Hokkaido Mathematical Journal

A conjugate system and tangential derivative norms on parabolic Bergman spaces

Yosuke HISHIKAWA, Masaharu NISHIO, and Masahiro YAMADA

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The $\alpha$-parabolic Bergman space ${\boldsymbol b}_{\alpha}^{p}(\lambda)$ is the Banach space of solutions of the parabolic equation $L^{(\alpha)} = \partial/\partial t+(-\Delta_{x})^{\alpha}$ on the upper half space $H$ which have finite $L^{p}(H,t^{\lambda}dV)$ norms, where $t^{\lambda}dV$ is the weighted Lebesgue volume measure on $H$. It is known that ${\boldsymbol b}^{p}_{1/2}(\lambda)$ coincide with the harmonic Bergman spaces. In this paper, we introduce the extension of notion of conjugate functions of ${\boldsymbol b}_{\alpha}^{p}(\lambda)$-functions and study their properties. As an application, we give estimates of tangential derivative norms on ${\boldsymbol b}_{\alpha}^{p}(\lambda)$.

Article information

Hokkaido Math. J., Volume 39, Number 1 (2010), 85-114.

First available in Project Euclid: 19 May 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K05: Heat equation
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 42A50: Conjugate functions, conjugate series, singular integrals

conjugate function tangential derivative heat equation parabolic operator of fractional order Bergman space


HISHIKAWA, Yosuke; NISHIO, Masaharu; YAMADA, Masahiro. A conjugate system and tangential derivative norms on parabolic Bergman spaces. Hokkaido Math. J. 39 (2010), no. 1, 85--114. doi:10.14492/hokmj/1274275021. https://projecteuclid.org/euclid.hokmj/1274275021

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