Vertical maximal functions on manifolds with ends

Himani Sharma; Adam Sikora

doi:10.1007/s00028-024-00981-8

Vertical maximal functions on manifolds with ends

Himani Sharma, Adam Sikora^*

^*Corresponding author for this work

School of Mathematical and Physical Sciences

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Abstract

We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form Rⁿⁱ×M_i. We investigate the family of vertical resolvent {t∇(1+tΔ)^-m}_t>0, where m≥1. We show that the family is uniformly continuous on all L^p for 1 ≤ p ≤ min_in_i. Interestingly, this is a closed-end condition in the considered setting. We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1, 1) for p=1. The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if 1 < p < min_in_i, and not at p = min_in_i.

Original language	English
Article number	49
Pages (from-to)	1-31
Number of pages	31
Journal	Journal of Evolution Equations
Volume	24
Issue number	3
DOIs	https://doi.org/10.1007/s00028-024-00981-8
Publication status	Published - Sept 2024

Bibliographical note

Keywords

Fefferman-Stein vector-valued maximal functions
Horizontal maximal functions
Laplace-Beltrami operator
Manifolds with ends
R-boundedness
Riesz transform
Vertical maximal functions
Vertical square functions

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10.1007/s00028-024-00981-8Licence: CC BY

Publisher version (open access)Final published version, 477 KBLicence: CC BY

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title = "Vertical maximal functions on manifolds with ends",

abstract = "We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form Rni×Mi. We investigate the family of vertical resolvent {t∇(1+tΔ)-m}t>0, where m≥1. We show that the family is uniformly continuous on all Lp for 1 ≤ p ≤ minini. Interestingly, this is a closed-end condition in the considered setting. We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1, 1) for p=1. The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if 1 < p < minini, and not at p = minini.",

keywords = "Fefferman-Stein vector-valued maximal functions, Horizontal maximal functions, Laplace-Beltrami operator, Manifolds with ends, R-boundedness, Riesz transform, Vertical maximal functions, Vertical square functions",

author = "Himani Sharma and Adam Sikora",

note = "{\textcopyright} The Author(s) 2024. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.",

year = "2024",

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doi = "10.1007/s00028-024-00981-8",

language = "English",

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TY - JOUR

T1 - Vertical maximal functions on manifolds with ends

AU - Sharma, Himani

AU - Sikora, Adam

N1 - © The Author(s) 2024. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

PY - 2024/9

Y1 - 2024/9

N2 - We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form Rni×Mi. We investigate the family of vertical resolvent {t∇(1+tΔ)-m}t>0, where m≥1. We show that the family is uniformly continuous on all Lp for 1 ≤ p ≤ minini. Interestingly, this is a closed-end condition in the considered setting. We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1, 1) for p=1. The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if 1 < p < minini, and not at p = minini.

AB - We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form Rni×Mi. We investigate the family of vertical resolvent {t∇(1+tΔ)-m}t>0, where m≥1. We show that the family is uniformly continuous on all Lp for 1 ≤ p ≤ minini. Interestingly, this is a closed-end condition in the considered setting. We prove that the corresponding maximal function is bounded in the same range except that it is only weak-type (1, 1) for p=1. The Fefferman-Stein vector-valued maximal function is again of weak-type (1, 1) but bounded if and only if 1 < p < minini, and not at p = minini.

KW - Fefferman-Stein vector-valued maximal functions

KW - Horizontal maximal functions

KW - Laplace-Beltrami operator

KW - Manifolds with ends

KW - R-boundedness

KW - Riesz transform

KW - Vertical maximal functions

KW - Vertical square functions

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UR - http://purl.org/au-research/grants/arc/DP200101065

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DO - 10.1007/s00028-024-00981-8

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SN - 1424-3199

VL - 24

SP - 1

EP - 31

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 3

M1 - 49

ER -

Vertical maximal functions on manifolds with ends

Abstract

Bibliographical note

Keywords

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Nonlinear harmonic analysis and dispersive partial differential equations

Cite this

Vertical maximal functions on manifolds with ends

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

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Nonlinear harmonic analysis and dispersive partial differential equations

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