Endpoint estimates for riesz transform on manifolds with ends

Dangyang He

doi:10.1007/s10231-024-01482-8

Endpoint estimates for riesz transform on manifolds with ends

Dangyang He^*

^*Corresponding author for this work

School of Mathematical and Physical Sciences

Research output: Contribution to journal › Article › peer-review

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Abstract

We consider a class of non-doubling manifolds M consisting of finite many “Euclidean” ends, where the Euclidean dimensions at infinity are not necessarily all the same. In [17], Hassell and Sikora proved that the Riesz transform on M is of weak type (1, 1), bounded on L^p if and only if 1 < p < n_∗, where n_∗=min_kn_k. In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space L^n∗,1and unbounded from L^n∗,p→L^n∗,q for all 1 < p < ∞ and p ≤ q ≤∞.

Original language	English
Pages (from-to)	245-259
Number of pages	15
Journal	Annali di Matematica Pura ed Applicata
Volume	204
Issue number	1
DOIs	https://doi.org/10.1007/s10231-024-01482-8
Publication status	Published - Feb 2025

Bibliographical note

Keywords

Endpoint estimates
Manifolds with ends
Riesz transform

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N2 - We consider a class of non-doubling manifolds M consisting of finite many “Euclidean” ends, where the Euclidean dimensions at infinity are not necessarily all the same. In [17], Hassell and Sikora proved that the Riesz transform on M is of weak type (1, 1), bounded on Lp if and only if 1 < p < n∗, where n∗=minknk. In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space Ln∗,1 and unbounded from Ln∗,p→Ln∗,q for all 1 < p < ∞ and p ≤ q ≤∞.

AB - We consider a class of non-doubling manifolds M consisting of finite many “Euclidean” ends, where the Euclidean dimensions at infinity are not necessarily all the same. In [17], Hassell and Sikora proved that the Riesz transform on M is of weak type (1, 1), bounded on Lp if and only if 1 < p < n∗, where n∗=minknk. In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space Ln∗,1 and unbounded from Ln∗,p→Ln∗,q for all 1 < p < ∞ and p ≤ q ≤∞.

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KW - Riesz transform

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JF - Annali di Matematica Pura ed Applicata

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ER -

Endpoint estimates for riesz transform on manifolds with ends

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