Abstract
In this paper, we study higher-order Riesz transforms associated with the inverse Gaussian measure given byπn/2ex2dxonℝn. We establishLpℝn,ex2dx-boundedness properties and obtain representations as principal values singular integrals for the higher-order Riesz transforms. New characterizations of the Banach spaces having the UMD property by means of the Riesz transforms and imaginary powers of the operator involved in the inverse Gaussian setting are given.
Highlights
Our setting is Rn endowed with the measure γ−1 whose density with respect to the Lebesgue measure is πn/2ejxj2, x ∈ Rn
The study of harmonic analysis operators in ðRn, γ−1Þ was began by Salogni [1]
The aim of this paper is to study LpðRn, γ−1Þ-boundedness properties of higher-order Riesz transforms in the inverse Gaussian setting
Summary
Our setting is Rn endowed with the measure γ−1 whose density with respect to the Lebesgue measure is πn/2ejxj , x ∈ Rn. He showed that, for λ ≥ 1, the shifted first-order Riesz transform ∇ðA + λIÞ−1/2 is bounded from L1ðRn, γ−1Þ into L1,∞ðRn, γ−1Þ These operators are studied on new Hardy type H1-spaces. We characterize the Banach spaces with the UMD property by using Riesz transforms in the inverse Gaussian setting. In order to the operator Aiσ is bounded from L2ðRn, γ−1Þ ⊗ X into itself as subspace of L2ðRn, γ−1, XÞ, we need to impose some additional property to the Banach space X. We are going to characterize the UMD Banach spaces as those Banach spaces for which Aiσ can be extended from ðL2ðRn, γ−1Þ ∩ LpðRn, γ−1ÞÞ ⊗ X to LpðRn, γ−1, XÞ as a bounded operator from LpðRn, γ−1, XÞ into itself, when 1 < p < ∞, and from L1ðRn, γ−1, XÞ into L1,∞ðRn, γ−1, XÞ. Throughout this paper, C and c denote positive constants that can change in each occurrence
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
