- Open Access
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.
- Bessel equation
- generalized shift operator
- mean value theorem
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 :
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.
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.
- Stein EM, Weiss G: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton; 1971.Google Scholar
- Muckenhoupt B, Stein EM: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 1965, 118: 17-92.MathSciNetView ArticleGoogle Scholar
- Kipriyanov IA: Fourier Bessel transformations and imbedding theorems. Trudy Math. Inst. Steklov 1967, 89: 130-213. (in Russian)Google Scholar
- Kipriyanov IA, Klyuchantsev MI: On singular integrals generated by the generalized shift operator II. Sib. Mat. Zh. 1970, 11: 1060-1083. (in Russian)Google Scholar
- Aliev IA, Gadjiev AD: Weighted estimates for multidimensional singular integrals generated by the generalized shift operator. Mat. Sb. 1994,77(1):37-55. 10.1070/SM1994v077n01ABEH003428MathSciNetView ArticleGoogle Scholar
- Guliev VS: Sobolev theorems for B -Riesz potentials. Dokl. Akad. Nauk, Ross. Akad. Nauk 1998,358(4):450-451. (in Russian)MathSciNetGoogle Scholar
- Aliev IA: On Riesz transformations generated by generalized shift operator. Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Teh. Mat. Nauk 1987,8(1):7-13.Google Scholar
- Ekincioglu I, Yıldırım H, Akın O: The mean value theorem for Laplacean-Bessel equation. II. In Invited Lecture Delivered at the Seventh International Colloquium on Differential Equations. Academic Publications, Plovdiv; 1996:29-37.Google Scholar
- Ekincioglu I, Ozkın IK: On higher order Riesz transformations generated by generalized shift operator. Turk. J. Math. 1997, 21: 51-60.Google Scholar
- Gadjiev AD, Guliev EV: Two weighted inequality for singular integrals in Lebesgue spaces associated with the Laplace-Bessel differential operator. Proc. A. Razmadze Math. Inst. 2005, 138: 1-15.MathSciNetGoogle Scholar
- Levitan BM: Bessel function expansions in series and Fourier integrals. Usp. Mat. Nauk 1951,6(2):102-143. (in Russian)MathSciNetGoogle Scholar
- Kipriyanov IA: Singular Elliptic Boundary Value Problems. Nauka, Moscow; 1997. (in Russian)Google Scholar
- Kipriyanov IA: Boundary value problems for elliptic partial differential operators. Sov. Math. Dokl. 1970, 11: 1416-1419.Google Scholar
- Cheng M: On a theorem of Nicolesco and generalized Laplace operators. Proc. Am. Math. Soc. 1951, 2: 77-86. 10.1090/S0002-9939-1951-0041292-XView ArticleGoogle Scholar
- Nicolesco, M: Les fonctions polyharmoniques. Actualitiés Sci. Ind. 4 (1936)Google Scholar
- Pizetti P: Sul significato geometrico del secondo parametro differenziale di una funzione sopra una superficie qualunque. Rend. Lincei 1909, 18: 309-316.Google Scholar
- Picone M: Nuovi indirizzi di ricerca teoria e nel calcolo soluzioni di talune equazioni lineari alle derivate parziali della fisica-matematica. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1936, 4: 213-288.MathSciNetGoogle Scholar
- Kipriyanova, NI: Mean value theorems for polyharmonic equations. Non-classic problems. Math. Phys. Eq. NovoSibirsk, 81-85 (1984) (in Russian)Google Scholar
- Kipriyanova, NI: The Mean Value Formula for Singular Differential Operator With Order Second. Differential Equations 121, 11 (1985)Google Scholar
- Betancor JJ, Castro AJ, Curbelo J: Harmonic analysis operators associated with multidimensional Bessel operators. Proc. R. Soc. Edinb., Sect. A 2012,142(5):945-974. 10.1017/S0308210511000643MathSciNetView ArticleGoogle Scholar
- Levitan BM: The Theory of Generalized Translation Operators. Nauka, Moscow; 1973. (in Russian)Google Scholar
- Guliev VS: Sobolev’s theorem for the anisotropic Riesz-Bessel potential in Morrey-Bessel spaces. Dokl. Akad. Nauk, Ross. Akad. Nauk 1999,367(2):155-156. (in Russian)MathSciNetGoogle Scholar
- Guliev VS: Some properties of the anisotropic Riesz-Bessel potential. Anal. Math. 2000,26(2):99-118. 10.1023/A:1005632315360MathSciNetView ArticleGoogle Scholar
- Guliev VS, Garakhanova NN, Zeren Y: Pointwise and integral estimates for the Riesz B -potential in terms of B -maximal and B -fractionally maximal functions. Sib. Mat. Zh. 2008,49(6):1263-1279. (in Russian). Translation in Sib. Math. J. 49(6), 1008-1022 (2008)MathSciNetView ArticleGoogle Scholar
- Guliev VS, Garakhanova NN: The Sobolev-II’in theorem for the Riesz B -potential. Sib. Mat. Zh. 2009,50(1):63-74. (in Russian). Translation in Sib. Math. J. 50(1), 49-59 (2009)MathSciNetView ArticleGoogle Scholar
- Kipriyanov IA, Lyakhov LN: Multipliers of the mixed Fourier-Bessel transform. Dokl. Akad. Nauk, Ross. Akad. Nauk 1997,354(4):449-451. (in Russian)MathSciNetGoogle Scholar
- Lyakhov LN: Multipliers of the mixed conversion of Fourier-Bessel. Proc. Steklov Inst. Math. 1997, 214: 234-249. (in Russian)MathSciNetGoogle Scholar
- Lyakhov LN, Raykhelgauz LB:Singular heat equation with -Bessel operator. Fundamental solutions. Problems in mathematical analysis. No. 67. J. Math. Sci. (N.Y.) 2013,188(3):285-292. 10.1007/s10958-012-1127-2MathSciNetView ArticleGoogle Scholar
- Stempak K, Torrera JL: Higher Riesz transforms and imaginary powers associated to the harmonic oscillator. Acta Math. Hung. 2006, 111: 43-64. 10.1007/s10474-006-0033-9View ArticleGoogle Scholar
- Stein EM: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton; 1970.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.