B \\ μB≠0 \end{matrix}}(μB)^{γ−1}(σB)^{\frac{1}{p′}} (ωB)^{\frac{1}{q}} < ∞.

Theorems 3.2 and 3.3 in fact were previously proved in [2] under the assumption that the space X possesses group structure, while Theorem 3.4 was proved for homogeneous type spaces under a stronger restriction imposed on the measure σ.

Remark. The above-formulated theorems remain valid also in the case where X = R^{1} is an arbitrary positive Borel measure. This follows from the fact that we proved Theorem A for such measures and from the general conclusion that the well-known technique is applicable for maximal functions with respect to a Borel measure on the real axis (see, for instance, [9]).

4. A Strong Type Two-Weight Problem for Fractional Maximal Functions in a Half-Space

This section deals with “Hörmander type” fractional maximal functions. Let X̂ = X × [0, ∞) and for (x, t) ∈ X̂

M̃_{γ}(fdσ)(x, t) = sup(μB)^{γ−1} ∫_{B} |f(y)|dσ(y),  (4.1)

where 0 ≤ γ < 1, σ is a Borel measure in X and the supremum is taken over all balls B ⊂ X containing x and having a radius greater than \frac{t}{2}.

We have

Theorem 4.1. If 1 < p < q < ∞, 0 < γ < 1, and X is an arbitrary homogeneous type space, then the inequality

( ∫ ; _{X} (M̃_{γ}(fdσ))(x, t))^{q}dω(x, t))^{\frac{1}{q}} ≤ c( ∫_{X} |f(x)|^{p}dσ(x))^{\frac{1}{p}} (4.2)

holds for all f with c independent of f iff

sup_{\begin{matrix}x∈X \\ r>0 \end{matrix}} (σB(x, 2N_{0}r))^{\frac{1}{p′}} ×

×( ∫_{_{X}\_{B}(x,r)} (μB(x, d(x, y) + t))^{(γ−1)q}dω(x, t))^{\frac{1}{q}} < ∞,  (4.3)

where B̂(x, r) = B(x, r) × [0, r).

Proof. The validity of the theorem follows from the main theorem using the following arguments. A similar idea is used in [18].

On the product space X × R we define the quasi-metric as

d̂((x, t), (y, s)) = max {d(x, y), |s − t|}

and the measure μ̂ = μ ⊕ δ_{0}, where δ_{0} is the Dirac measure concentrated at zero.

Note that the center of a ball from X̂ with respect to d̂ lies in X̂ and is obtained by the intersection of the respective ball in X × R with X̂. Since B̂((x, t), r) = B(x, r) × (t − r, t + r) ∩ X̂, we have

μ̂B̂((x, t), r) = μB(x, r) for r ≥ t,
μ̂B̂((x, t), r) = 0 for r < t.

Therefore (X̂, d̂, μ̂) is a homogeneous type space. It can be easily verified that the maximal function M̂ is actually a function of the form

M_{γ}(fdσ)(x, t) = sup (μB̂((z, t), r))^{γ−1} ∫_{_{B}((z,τ),r)} |f(y)|dσ̂,

where σ̂ = σ ⊕ δ_{0}.

Hence to derive Theorem 4.2 from the main theorem it is sufficient to show that in the considered case condition (2.11) can be reduced to condition (4.3). For this, in turn, since μ̂B̂((x, τ), r) ≠ 0 for τ < r and B̂((x, τ), r) = B(x, r) × [0, r + τ) it is sufficient to show that

μ̂B̂((x, τ), d̂((x, τ), (y, t))) ∼ μB(x, d(x, y) + t)
for τ < r < d̂((x, τ), (y, t)) = max {d(x, y), |τ − t|}.

Indeed,

d(x, y) + t ≤ d(x, y) + |τ − t| + τ ≤ d(x, y) + |τ − t| + r ≤ 3d̂((x, τ), (y, t)).

Since 0 < τ < max{d(x, y), |τ − t|}, we have {d(x, y), |τ − t|} < d(x, y) + t, i.e.,

d(x, y) + t ∼ d̂((x, τ), (y, t)).

Therefore by virtue of the doubling condition

μ̂B̂((x, t), d̂((x, t), (y, s))) =
= μB(x, d̂((x, τ), (y, t))) ∼ μB(x, d(x, y) + t).

Theorem 4.2. Let X be an arbitrary homogeneous type space, 1 < p ≤ q < ∞, 0 ≤ γ < 1. Then (4.2) holds for all f with c independent of f iff

( ∫ ; ˆ_{B} (M_{γ}(χ_{B}dσ)(x, t))^{q}dω(x, t))^{\frac{1}{q}} ≤ c_{1}(σB)^{\frac{1}{P}} ,  (4.4)

where ˆ B̂ = B × [0, 2 rad B).

