## Journal of the Mathematical Society of Japan

### Conjugate functions on spaces of parabolic Bloch type

#### Abstract

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.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 2 (2013), 487-520.

Dates
First available in Project Euclid: 25 April 2013

https://projecteuclid.org/euclid.jmsj/1366896642

Digital Object Identifier
doi:10.2969/jmsj/06520487

Mathematical Reviews number (MathSciNet)
MR3055594

Zentralblatt MATH identifier
1327.35405

#### Citation

HISHIKAWA, Yôsuke; NISHIO, Masaharu; YAMADA, Masahiro. Conjugate functions on spaces of parabolic Bloch type. J. Math. Soc. Japan 65 (2013), no. 2, 487--520. doi:10.2969/jmsj/06520487. https://projecteuclid.org/euclid.jmsj/1366896642

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