Open Access
April, 2013 Conjugate functions on spaces of parabolic Bloch type
Yôsuke HISHIKAWA, Masaharu NISHIO, Masahiro YAMADA
J. Math. Soc. Japan 65(2): 487-520 (April, 2013). DOI: 10.2969/jmsj/06520487


Let $H$ be the upper half-space of the $(n+1)$-dimensional Euclidean space. Let 0 < $\alpha \le 1$ and $m(\alpha)=\min \{1, 1/(2\alpha) \}$. For $\sigma$ > $-m(\alpha)$, the $\alpha$-parabolic Bloch type space ${\cal B}_{\alpha}(\sigma)$ on $H$ is the set of all solutions $u$ of the equation $( \partial/\partial t+(-\Delta_{x})^{\alpha} )u=0$ with finite Bloch norm $\| u \|_{{\cal B}_{\alpha}(\sigma)}$ of a weight $t^{\sigma}$. It is known that ${\cal B}_{1/2}(0)$ coincides with the classical harmonic Bloch space on $H$. We extend the notion of harmonic conjugate functions to functions in the $\alpha$-parabolic Bloch type space ${\cal B}_{\alpha}(\sigma)$. We study properties of $\alpha$-parabolic conjugate functions and give an application to the estimates of tangential derivative norms on ${\cal B}_{\alpha}(\sigma)$. Inversion theorems for $\alpha$-parabolic conjugate functions are also given.


Download Citation

Yôsuke HISHIKAWA. Masaharu NISHIO. Masahiro YAMADA. "Conjugate functions on spaces of parabolic Bloch type." J. Math. Soc. Japan 65 (2) 487 - 520, April, 2013.


Published: April, 2013
First available in Project Euclid: 25 April 2013

zbMATH: 1327.35405
MathSciNet: MR3055594
Digital Object Identifier: 10.2969/jmsj/06520487

Primary: 35K05
Secondary: 32A18 , 42A50

Keywords: Bloch space , conjugate function , parabolic operator of fractional order

Rights: Copyright © 2013 Mathematical Society of Japan

Vol.65 • No. 2 • April, 2013
Back to Top