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Pointwise convergence to initial data of heat and Laplace equations


Authors: Gustavo Garrigós, Silvia Hartzstein, Teresa Signes, José Luis Torrea and Beatriz Viviani
Journal: Trans. Amer. Math. Soc. 368 (2016), 6575-6600
MSC (2010): Primary 42C10, 35C15, 33C45, 40A10
DOI: https://doi.org/10.1090/tran/6554
Published electronically: January 13, 2016
MathSciNet review: 3461043
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Abstract: Let $ L$ be either the Hermite or the Ornstein-Uhlenbeck operator on $ \mathbb{R}^d$. We find optimal integrability conditions on a function $ f$ for the existence of its heat and Poisson integrals, $ e^{-tL}f(x)$ and $ e^{-t\sqrt L}f(x)$, solutions respectively of $ U_t = -LU$ and $ U_{tt} = LU$ on $ \mathbb{R}^{d+1}_+$ with initial datum $ f$. As a consequence we identify the most general class of weights $ v(x)$ for which such solutions converge a.e. to $ f$ for all $ f\in L^p(v)$, and each $ p\in [1,\infty )$. Moreover, if $ 1\!<\!p\!<\!\infty $ we additionally show that for such weights the associated local maximal operators are strongly bounded from $ L^p(v)\to L^p(u)$ for some other weight $ u(x)$.


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Additional Information

Gustavo Garrigós
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
Email: gustavo.garrigos@um.es

Silvia Hartzstein
Affiliation: IMAL (UNL-CONICET) y FIQ (Universidad Nacional del Litoral), Güemes 3450, 3000 Santa Fe, Argentina
Email: shartzstein@santafe-conicet.gov.ar

Teresa Signes
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
Email: tmsignes@um.es

José Luis Torrea
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, ICMAT-CISC- UAM-UCM-UC3M, 28049, Madrid, Spain
Email: joseluis.torrea@uam.es

Beatriz Viviani
Affiliation: IMAL (UNL-CONICET) y FIQ (Universidad Nacional del Litoral), CCT CONICET Santa Fe Colectora Ruta Nac. N168, Paraje El Pozo, 3000 Santa Fe, Argentina
Email: viviani@santafe-conicet.gov.ar

DOI: https://doi.org/10.1090/tran/6554
Keywords: Hermite operator, Ornstein-Uhlenbeck, Poisson integral, weighted inequalities
Received by editor(s): November 18, 2013
Received by editor(s) in revised form: August 15, 2014, and August 29, 2014
Published electronically: January 13, 2016
Additional Notes: The first author was partially supported by grants MTM2010-16518, MTM2013-40945-P and MTM2014-57838-C2-1-P (Spain). The third author was partially supported by grants MTM2013-42220-P and Fundación Séneca 19378/PI/14 (Murcia, Spain). The fourth author was partially supported by Grant MTM2011-28149-C02-01 (Spain). The second and fifth authors were partially supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad Nacional del Litoral (Argentina).
Article copyright: © Copyright 2016 American Mathematical Society