Journal of the Mathematical Society of Japan

Conjugate functions on spaces of parabolic Bloch type

Yôsuke HISHIKAWA, Masaharu NISHIO, and Masahiro YAMADA

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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

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

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K05: Heat equation
Secondary: 42A50: Conjugate functions, conjugate series, singular integrals 32A18: Bloch functions, normal functions

conjugate function Bloch space parabolic operator of fractional order


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.

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  • Y. Hishikawa, Fractional calculus on parabolic Bergman spaces, Hiroshima Math. J., 38 (2008), 471–488.
  • Y. Hishikawa, The reproducing formula with fractional orders on the parabolic Bloch space, J. Math. Soc. Japan, 62 (2010), 1219–1255.
  • Y. Hishikawa, M. Nishio and M. Yamada, A conjugate system and tangential derivative norms on parabolic Bergman spaces, Hokkaido Math. J., 39 (2010), 85–114.
  • Y. Hishikawa and M. Yamada, Function spaces of parabolic Bloch type, Hiroshima Math. J., 41 (2011), 55–87.
  • H. Koo, K. Nam and H. Yi, Weighted harmonic Bergman functions on half-spaces, J. Korean Math. Soc., 42 (2005), 975–1002.
  • M. Nishio, K. Shimomura and N. Suzuki, $\alpha$-parabolic Bergman spaces, Osaka J. Math., 42 (2005), 133–162.
  • M. Nishio, N. Suzuki and M. Yamada, Toeplitz operators and Carleson measures on parabolic Bergman spaces, Hokkaido Math. J., 36 (2007), 563–583.
  • W. C. Ramey and H. Yi, Harmonic Bergman functions on half-spaces, Trans. Amer. Math. Soc., 348 (1996), 633–660.
  • E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^{p}$-spaces, Acta Math., 103 (1960), 25–62.
  • M. Yamada, Harmonic conjugates of parabolic Bergman functions, In: Potential Theory in Matsue, (eds. H. Aikawa, T. Kumagai, Y. Mizuta and N. Suzuki), Adv. Stud. Pure Math., 44, Math. Soc. Japan, Tokyo, 2006, pp.,391–402.