^{1, 2}

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Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators

Our aim is to derive necessary and sufficient conditions on weight pairs governing the boundedness of the following potential operators:

Historically, necessary and sufficient condition on a weight function

In the paper [

It should be emphasized that the two-weight problem for the Hardy-type transforms and fractional integrals with single kernels has been already solved. For the weight theory and history of these operators in classical Lebesgue spaces, we refer to the monographs [

The monograph [

Unfortunately, in the case of double potential operator, we assume that the right-hand weight is of product type and the left-hand one satisfies the doubling condition with respect to one of the variables. Even under these restrictions the two-weight criteria are written in terms of several conditions on weights. We hope to remove these restrictions on weights in our future investigations.

Some of the results of this paper were announced without proofs in [

Finally we mention that constants (often different constants in the same series of inequalities) will generally be denoted by

We say that a function

Let

The next statement regarding two-weight criteria for the Hardy operator

Let

Suppose that

Let

The following statement was proved in [

Let

Let

If

In the case when

The next result deals with the case

Let

Let

the following four conditions hold simultaneously:

In particular, Theorem C yields the trace inequality criteria on the cone of nonincreasing functions.

Let

the boundedness of

In this section we discuss the two-weight problem for the operator

The following relation holds for nonnegative and nonincreasing function

We follow the proof of Proposition 3.1 of [

Observe that if

Finally we have the upper estimate for

The lower estimate is obvious because

In the next statement we assume that

Let

Taking Proposition A into account (for

We have

Now we formulate the main results of this section.

Let

Let

By using the representation

Let

By Theorem

The estimates

Let

It is easy to check that if

Let

Observe that by Remark

Let

Indeed, for

In this section we discuss two-weight criteria for the potentials with product kernels

To derive the main results, we introduce the following multiple potential operators:

One says that a locally integrable a.e. positive function

Analogously is defined the class of weights

If

Analogously,

Theorem C implies the next statement.

Let the conditions of Theorem C be satisfied.

If

If

If

The proof of this statement follows by using the arguments of the proof of Corollary

The following result concerns with the two-weight criteria for the two-dimensional operator

Let

Suppose that

Let

Let us introduce the following multiple integral operators:

Now we prove some auxiliary statements.

Let

The operator

The operator

The operator

The operator

Let

First suppose that

It is obvious that

On the other hand, (

If

To prove, for example, (iii), we choose the sequence

Let

The operator

The operator

The operator

The operator

Let

If

If

If

By using Proposition A we see that the operator

By using Theorem D, (i) and (ii) follow immediately.

To prove (iii) we show that if

For necessity, let us see, for example, that (

The following statements give analogous statement for the mixed-type operator

Let

The operator

The operator

We prove

First we show that the two-sided pointwise relation

Further, it is easy to check that

Hence, since the boundedness of

Let the conditions of Theorem

if

if

if

(i) Taking into account the arguments used in Theorem

(ii) It can be checked that (

The proof of the next statement is similar to that of Proposition

Let the conditions of Theorem

if

if

if

Now we are ready to discuss the operators

Let

Let

Let

The next statement shows that the two-weight inequality for

Let

The first author was partially supported by the Shota Rustaveli National Science Foundation Grant (Project no. GNSF/ST09_23_3-100). A part of this work was carried out at the Abdus Salam School of Mathematical Sciences, GC University, Lahore. The authors are thankful to the Higher Education Commission, Pakistan, for the financial support. The first author expresses his gratitude to Professor V. M. Kokilashvili for drawing his attention to the two-weight problem for potentials with product kernels.