Abstract
In this paper, we define the Riesz transform on the dual of the Laguerre hypergroup associated with Plancherel measure and we give some properties for this transform. Also we investigate $L^p$-boundedness of this operator and under this definition, the higher order Riesz transform is given. Moreover we establish that the Riesz transform can be extended as a principal value singular integral operator and it is a multiplier operator under the Fourier transform.
Highlights
We define the Riesz transform on the dual of the Laguerre hypergroup associated with Plancherel measure and we give some properties for this transform
We investigate Lp-boundedness of this operator and under this definition, the higher order Riesz transform is given
1970) suggested the study of analogues of the fundamental operators in the classical harmonic analysis as well as in the theory of partial differential equations, such as Riesz transforms, conjugate Poisson integrals, multipliers, fractional integrals, maximal functions, square functions, ..., extensively in the literature in many different aspects and in a context of discrete or continuous expansions with respect to eigenfunctions of self-adjoint and positive differential operators that can be considered in the context of the Laguerre operator Lα
Summary
M. Stein, 1970) suggested the study of analogues of the fundamental operators in the classical harmonic analysis as well as in the theory of partial differential equations, such as Riesz transforms, conjugate Poisson integrals, multipliers, fractional integrals, maximal functions, square functions, ..., extensively in the literature in many different aspects and in a context of discrete or continuous expansions with respect to eigenfunctions of self-adjoint and positive differential operators that can be considered in the context of the Laguerre operator Lα. Throughout this paper, C will always represent a positive constant, not necessarily the same in each occurrence
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