The weak-type (1, 1) of LlogL homogeneous convolution operators

Terence Tao

Type: Article

Publication Date: 1999-01-01

Citations: 32

DOI: https://doi.org/10.1512/iumj.1999.48.1784

Abstract

We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L log L on the unit annulus is bounded on L p for 1 < p < ∞ and is of weak-type (1, 1), generalizing the result of Seeger [9].The proof is in a similar spirit to that of Christ and Rubio de Francia [3].

Locations

Indiana University Mathematics Journal - View - PDF
arXiv (Cornell University) - View
Indiana University Mathematics Journal - View - PDF
arXiv (Cornell University) - View

Similar Works

Action	Title	Year	Authors
+	The weak-type (1,1) of L \log L homogeneous convolution operators	1999	Terence Tao
+	Some Estimates for Convolution Operators with Kernels of Type (l, r) on Homogeneous Groups	2005	Yin Sheng Jiang Ming Ju Liu Lin Tang
+	Some Estimates for Convolution Operators with Kernels of Type （l, r） on Homogeneous Groups	2005	Yin Yin Sheng Jiang Jiang Ming Ming Jü Liu Liu Lin - Tang
+	Local Sobolev–Morrey estimates for nondivergence operators with drift on homogeneous groups	2013	Xiaojing Feng Pengcheng Niu
+	Convolution and Homogeneous Spaces	2009	Rajab Ali Kamyabi Gol N. Tavallaei
+	Convolution and Homogeneous Spaces	2008	Rajab Ali Kamyabi Gol Null Null
+ PDF	Homogeneous Banach Spaces as Banach Convolution Modules over M(G)	2022	Hans G. Feichtinger
+	Spectral analysis for convolution operators on locally compact groups	2007	Marius Măntoiu Rafael Tiedra de Aldecoa
+	On the multiplication and convolution of homogeneous distributions	1990	Peter Wagner
+	The Schwartz Space and Homogeneous Distributions on H-Type Groups	2001	Francesca Astengo Bianca Di Blasio
+ PDF	Homogeneous models of convolution on the line	1994	V. �. Vol'kovich
+	Homogeneous Function Spaces	1983	Hans Triebel
+	Type 1,1-operators on spaces of temperate distributions	2010	Jon Johnsen
+ PDF Chat	Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups	2017	Nathaniel Eldredge
+ PDF Chat	Product of functions in BMO and H 1 in non-homogeneous spaces	2011	Justin Feuto
+ PDF	A Gaussian bound for convolutions of functions on locally compact groups	2006	Nick Dungey
+ PDF	On the relation between Gaussian measures and convolution semigroups of local type.	1975	Christian Berg
+ PDF Chat	Spectral multiplier theorems of Euclidean type on new classes of two-step stratified groups	2014	Alessio Martini Detlef Müller
+	Weighted Sobolev–Morrey estimates for hypoelliptic operators with drift on homogeneous groups	2015	Yuexia Hou Pengcheng Niu
+ PDF	A convolution approximation property for $L^{1}$(G)	1976	Louis Pigno

Works That Cite This (33)

Action	Title	Year	Authors
+ PDF	Noncommutative maximal operators with rough kernels	2024	Xudong Lai
+	Rough bilinear singular integrals	2017	Loukas Grafakos Danqing He Petr Honzík
+	Parabolic Littlewood-Paley g-function with rough kernel	2008	Qing Ying Xue Yong Ding Kôzô Yabuta
+	Method of rotations for bilinear singular integrals	2011	Geoff Diestel Loukas Grafakos Peter Honzik Zengyan Si Erin Terwilleger
+	Weighted weak type (1,1) estimates for singular integrals with non-isotropic homogeneity	2015	Shūichi Satō
+ PDF	Estimates for singular integrals on homogeneous groups	2012	Shūichi Satō
+	An Endpoint Estimate for Rough Maximal Singular Integrals	2018	Petr Honzík
+	Asymptotic behavior of 𝐿^{𝑝} estimates for a class of multipliers with homogeneous unimodular symbols	2022	Aleksandar Bulj Vjekoslav Kovač
+ PDF	A note on singular integrals with angular integrability	2019	Feng Liu
+	A Weak-Type Estimate for Commutators	2011	Loukas Grafakos Petr Honzík

Works Cited by This (7)

Action	Title	Year	Authors
+	A theory of entropy in Fourier analysis	1978	Robert Fefferman
+	Harmonic analysis on nilpotent groups and singular integrals. II. Singular kernels supported on submanifolds	1988	Fulvio Ricci Elias M. Stein
+	The Kakeya Maximal Function and the Spherical Summation Multipliers	1977	Antonio Córdoba
+ PDF	Weak (1,1) boundedness of singular integrals with nonsmooth kernel	1988	Steve Hofmann
+	Weak type (1, 1) bounds for rough operators, II	1988	Michael Christ J.-L. Rubio de Francia
+ PDF	Singular integral operators with rough convolution kernels	1996	Andreas Seeger
+	Weak (1, 1) Boundedness of Singular Integrals with Nonsmooth Kernel	1988	Steve Hofmann