Harmonic Bergman Functions on Half-Spaces

Ramey, Wade; Yi, HeungSu

doi:10.1090/S0002-9947-96-01383-9

Harmonic Bergman Functions on Half-Spaces
HTML articles powered by AMS MathViewer

by Wade C. Ramey and HeungSu Yi PDF

Trans. Amer. Math. Soc. 348 (1996), 633-660 Request permission

Abstract:

We study harmonic Bergman functions on the upper half-space of $\mathbf {R}^n$. Among our main results are: The Bergman projection is bounded for the range $1< p < \infty$; certain nonorthogonal projections are bounded for the range $1\leq p < \infty$; the dual space of the Bergman $L^1$-space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range $1\leq p < \infty$; the Bergman norm is equivalent to a “normal derivative norm” as well as to a “tangential derivative norm”.

References

Lars V. Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math. (2) 74 (1961), 171–191. MR 204641, DOI 10.2307/1970309
H. Ajmi and W. Ramey, Harmonic Bloch functions on the upper half space (to appear).
Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR 958569
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Frank Forelli and Walter Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593–602. MR 357866, DOI 10.1512/iumj.1974.24.24044
G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. Reine Angew. Math. 167 (1931), 405–423
H. S. Shapiro, Global geometric aspects of Cauchy’s problem for the Laplace operator, research report TRITA-MAT-1989-37, Royal Inst. Tech., Stockholm.
A. L. Shields and D. L. Williams, Bonded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302. MR 283559, DOI 10.1090/S0002-9947-1971-0283559-3
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990. MR 1074007