A Note about Young’s Inequality with Different Measures

Abstract

The key purpose of this paper is to work on the boundedness of generalized Bessel–Riesz operators defined with doubling measures in Lebesgue spaces with different measures. Relating Bessel decaying the kernel of the operators is satisfying some elementary properties. Doubling measure, Young's inequality, and Minköwski’s inequality will be used in proofs of boundedness of integral operators. In addition, we also explore the relation between the parameters of the kernel and generalized integral operators and see the norm of these generalized operators which will also be bounded by the norm of their kernel with different measures.

1. Introduction

Suppose we are given , with . Fractional integral operator, one of integral operators, is often studied since early last century. This operator is defined byfor every , with . Here, is called fractional integral kernel or Riesz kernel [1].

Studies about Riesz potentials were started since 1920’s. Hardy-Littlewood [2,3] proved the boundedness of Riesz potentials on Lebesgue spaces for . After 50’s, Hardy-Littlewood and Sobolev [4] proved the boundedness of for . [5] (see also D. Edmunds [[6], Chapter 6]) Kokilashvili had a complete description of nondoubling measure guaranteeing the boundedness of fractional integral operator from to , . We notice that this result was derived in [7] for potentials on Euclidean spaces. In [4], theorems of Sobolev and Adams type for fractional integrals defined on quasimetric measure spaces were established.

Some two-weight norm inequalities for fractional operators on with nondoubling measure were studied in [8]. The boundedness of the Riesz potential in Lebesgue and Morrey spaces defined on Euclidean spaces was studied in Peter and Adams’s paper [9,10]. The same problem for fractional integrals on with nondoubling measure was investigated by Sawano in [11]. Eridani ([12], Theorem 4, Theorem 3.1, Theorem 3.3) established the boundedness of fractional integral operators and mention the necessary and sufficient conditions for the boundedness of maximal operators.

Since 1930s, some researchers [2,3] have studied the boundedness of on some function spaces.

Theorem 1 (Hardy-Littlewood-Sobolev) (see [4]). If , with , then there exists such thatFrom now on, will be serve as a positive constant, not necessarily the same one.
The purpose of this paper is to work on the boundedness of generalized Bessel–Riesz operators defined in Lebesgue spaces with different measures. Role of Young’s inequality and Minköwski’s inequality will be used in proofs of boundedness of integral operators.
Here, we defineand , as a collection of such that . Next, for a given , withWe define Bessel-Riesz operator asFor every , with . Here, is called Bessel–Riesz kernel.
The origin of Bessel–Riesz operators is Schröndinger equation. The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. In quantum mechanics, the analogue of Newton’s law is Schrödinger’s equation. In 1999, Kazuhiro Kurata, Seiichi Nishigaki, and Satoko Sugano [13] studied boundedness of integral operators on Lebesgue and generalized Morrey spaces and its application to estimate in Morrey spaces for the Schrödinger operator with nonnegative (reverse Hölders class) and small perturbed potentials .
In a recent previous year 2016, Idris et al. ([14], Theorem 6, pp. 3) presented the boundedness of Bessel–Riesz operators. They obtained results that were similar to Chiarenza–Frasca’s result [15] for the boundedness of fractional integral operators.
Eridani et al. [12] presented the boundedness of fractional integral operators defined on quasimetric measure spaces. Moreover, Idris et al. [14] have investigated the boundedness of generalized Morrey spaces with weight and presented the boundedness of these operators on Lebesgue spaces and Morrey spaces for Euclidean spaces.
Since Euclidean spaces are the simplest example of measure metric spaces. Kurata et al. [16] have investigated the boundedness of Bessel–Riesz operators on generalized Morrey spaces with weight. The boundedness of these operators on Lebesgue spaces in Euclidean settings will be proved using Young’s inequality and Minköwski’s inequality.
Moreover, we will also find the norm of the generalized Bessel–Riesz operators bounded by the norm of the kernels. Saba et al. [17] used Young’s inequality, to prove the boundedness of Bessel–Riesz operators on Lebesgue spaces in measure metric spaces, which are easy consequences of Young's inequality. The second consequence after the fact Bessel–Riesz operator is bounded on Lebesgue spaces; we entered to next phenomena of Morrey spaces. In Young's inequality, we have the best constant known as 1. But at this point, we still have no information about the best constant in Morrey spaces. So in this paper, we move towards generalized Bessel–Riesz operators in Lebesgue spaces.
We will also mention the case when the measure satisfies the doubling condition. The derived conditions are simultaneously necessary and sufficient for appropriate inequalities that were derived in [18,19]. We also have the following result [14] about the boundedness of such an operator as follows:

Theorem 2. (Idris–Gunawan–Lindiarni–Eridani). If and , then there exists such thatFor some functions and , we defineWe know that is a generalization of and . Moreover, the above result is a particular case of the following [20].

