Hiroshima Mathematical Journal

Function spaces of parabolic Bloch type

Yôsuke Hishikawa and Masahiro Yamada

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The $L^{(\alpha)}$-harmonic function is the solution of the parabolic operator $L^{(\alpha)}= \partial_{t}+(-\Delta_{x})^{\alpha}$. We study a function space $\widetilde{{\cal B}}_{\alpha}(\sigma)$ consisting of $L^{(\alpha)}$-harmonic functions of parabolic Bloch type. In particular, we give a reproducing formula for functions in $\widetilde{{\cal B}}_{\alpha}(\sigma)$. Furthermore, we study the fractional calculus on $\widetilde{{\cal B}}_{\alpha}(\sigma)$. As an application, we also give a reproducing formula with fractional orders for functions in $\widetilde{{\cal B}}_{\alpha}(\sigma)$. Moreover, we investigate the dual and pre-dual spaces of function spaces of parabolic Bloch type.

Article information

Hiroshima Math. J., Volume 41, Number 1 (2011), 55-87.

First available in Project Euclid: 31 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K05: Heat equation
Secondary: 31B10: Integral representations, integral operators, integral equations methods 32A18: Bloch functions, normal functions

Bloch space parabolic operator of fractional order reproducing formula


Hishikawa, Yôsuke; Yamada, Masahiro. Function spaces of parabolic Bloch type. Hiroshima Math. J. 41 (2011), no. 1, 55--87. doi:10.32917/hmj/1301586290. https://projecteuclid.org/euclid.hmj/1301586290

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