Parabolic function spaces with mixed norm
Author:
V. R. Gopala Rao
Journal:
Trans. Amer. Math. Soc. 246 (1978), 451461
MSC:
Primary 46E35
MathSciNet review:
515551
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Abstract: The spaces of parabolic Bessel potentials were introduced by B. F. Jones and R. J. Bagby. We prove a Sobolevtype imbedding theorem for (multinormed versions of ) when is a positive integer k, , . In particular this theorem holds for , since . We use the concepts of parabolic Riesz transforms and halftime derivatives introduced by us elsewhere.
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 R. J. Bagby and D. R. Adams, Translationdilation invariant estimates for Riesz potentials, Indiana Univ. Math. J. 23 (1974), 10511067. MR 0348471 (50:969)
 [3]
 A. Benedek and R. Panzone, The space with mixed norm, Duke Math. J. 28 (1961), 301324. MR 0126155 (23:A3451)
 [4]
 A. Benedek, Spaces of differentiable functions and distributions, with mixed norm, Rev. Un. Mat. Argentina. 32 (1964), 321. MR 0167838 (29:5104)
 [5]
 L. Hormander, Linear partial differential operators, SpringerVerlag, Berlin and New York, 1969. MR 0248435 (40:1687)
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 B. F. Jones, Jr., Lipschitz spaces and the heat equation, J. Math. Mech. 18 (1968), 379410. MR 0511929 (58:23543)
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 , Singular integrals and a boundary value problem for the heat equation, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, R. I., 1967, pp. 196207. MR 0235432 (38:3741)
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 O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R. I., 1968.
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 P. I. Lizorkin, Multipliers of Fourier integrals and bounds of convolution in spaces with mixed norms, Math. USSRIzv. 4 (1970), 225254.
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 V. R. Gopala Rao, A characterization of parabolic function spaces, Amer. J. Math. 99 (1977), 985993. MR 0500114 (58:17815)
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 C. H. Sampson, A characterization of parabolic Lebesgue spaces, Thesis, Rice Univ., 1968.
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 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197805155511
PII:
S 00029947(1978)05155511
Keywords:
Parabolic Bessel potentials,
parabolic Riesz transforms,
halfderivative,
imbedding theorem
Article copyright:
© Copyright 1978
American Mathematical Society