Theorem 4.3. If 1 < p < q < ∞, 0 < γ < 1, σ ∈ (RD), then (4.2) holds iff

sup(μB)^{γ−1}(σB)^{\frac{1}{p′}} (ω ˆB̂)^{\frac{1}{q}} < ∞,  (4.5)

where the supremum is taken with respect to all balls B.

It should be said that theorems of the type of Theorems 4.1 and 4.2 have previously been known only for homogeneous type spaces with a group structure [10], [11]. In the latter papers instead of condition (4.3) there figures a condition which actually implies the simultaneous fulfillment of conditions (4.3) and (4.5). As for Theorem 4.3, it is proved in [10], [11] but under a more restrictive doubling condition for the measure σ.

We should also note papers [12], [13], where some more particular results are obtained.

5. A Strong Type Two-Weight Problem for One-Sided Fractional Maximal Functions

The method developed in the preceding sections enables one to solve the following strong type two-weight problem:

M_{γ}^{+}(f)(x) = sup_{h>0} h^{γ−1} ∫_{x}^{x+h}|f(t)|dt,

and

M_{γ}^{−}(f)(x) = sup_{h>0} h^{γ−1} ∫_{x−h}^{x} |f(t)|dt,

where x > 0 and 0 < γ < 1.

To this end it is sufficient to follow the proof of the main theorem and apply an analog of Theorem 2.1 for one-sided maximal functions (see, for instance, [17, Theorem 2]), Theorem 1.6 from [7], and also the fact that the Riemann-Liouville operator R_{γ} and the Weyl operator W_{γ} control the one-sided maximal functions

M_{γ}^{−}(f)(x) ≤ R_{γ}(|f|)(x)

and

M_{γ}^{+}(f)(x) ≤ W_{γ}(|f|)(x),

where

R_{γ}(f)(x) = ∫_{0}^{x}(x − t)^{γ−1}f(t)dt

and

W_{γ}(f)(x) = ∫_{x}^{∞}(t − x)^{γ−1}f(t)dt,

where 0 < γ < 1, x > 0.

We thus come to the validity of the following two statements:

Theorem 5.1. Let 1 < p < q < ∞, 0 < γ < 1. For the inequality

( ∫_{0}^{∞}(M_{γ}^{+}(f)(x))^{q}v(x)dx)^{\frac{1}{q}} ≤ c( ∫_{0}^{∞}|f(x)|^{p}w(x)dx)^{\frac{1}{p}}

with the positive constant c independent of f to hold, it is necessary and sufficient that the condition

sup_{\begin{matrix}a,h \\ 0<h<a \end{matrix}} ( ∫_{a−h}^{a+h}w^{1−p′}(y)dy)^{\frac{1}{p′}} ( ∫_{a+h}^{∞} \frac{v(y)}{(y − a)^{(1−γ)q}} dy)^{\frac{1}{q}} < ∞

be fulfilled.

Theorem 5.2. Let 1 < p < q < ∞, 0 < γ < 1. For the inequality

( ∫_{0}^{∞}(M_{γ}^{−}(f)(x))^{q}v(x)dx)^{\frac{1}{q}} ≤ c( ∫_{0}^{∞}|f(x)|^{p}w(x)dx)^{\frac{1}{p}}

with the positive constant c independent of f to hold, it is necessary and sufficient that the condition

sup_{\begin{matrix}a,h \\ 0<h<a \end{matrix}} ( ∫_{a−h}^{a+h}w^{1−p′}(y)dy)^{\frac{1}{p′}} ( ∫_{0}^{a−h}\frac{v(y)}{(a − y)^{(1−γ)q}} dy)^{\frac{1}{q}} < ∞

be fulfilled.

The solution of a weak type two-weight problem for one-sided maximal functions was obtained as a particular case of a more general theorem in [7] (see also [8]).

6. Criteria of Two-Weighted Inequalities for Integral Transforms with Positive Kernels

In this section it will be assumed that the space X is such that there exists a number L > 1 such that any ball B(x, r) contains at most L points x_{i} such that

d(x_{i}, x_{m}) > \frac{r}{2} .

The assumption that a measure μ given in X satisfies the doubling condition guarantees the fulfillment of the above-formulated condition (see [15], Lemma 1.1).

For such space X the following statements are valid.