Theorem 3. (Young’s inequality). If and , then there exists such thatKurata et al. [16] have shown that is bounded on generalized Morrey spaces where is any real functions. For applications of the above operators in Euclidean spaces setting, see [16].

2. The Kernel

For , and , we define functions , with the following conditions:

From we will have

That is, there exists such that

Since

then and are equivalent, with the following explanations.

It is easy to see that . Suppose is true. Then, for , we have

On the other hand, for , thenand we already prove that both of the conditions are equivalent.

We define

Using with , and any , then

By , then , for .

If , then and . On the other hand, we can also have for , then , and .

Finally, we will have

After the above estimates, we conclude.

Lemma 1. For , we will haveWith the same technique as used to estimate the kernel defined by equation (15) or every , we also haveAs a classical example, we can consider the following:If we considerthen as a consequences of Young’s inequality, we also have the following.

Lemma 2. If and , then there exists such thatFrom the above lemma, there exists such thatSuppose is an arbitrary measure on . We define (growth condition), if and only if there exists such thatror every open balls in . For more information about this kind of measure, see [21].
Based on the above definitions, we try to estimateFor , and , we consider the following:and alsowhere letter shall always denote a constant, not necessarily the same one.
At this point, for , and , we define

3. Main Results

For any measure on , any measurable functions , and , we define

Before we state our main results, about the boundedness of , we consider the following simple result [13].

Lemma 3 (Minköwski’s inequality). Suppose , and we are given . For any measure and on , thenNow, we state the following:

Theorem 4. Suppose and is any measure on .
If and , then there exists such that

Proof. By Minköwski’s inequality, with , thenAs a corollary of the above result, we also have

Corollary 1. Suppose and is any measure on .
If and , then there exists such thatSuppose we have theIt is noted thatAfter the above simple calculations, we have

Theorem 5. Suppose and any measure on such thatfor some and for every open balls on .
If and , then there exists such that

Proof. With Hölder’s inequality, we start with the following:If we definethen for every , we come toNow, we want to estimate the right hand side, especially, for and , we will haveWe start withand alsoUp to now, for every , we already haveBy the previous fact, we know thatand after this, we come toAnd finally,Since then we are done.
As a corollary, we also have:

Corollary 2. Suppose and any measure on such thatfor some and for every open balls on .
If and , then there exists such that

4. Pointwise Multiplier Operators

Suppose and , with and . Then, by Hölder inequality, we will have .

Next, we consider a pointwise multipliers operator by

So, from Hölder inequality, if , then is a bounded operators from to , with .

Another example is the following equation. Suppose, we consider a fractional integral operators , and we define

Again, since and , in view of Hölder inequality, then

So, if , then is a bounded operator.

From our main results, we also have:

Corollary 3. Suppose and is any measure on .
If , and , thenis a bounded operator. That is, for every , there exists such that

Corollary 4. Suppose and any measure on such thatfor some and for every open balls on .
If , and , thenis a bounded operator. That is, for every , there exists such thatNext, this will be served as our last corollary.

Corollary 5. Suppose and any measure on such thatfor some and for every open balls on .
If , and , thenis a bounded operator. That is, there exists such that

5. Conclusions

From the result of this study, we have seen the boundedness of the generalized Bessel–Riesz operators in Lebesgue spaces, in the framework of different measures (Theorem 3.1, Theorem 5) by using, as tools, an estimate of the Lebesgue norm of the Bessel–Riesz kernel (Theorem 2), the Young inequality (Theorem 3), and functions with doubling conditions. In the future, we shall continue this study to prove the boundedness of generalized Bessel–Riesz operators on Morrey spaces and generalized Morrey spaces.

Data Availability

The data used to support the findings of this study are currently under embargo while the research findings are commercialized. Requests for data, after the publication of this article, will be considered by the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Part of this work was carried out during the second author’s visit to the Faculty of Engineering Sciences, Bahria University Islamabad in 2019, under Staff Exchange Program, Universitas Airlangga 2019. Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya, Indonesia.