Revista de la Unión Matemática Argentina - Weighted inequalities for generalized fractional operators

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## versão On-line ISSN 1669-9637

### Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dez. 2008

Weighted inequalities for generalized fractional operators

María Silvina Riveros

Abstract. In this note we present weighted Coifman type estimates, and two-weight estimates of strong and weak type for general fractional operators. We give applications to fractional operators given by an homogeneous function, and by a Fourier multiplier. The complete proofs of these results appear in the work [5] done jointly with Ana L. Bernardis and María Lorente.

2000 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Fractional integrals, Hörmander's condition of Young type, Muckenhoupt weights, two-weight estimates.
The author is partially supported by CONICET, Agencia Nación, and SECYT-UNC

1. Introduction and preliminaries

I would like to dedicate this note in memory of Dr Carlos Segovia. First we will give some basic definitions and preliminaries needed to state the results. Let us recall some of the background on Orlicz spaces. (See [24] and [21] to complete this topic.)

A function is a Young function if it is continuous, convex, increasing and satisfies and as .
Given a Young function , we define the -mean Luxemburg norm of a function on a cube (or a ball) in by

 (1.1)

It is well known that if for all then , for all cubes and functions . Thus, the behavior of for is not important. If , that is there are constants such that for , the latter estimate implies that .
Each Young function has an associated complementary Young function satisfying

for all . There is a generalization of Hölder's inequality

 (1.2)

A further generalization of Hölder's inequality (see [21]) that will be useful later is the following: If and are Young functions and

then
 (1.3)

When we understand that if and otherwise. Then is not a Young function, but and the latter inequalities make sense if one of the functions is or .

For each locally integrable function and , the fractional maximal operator associated to the Young function is defined by

For we write instead of . When then is the classical fractional maximal operator. For and we obtain , the Hardy-Littlewood maximal operator. Consider the case , for
the maximal operators associated to these Young functions are:
If , , then is pointwise equivalent to , where It is also easy to check that if and , then

The good weights for are those in the classes of Muckenhoupt (see [19] and also [26] and [18] for the one-sided case).

The good weights for the maximal operator are the classes. It is proved in [20] ( see [1] for the one sided version) that if and only if , for , , where

for all cube .

Also observe that for the case , is equivalent to say that .

Let us define a generalization of the Hörmander condition, for a given kernel . We used the notation: for and

Definition 1.1. Let be a Young function and let . The kernel is said to satisfy the -Hörmander type condition, we write , if there exist , such that for any and

We say that if satisfies the previous condition with in place of .

Definition 1.2. The kernel is said to satisfy the condition, if there exist , such that

Observe that when we obtain that defined in [16].

If , for , then we write . This condition appears implicitly in [12]. On the other hand, since for we have that . Also, it is easy to see that .

Suppose that is an operator given by convolution with a kernel which satisfies some regularity condition and suppose that we know some behavior of with respect to the Lebesgue's measure (weak or strong type inequalities for ). Sometimes, in order to know how is the behavior of when we change the measure, (i.e., when we consider the measure where is a weight, ()) the following inequality is useful (we call it a Coifman type inequality)

 (1.4)

Here is a maximal operator related to the operator which is normally easier to deal with. In general, is strongly related with the kernel and its size is inverse to the smoothness of : the rougher the kernel, the bigger the maximal.

For a Calderón-Zygmund singular integral operator (i.e., , see Definition 1.2, for ) inequality (1.4) holds with , the Hardy-Littlewood maximal function, , and (see [8]).

If is a singular integral operator with less regular kernel, (see [13]) for example if the kernel satisfies an -Hörmander condition (Definition 1.1, for and ), then inequality (1.4) holds with , with , for all , and (see [25]).

For a Young function , the -Hörmander condition is introduced in [16], which generalized in the scale of the Orlicz spaces the -Hörmander condition. In [16] the authors showed that, if the kernel (Definition 1.1, for ), then inequality (1.4) holds with , where is the complementary function of , for all , and .

The differential transform operator was studied in [11] and [3]. In [14] it is proved an inequality of the type (1.4), by showing that the kernel satisfies the -Hörmander condition for (). Therefore, this operator satisfies inequality (1.4) with the maximal operator , where (actually they obtain a smaller operator since is a one-sided operator, because ).

The Coifman type inequality allows us to obtain, for general linear operators, two-weight inequalities of the type

 (1.5)

for and in the endpoint case ,

 (1.6)

for every weight , with no assumptions on . The operators are again suitable maximal operators related with and not necessarily the same for inequalities (1.4), (1.5) and (1.6).

There is a great amount of works that deal with inequalities of the type (1.5) and (1.6). When is a Calderón-Zygmund operator (with kernel ), inequality (1.5) holds with , where is the integer part of (see [22]). In the endpoint case , inequality (1.6) for Calderón-Zygmund operators hold with , for any , where is the maximal function associated to the Young function . This result was proved by Carlos Pérez in [22]. For a singular integral associated to a kernel K satisfying a general Hörmander's condition given by a Young function , the corresponding results, that include as particular cases those of C. Pérez, has been proved in [14] and [15].