Lemma B ([16], p. 623). Let Ω be an open bounded subset of X and c ≥ 1. Then there exists a sequence of balls B_{j} = B(x_{j}, r_{j}) such that (i)

Ω = ⋃_{j=1}^{∞} \overline{B}_{j} = ⋃_{j=1}^{∞} B_{j},

where \overline{B}_{j} = B(x_{j}, cr_{j}); (ii) there exists a positive number M = M(c, L, a_{0}, a_{1}) such that

∑_{j=1}^{∞} χ_{\overline{B}_{j}} (x) ≤ M;

(iii) (X\Ω) ∩ B^{=}_{j} ≠ ∅ for each B^{=}_{j} = B(x_{j}, 3ca_{1}r_{j});

Theorem 6.1. Let 1 < p ≤ q < ∞, let v and μ be locally finite measures in X, and let the kernel k satisfy condition (V). Then the following conditions are equivalent:

(i) there exists c_{1} > 0 such that

v\{x ∈ X : K(f)(x) > λ\} ≤ c_{1}λ^{−q}( ∫_{X} f^{p}(x)dμ)^{\frac{q}{p}}  (6.1)

for any λ > 0 and measurable nonnegative f; (ii) there exists c_{2} > 0 such that

( ∫_{B} (K^{∗}(χ_{B}dv))^{p′}dμ)^{\frac{1}{p′}} ≤ c_{2}(vB)^{\frac{1}{q′}} .  (6.2)

The theorem also hold in the case p = 1, 1 < q < ∞ provided that condition (6.2) is replaced by the condition

exp_{B} K^{∗}(χ_{B}dv)(x) ≤ c_{2}(vB)^{\frac{1}{q′}}

for each ball B.

Proof. Let f be a bounded nonnegative function with a compact support. For given λ > 0 we set

Ω_{λ} = {x ∈ X : K(f)(x) > λ}.

Let B_{j} = B(x_{j}, r_{j}) be the sequence of balls from Lemma B corresponding to the number c = 2a_{1}. Then there exists a constant c_{3} > 0 such that (see, for instance, [17])

K(χ_{X\cB_{j}} f)(x) ≤ c_{3}λ  (6.3)

for any x ∈ B_{j}, where cB_{j} = B(x_{j}, cr_{j}). Hence K(χ_{X\cB_{j}} f)(x) > c_{3}λ for any x ∈ B_{j} ∩ Ω_{2c_{3}λ}.

442 A. GOGATISHVILI AND V. KOKILASHVILI

Next,

vΩ_{2c_{3}λ} ≤ ∑_{j=1}^{∞} v(B_{j} ∩ Ω_{2c_{3}λ}) =

= ∑_{j∈F} + ∑_{j∈G} v(B_{j} ∩ Ω_{2c_{3}λ}) = I_{1} + I_{2},

where

F = {j : v(B_{j} ∩ Ω_{2c_{3}λ}) > βv(cB_{j})}

and

G = {v(B_{j} ∩ Ω_{2c_{3}λ}) ≤ βv(cB_{j})},

while the number β will be chosen below so that 0 < β < 1.

Applying the Hölder inequality and condition (6.2) we obtain

c_{3}λv(cB_{j}) ≤ c_{3}β^{−1}λv(B_{j} ∩ Ω_{2c_{3}λ}) ≤

≤ β^{−1} ∫_{B_{j}} ( ∫_{cB_{j}} k(x, y)f(y)dμ(y))dv(x) ≤

≤ β^{−1} ∫_{cB_{j}} ( ∫_{B_{j}} k(x, y)dv(x))f(y)dμ(y) ≤

≤ β^{−1}( ∫_{cB_{j}} ( ∫_{B_{j}} k(x, y)dv(x))^{p′}dμ(y))^{\frac{1}{p′}} ( ∫_{B_{j}} f^{p}(y)dμ(y))^{\frac{1}{p}} ≤

≤ β^{−1}c_{2}v(cB_{j})^{\frac{1}{q′}} ( ∫_{cB_{j}} f^{p}(y)dμ)^{\frac{1}{p}} .

Therefore

λ^{q}v(cB_{j}) ≤ (c_{3}^{−1}β^{−1}c_{2})^{q}( ∫_{cB_{j}} |f(y)|^{p}dμ(y))^{\frac{q}{p}} .

The summation of the latter inequality gives

I_{1} ≤ λ^{−q}(c_{3}^{−1}β^{−1}c_{2})^{q}( ∫_{X} |f(y)|^{p}dμ)^{\frac{q}{p}} .

