High order Riesz transforms and mean value formula for generalized translate operator
© Ekincioglu et al.; licensee Springer. 2014
Received: 5 December 2013
Accepted: 27 March 2014
Published: 11 April 2014
In this paper, the mean value formula depends on the Bessel-generalized shift operator corresponding to the solutions of the boundary value problem related to the multidimensional Bessel operator are studied. In addition, Riesz transforms related to the multidimensional Bessel operators are studied. Since a Bessel-generalized shift operator is a translation operator corresponding to the multidimensional Bessel operator, we construct a family of by using a Bessel-generalized shift operator. Finally, we analyze weighted inequalities involving .
MSC:31B05, 35G10, 34B30, 47F05.
Singular integral operators are playing an important role in harmonic analysis, the theory of functions and partial differential equations. Singular integrals are associated with the Laplace-Bessel differential operator, which is known as an important operator in analysis and its applications; these have been the research areas of many mathematicians such as Muckenhoupt and Stein [1, 2], Kipriyanov and Klyuchantsev [3, 4], Aliev and Gadjiev , Guliev , Gadjiev. Also, singular integral operators related to the generalized shift operator were studied by  and [8–10] and others.
The Bessel-generalized shift operator is one of the most important generalized shift operator on the half line , [11, 12]. The Bessel-generalized translation is used, while studying various problems connected with Bessel operators . Fourier-Bessel harmonic analysis, i.e. the part of harmonic analysis addressing various problems on Bessel (Hankel) integral transforms, is closely connected with Bessel-generalized shift operator.
It is well known that the fundamental solutions of the classical stationary of mathematical physics (the harmonic equation, polyharmonic equation, and Helmholtz equation) are radial functions. Therefore, it is natural to seek these solutions as solutions of ordinary differential equations. However, since the spherical coordinate transformation transforms an equation with the Laplace operator in into an ordinary differential equation with the singular Bessel differential operator, interest arose (probably, a long time ago) in studying methods for constructing the fundamental solutions of singular ordinary differential equations with the Bessel operator in place of the second derivative. In this connection, it might be very useful to prove a theorem on the fundamental solution of an ordinary differential equation involving the Bessel operator with constant coefficients, similar to the well-known theorem on the fundamental solution of an ordinary differential equation.
The result obtained on the fundamental solution of an ordinary differential operator with the Bessel operator has allowed us to analyze equations with the singular differential operator where the different indices act with respect to part of the variables and these indices may take negative values. The latter fact is essential because the tools used in problems of this kind (the Poisson operator and generalized shift operator of integral nature) are defined only for . To find the fundamental solution with a singularity at an arbitrary point, we use the generalized shift operator that acts with respect to the radial variable. Note also that the mixed-type generalized shift operator, which is conventionally used in such problems, coincides with the radial shift operator on radial functions provided that . The fundamental solutions (of the B-harmonic and B-polyharmonic equations and of the singular Helmholtz equation) found coincide with the known solutions when and with the classical solutions when .
It is well known that harmonic functions satisfy various mean value theorems, which may be considered as generalizations of the Gauss mean value theorem. There have been a number of studies on mean value theorems. Cheng  obtained a converse for a different mean value expression. Nicolesco  gave an expression in terms of certain iterated means and showed that a converse was also true. The mean value theorems for harmonic functions have also been studied by Pizetti , Picone , Ekincioglu , and Kipriyanova  and .
The solutions of the boundary value problems for Laplace operator are related to the ordinary shift operator. Also, the solutions of boundary value problems for Laplace-Bessel and Bessel operators are corresponding to the generalized shift operator and Bessel-generalized shift operator, respectively.
In this paper, singular integral operators generated by a Bessel-generalized shift operator are studied. In addition, the mean value formulas related to the Bessel-generalized shift operator are given.
Riesz-Bessel singular integral operators related to generalized shift operator for Laplace-Bessel operator were showed in  and . The authors used the mean value theorem related to the generalized shift operator. The mean value formula for the equations and were obtained by  and , respectively, where .
The convolution (1.3) is known as a B-convolution. We note some properties for the B-convolution and the Bessel-generalized shift operator:
If , is a bounded function, , andthen
From the above result, we have the following equality for :
2 The high order Riesz-Bessel transforms associated with Bessel-generalized shift operator
In this section, we consider a Bessel-generalized shift operator related to the multidimensional Bessel differential operator. Then we give the Fourier-Bessel transformation of a homogeneous polynomial which obeys the Bessel equations. Finally, we define the high order Riesz-Bessel transforms related to Bessel-generalized shift operator and so we show that high order Riesz-Bessel transforms obey the condition of classical Riesz transforms, that is, these operators extend to high order Riesz-Bessel transforms .
It follows from the general theory of singular integrals that Riesz transforms are bounded on for all . In this paper we extend this result to the context of Bessel theory where a similar operator is already defined. It has been noted that the difficulty arises in the application of the classical -theory of Calderon-Zygmund, since Riesz transforms are singular integral operators. In this paper we describe how this theory can be adapted in a Bessel setting and give an -result for high order Riesz transforms for all .
We have , , the range over the homogeneous harmonic polynomials the latter arise in special case . Those for , we call the high order Riesz-Bessel transform where we refer to k as the degree of the high order Riesz Bessel transform . They can also be characterized by their invariance properties.
Let be homogeneous polynomial of degree k in . We shall say that P is elliptic if vanishes only at the origin. For any polynomial P we consider also its corresponding differential polynomial. Thus if we write , where and with the monomials (see ).
The mean value theorem for multidimensional Bessel differential operators is very convenient for obtaining multidimensional singular integral operators generated by a Bessel-generalized shift operator. Therefore, we studied the mean value formula related to the Bessel-generalized shift operator for the solutions of the boundary value problem for the multidimensional Bessel operator .
The Bessel-generalized shift operator is one of the most important generalized shift operator on the half line [11–13]. The Bessel-generalized translation is used while studying various problems connected with Bessel operators. Fourier-Bessel harmonic analysis, i.e. the part of harmonic analysis addressing various problems on Bessel (Hankel) integral transforms, is closely connected with the Bessel-generalized shift operator.
3 The mean value formula
In this section, we determine the mean value formula for the Bessel-generalized shift operator. Let be an n-dimensional Euclidian space and , be vectors in . Then . Denote and , . We assume that and is its boundary. In this paper, we are mainly concerned with the mean value theorem. Now, we relate this concept in the following theorem.
Thus the proof is completed. □
and so we obtain the desired conclusion. □
We come now to what has been our main goal in this paper.
where () and is a homogeneous polynomial of degree k in which satisfies .
One of the important applications of the high order Riesz transforms is that they can be used to mediate between various combinations of partial derivatives of a function. We shall here content ourselves with two very simple illustrations, which examples have an interest on their own and have already the characteristic features of a general type of estimate which can be made in the theory of elliptic differential operators.
with independent of f.
We also have the following boundedness of the high order Riesz-Bessel transform.
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