In 1974, Muckenhoupt and Wheeden [20] proved inequality (1.4) for the classical Riesz potential and the fractional maximal function , defined for and locally integrable function by

Observe that using the mean value theorem we can prove that, the kernel of the fractional integral, , , belongs to . For , inequality (1.5) holds with (this result is also due to Carlos Pérez, see [23]). On the other hand, inequality (1.6) for holds with in [6] (see also [2]).

There are fractional integrals with less regular kernel than the Riesz transform (see for example [7], [12], [27], [20], [9], [10]). Suppose that is homogeneous of degree zero and , where denotes the sphere of and . Define the fractional integral associated to by

In [10] inequalities with weights were established for this operator, which generalized the corresponding inequalities for given by Muckenhoupt and Wheeden in [20]. In a more general context and with an aditional condition in , that is, satisfying the -Dini smoothness condition, Segovia and Torrea [27], studied the good weights for this operator and its commutators, using extrapolation theorems.

In this note we state and briefly sketch the proofs of the corresponding results for general fractional integrals , , given by convolution with a kernel which satisfy a condition, for appropriate Young functions (see Theorems 2.1, 2.3 and 2.5).

From now on, for , will be a fractional operator bounded from to , for all satisfying .

2. Statements of the Results

Theorem 2.1. Let be a fractional operator given by a kernel . Suppose is of weak-type .

If be a Young function and , then for any and ,

 (2.1)

Moreover, if the kernel is supported in , then for any , , it follows that (2.1) holds with in place of where .

Remark 2.2. Observe that we can apply the theorem to and (respectively) obtaining the result in [20] and [17], for .

Proof. To prove this Theorem we use the sharp operator of . Given and a cube , decompose , where and . For we use Kolmogorov and that is of weak-type . For the global part we use that is the convolution with the kernel and the generalized Hölders inequality. □

Theorem 2.3. Let be a Young function and . Suppose that there exist Young functions , such that and with .  Let be a linear operator such that its adjoint satisfies

 (2.2)

for all and . Then, for and for any weight ,

 (2.3)

Remark 2.4. For the applications below, and since all our operators are of convolution type, proving (2.2) for or turns out to be equivalent.

Proof. To prove this Theorem we use duality and apply Theorem 2.1. To do this we need the fact that the weight belongs to , for all and any . For the maximal operators that appears in this proof, has been proved in [4]. □

Theorem 2.5. Let be a fractional operator. Suppose that there exists such that for any , there exists a Young function satisfying

 (2.4)

for all weight . If , then for any weight ,

 (2.5)

for all .

3. Applications

3.1. Fractional integrals associated to a homogeneous function. Denote by the unit sphere on . For , we write . Let us consider . This function can be extended to as (abusing on the notation we call both functions ). Thus is a function homogeneous of degree . Let , and let be a Young function such that is also a Young function. Let and satisfying the -Dini smoothness condition, i.e.,

 (3.1)

where

Consider the fractional operator
Using Hölder's inequality with and it is easy to see that implies . Then, by the result in [7], is of weak type , with respect to the Lebesgue's measure and is bounded from to , whenever , . We can prove (as in [14]) that the kernel satisfies the condition. Therefore Theorems 2.1, 2.3 and 2.5 can be applied to the operator .

In the particular case that with we get the following:

Theorem 3.1. Let be as above and satisfying the -Dini condition.

If and , then

 (3.2)

If and a weight, then

 (3.3)

If and is a weight, then

 (3.4)

In both cases and is small enough.

Proof. We only have to apply the theorems with the following Young functions: , , and , where is some small enough number that is related with . Observe that in part (c) we obtain on the right hand side, but it is easy to see that and . □

For as above, we obtain the following weighted inequality as in [10] (see also [27]).

Corollary 3.2. Suppose that we are under the same hypothesis as in Theorem 3.1. Let , and . Then

Proof. First of all observe that implies . Then by part (a) of Theorem 3.1

Since and (see [20]) we have that

3.2. Fractional integrals associated to a multiplier. Let . Given a function defined in we consider the multiplier operator defined a priori for functions in the Schwartz class by . Given and we say that if there exists a constant such that and

In [12], Kurtz proved that if and then is bounded from to , for and . If is the kernel of , he proved that for all and, as a consequence, he obtained the following Coifman type inequality: for all , and ,
 (3.5)

Now we can apply Theorems 2.3 and 2.5 to this operator.

Theorem 3.3. If and a weight, then

 (3.6)

and

 (3.7)

where and is small enough.

Observe that as is at our choice, we can write , for all . Therefore, we can write in (3.6) and (3.7).

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M. S. Riveros
FaMAF,
Universidad Nacional de Córdoba,
CIEM (CONICET),
5000 Córdoba, Argentina
sriveros@mate.uncor.edu

Recibido: 10 de abril de 2008
Aceptado: 5 de junio de 2008