Here we have used the fact that \frac{q}{p} > 1 and ∑_{j=1}^{∞} χ_{cB_{j}} ≤ M. Next for j ∈ G we obtain

v(B_{j} ∩ Ω_{2c_{3}λ}) ≤ βv(cB_{j}).

Therefore

I_{2} ≤ βMvΩ_{λ}.

Finally, we find that

vΩ_{2c_{3}λ} ≤ c_{4}λ^{−q}( ∫_{X} |f(y)|^{p}dμ)^{\frac{q}{p}} + βMvΩ_{λ}.

Multiplying the latter inequality by λ^{q} and taking the exact upper bound with respect to λ, 0 < λ < \frac{δ}{2c_{3}} , we obtain

sup_{0<λ<s} λ^{q}vΩ_{λ} ≤ c_{4}( ∫_{X} |f(y)|^{p}dμ)^{\frac{q}{p}} + (2c_{3})^{q}M sup_{0<λ<s} λ^{q}vΩ_{λ}.

If we set β = \frac{1}{2M(2c_{3})^{q}}, the latter inequlity will imply

sup_{0<λ<s} λ^{q}vΩ_{λ} ≤ c_{4}( ∫_{X} |f(y)|^{p}dμ)^{\frac{q}{p}} + \frac{1}{2} sup_{0<λ<s} λ^{q}vΩ_{λ}.

If the second term on the right-hand side is assumed to be finite, then from the latter inequlity we obtain

sup_{0<λ<s} λ^{q}vΩ_{λ} ≤ c_{4}( ∫_{X} |f(y)|^{p}dμ)^{\frac{q}{p}} .

Making s tend to infinity, we concluded that (6.1) holds.

It remains to show that

sup_{0<λ<s} λ^{q}vΩ_{λ} < ∞  (6.4)

for arbitrary finite s.

Let supp f ⊂ B, where B is a ball in X. It is obvious that

λ^{q}v(2a_{1}B) ≤ s^{q}v(2a_{1}B) < ∞.

Therefore it is suffiecient to show that

sup_{λ>0} λ^{q}v(Ω_{λ}\2a_{1}B) < ∞.

Let x ∈ Ω_{λ}\2a_{1}B and B_{x} = B(x_{B}, 2d(x_{B}, x)), where x_{B} is the center of B. For x′ ∈ B_{x} and y ∈ B we have

2a_{1}d(x_{B}, y) ≤ d(x_{B}, x) ≤ a_{1}(d(x_{B}, y) + d(y, x)).

Hence

d(x_{B}, y) ≤ a_{0}d(x, y).

Further for x′ ∈ B_{x} and y ∈ B we have

d(x′, y) ≤ a_{1}(d(x′, x_{B}) + d(x_{B}, y)) ≤ a_{1}a_{0}d(x, y) + 2a_{1}a_{0}d(x_{B}, x) ≤

≤ a_{1}a_{0}d(x, y) + 2a_{1}^{2}a_{0}(d(x_{B}, y) + d(y, x)) ≤
≤ a_{1}a_{0}d(x, y) + 4a_{1}^{2}a_{0}^{2}d(x, y) = (a_{1}a_{0} + 4a_{1}^{2}a_{0}^{2})d(x, y).

Now by virtue of the remark made at the beginning of the proof of Proposition 3.1 from [7] there exists a constant c′ such that

k(x, y) ≤ c′k(x′, y)

for any x′ ∈ B_{x} and y ∈ B. Therefore using (6.2) and the Hölder inequality we obtain

λvB_{x} ≤ vB_{x} ∫_{B} k(x, y)f(y)dμ(y) ≤

≤ ( ∫_{B} |f(y)|^{p}dμ(y))^{\frac{1}{p}} ( ∫_{B} (k(x, y)vB_{x})^{p′}dμ(y))^{\frac{1}{p′}} ≤

≤ c_{5}( ∫_{B} |f(y)|^{p}dμ(y))^{\frac{1}{p}} ( ∫_{B} ( ∫_{B_{x}} k(x′, y)dv(x′))^{p′}dμ(y))^{\frac{1}{p′}} ≤

≤ c_{6}( ∫_{B} |f(y)|^{p}dμ(y))^{\frac{1}{p}} (vB_{x})^{\frac{1}{q′}} .

Hence we conclude that

λ(vB_{x})^{\frac{1}{q}} ≤ c_{6}( ∫_{B} |f(y)|^{p}dμ(y))^{\frac{1}{p}} .

Since Ω_{λ}\2a_{1}B ⊂ ∪B_{x} and balls B_{x} have a common center, by virtue of the Fatou theorem the latter estimate implies that (6.4) is valid. Thus the implication (6.2) ⇒ (6.1) is proved for functions f with a compact support. One can easily pass over to the general case.

