Pointwise convergence to initial data of heat and Laplace equations
HTML articles powered by AMS MathViewer
- by Gustavo Garrigós, Silvia Hartzstein, Teresa Signes, José Luis Torrea and Beatriz Viviani PDF
- Trans. Amer. Math. Soc. 368 (2016), 6575-6600 Request permission
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 $u(x)$.References
- Ibraheem Abu-Falahah, Pablo Raúl Stinga, and José L. Torrea, A note on the almost everywhere convergence to initial data for some evolution equations, Potential Anal. 40 (2014), no. 2, 195–202. MR 3152161, DOI 10.1007/s11118-013-9351-z
- V. Aldaya, F. Cossío, J. Guerrero, and F. F. López-Ruiz, The quantum Arnold transformation, J. Phys. A 44 (2011), no. 6, 065302, 19. MR 2763443, DOI 10.1088/1751-8113/44/6/065302
- N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble) 11 (1961), 385–475 (English, with French summary). MR 143935
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- L. Carleson and P. Jones, Weighted norm inequalities and a theorem of Koosis, Mittag-Leffler Inst., report n. 2, 1981.
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- Silvia I. Hartzstein, José L. Torrea, and Beatriz E. Viviani, A note on the convergence to initial data of heat and Poisson equations, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1323–1333. MR 3008879, DOI 10.1090/S0002-9939-2012-11441-8
- E. Harboure, J. L. Torrea, and B. Viviani, On the search for weighted inequalities for operators related to the Ornstein-Uhlenbeck semigroup, Math. Ann. 318 (2000), no. 2, 341–353. MR 1795566, DOI 10.1007/PL00004424
- Bang-He Li, Explicit relation between the solutions of the heat and the Hermite heat equation, Z. Angew. Math. Phys. 58 (2007), no. 6, 959–968. MR 2363895, DOI 10.1007/s00033-007-7044-4
- L. Liu and P. Sjögren, A characterization of the Gaussian Lipschitz space and sharp estimates for the Ornstein-Uhlenbeck Poisson kernel, to appear in Rev. Matem. Iberoamericana.
- E. M. Nikišin, A resonance theorem and series in eigenfunctions of the Laplace operator, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 795–813 (Russian). MR 0343091
- Ebner Pineda and Wilfredo Urbina R., Non tangential convergence for the Ornstein-Uhlenbeck semigroup, Divulg. Mat. 16 (2008), no. 1, 107–124 (English, with English and Spanish summaries). MR 2587011
- José L. Rubio de Francia, Weighted norm inequalities and vector valued inequalities, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 86–101. MR 654181
- Peter Sjögren and J. L. Torrea, On the boundary convergence of solutions to the Hermite-Schrödinger equation, Colloq. Math. 118 (2010), no. 1, 161–174. MR 2600523, DOI 10.4064/cm118-1-8
- Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Pablo Raúl Stinga and José Luis Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092–2122. MR 2754080, DOI 10.1080/03605301003735680
- Krzysztof Stempak and José Luis Torrea, Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 (2003), no. 2, 443–472. MR 1990533, DOI 10.1016/S0022-1236(03)00083-1
- A. Tychonoff, Théorèmes d’unicité pour l’équation de la chaleur, Math. Sb. 42 (2) (1935), 199–216.
- D. V. Widder, The heat equation, Pure and Applied Mathematics, Vol. 67, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0466967
- Calvin H. Wilcox, Positive temperatures with prescribed initial heat distributions, Amer. Math. Monthly 87 (1980), no. 3, 183–186. MR 562921, DOI 10.2307/2321603
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
- 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).
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6575-6600
- MSC (2010): Primary 42C10, 35C15, 33C45, 40A10
- DOI: https://doi.org/10.1090/tran/6554
- MathSciNet review: 3461043