Abstract
We obtain parabolic Besov smoothness improvement for temperatures on cylindrical regions based on Lipschitz domains. The results extend those for harmonic functions obtained by S. Dahlke and R. DeVore using the wavelet description of Besov regularity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aimar, H., Gómez, I.: Smoothness improvement for temperatures in terms of the Besov regularity of initial and Dirichlet data. Preprint
Aimar, H., Gómez, I., Iaffei, B.: Parabolic mean values and maximal estimates for gradients of temperatures. J. Funct. Anal. 255(8), 2008 (1939–1956)
Aimar, H., Gómez, I., Iaffei, B.: On Besov regularity of temperatures. J. Fourier Anal. Appl. 16(6), 1007–1020 (2010)
Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)
Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representations of Functions and Imbedding Theorems. Scripta Series in Mathematics, vol. I+II. Winston, Washington (1978/1979). English transl.: Taibleson, M.H. (ed.)
Bownik, M.: The construction of r-regular wavelets for arbitrary dilations. J. Fourier Anal. Appl. 7(5), 489–506 (2001)
Dahlke, S.: Besov regularity for elliptic boundary value problems in polygonal domains. Appl. Math. Lett. 12(6), 31–36 (1999)
Dahlke, S., DeVore, R.A.: Besov regularity for elliptic boundary value problems. Commun. Partial Differ. Equ. 22(1–2), 1–16 (1997)
DeVore, R.A., Sharpley, R.C.: Maximal functions measuring smoothness. Mem. Am. Math. Soc. 47(293), viii+115 (1984)
Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)
Jakab, T., Mitrea, M.: Parabolic initial boundary value problems in nonsmooth cylinders with data in anisotropic Besov spaces. Math. Res. Lett. 13(5–6), 825–831 (2006)
Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130(1), 161–219 (1995)
Leisner, C.: Nonlinear wavelet approximation in anisotropic Besov spaces. Indiana Univ. Math. J. 52(2), 437–455 (2003)
Meyer, Y.: Wavelets and Operators. Cambridge Studies in Advanced Mathematics, vol. 37. Cambridge University Press, Cambridge (1992). Translated from the 1990 French original by D.H. Salinger
Peetre, J.: New Thoughts on Besov Spaces. Duke University Mathematics Series, vol. 1. Mathematics Department, Duke University, Durham (1976)
Schmeisser, H.-J.: Anisotropic spaces. II. Equivalent norms for abstract spaces, function spaces with weights of Sobolev–Besov type. Math. Nachr. 79, 55–73 (1977)
Schmeisser, H.-J., Triebel, H.: Anisotropic spaces. I. Interpolation of abstract spaces and function spaces. Math. Nachr. 73, 107–123 (1976)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)
Acknowledgement
The research was supported by CONICET, UNL and ANPCyT (Argentina).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ronald A. DeVore.
Rights and permissions
About this article
Cite this article
Aimar, H., Gómez, I. Parabolic Besov Regularity for the Heat Equation. Constr Approx 36, 145–159 (2012). https://doi.org/10.1007/s00365-012-9166-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-012-9166-y