The implication (6.1) ⇒ (6.1) is proved by Theorem 1.5 from [7].

Insignificant changes in the proof show that the theorem also holds for p = 1 and 1 < q < ∞.

For the above-mentioned space (X, d, μ) we will consider an upper halfspace of the product space X × R. We set X̂ = X × [0, ∞).

Let k : X × X × [0, ∞) be a positive measurable kernel satisfying the condition: there exists a constant c > 0 such that

k(x, y, t) ≤ ck(x′, y, t′)

for x, x′, y from X, t ≥ 0, t′ ≥ 0, satisfying the condition d(x, x′) + t′ ≤ 5N_{0}(d(x, y) + t).

Consider the integral operators

T (f)(x, t) = ∫_{X} k(x, y, t)f(y)dμ

and

T^{∗}(gdv)(y) = ∫ ; _{X} k(x, y, t)g(x, t)dv(x, t), y ∈ X,

where v is some locally finite measure in X̂.

Using the consideration of Section 2 from [7] we can deduce the validity of Theorem 1.3 from [18] in this more general case.

Note that in Theorem 6.1 the measure μ was not assumed to satisfy the doubling condition so that we used only the special geometric property of the space. The latter condition is fulfilled if the measure μ has the doubling property.

References

1. E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators. Studia Math. 75(1982), 1–11.

2. R. L. Wheeden, A characterization of some weighted norm inequalities for fractional maximal functions. Studia Math. 107(1993), 251–272.

3. M. Gabidzashvili, Weighted inequalities for anisotropic potentials. (Russian) Trudy Tbil. Mat. Inst. Razmadze Akad. Nauk Gruz. SSR 89(1986), 25–36.

4. V. Kokilashvili and M. Gabidzashvili, Two-weight weak type inequalities for fractional type integrals. Preprint No. 45, Math. Inst. Czech. Akad. Sci., Praha, 1989.

5. J. O. Strömberg and A. Torchinsky, Weighted Hardy spaces. Lect. Notes in Math., v. 1381, Springer Verlag, New York, 1989.

6. I. Genebashvili, A. Gogatishvili, and V. Kokilashvili, Criteria of twoweghted inequalities for integral transforms with a positive kernel and maximal functions. Dokl. Akad. Nauk RAN (to appear).

7. I. Genebashvili, A. Gogatishvili, and V. Kokilashvili, Solution of twoweight problems for integral transforms with positive kernels. Georg. Math. J. 3(1996), 319–347.

8. A. Gogatishvili and V. Kokilashvili, Criteria of weight inequalities for integral transforms defined on homogeneous type spaces. In: Topological Vector Spaces, Algebras and Related Areas. Longman. Pitman Research Notes in Mathematics 316(1994), 251–262.

9. B. Jawerth, Weighted inequalities for maximal operators: Linearization, localization and factorization. Amer. J. Math. 108(1986), 361–414.

10. R. L. Wheeden, Poincaré-Sobolev and isoperimetric inequalities, maximal functions and half-space estimates for the gradient. In: Nonlinear Analysis, Function Spaces and Applications, vol. 5, Proceedings. Prometheus Publishing House, 1994, 103–137.

11. E. T. Sawyer, R. L. Wheeden, and S. Zhao, Weighted norm inequalities for operators of potential type and fractional maximal functions. Potential Analysis (1995) (to appear).

12. F. J. Ruiz and J. L. Torrea, A unified approach to Carleson measures and A_{p} weights, II. Pacific J. Math. 120(1985) 189–197.

13. I. Genebashvili, Generalized fractional maximal functions and potentials in weighted spaces. Proc. A. Razmadze Math. Inst. 102(1993), 41–57.

14. E. J. Martin-Reyes and de la Torre, Two-weight norm inequalities for fractional one-sided maximal operators. Proc. Amer. Math. Soc. 117(1993), 483–499.

15. R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes. Lecture Notes in Math. 242, Springer-Verlag, Berlin and New York, 1971.

16. R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in Analysis. Bull. Amer. Math. Soc. 83(1977), 569–645.

17. E. T. Sawyer and R.L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous type spaces. Amer. J. Math. 114(1992), 813–875.

18. R. L. Wheeden and S. Zhao, Weak type estimates for operators of potential type. Preprint. 1994.

(Received 31.01.1995)

Authors’ address:

A. Razmadze Mathematical Institute

Georgian Academy of Sciences

1, M. Alexidze St., Tbilisi 380093,

Republic of Georgia