On the weighted 𝐿^{𝑝}-integrability of nonnegative ℳ-superharmonic functions

Zhao, Shi Ying

doi:10.1090/S0002-9939-1992-1101993-5

On the weighted $L^p$-integrability of nonnegative $\mathcal {M}$-superharmonic functions
HTML articles powered by AMS MathViewer

by Shi Ying Zhao PDF

Proc. Amer. Math. Soc. 115 (1992), 677-685 Request permission

Abstract:

A weighted ${L^p}$-integrability of nonnegative $\mathcal {M}$-superharmonic functions in the unit ball of ${\mathbb {C}^n}$ is studied in this paper. Our result is analogous to an earlier result of Armitage (J. London Math. Soc. (2) 4 (1971), 363-373) concerning the ${L^p}$-integrability of superharmonic functions for balls in ${\mathbb {R}^d}$. An example is given to show the sharpness of the result. Also, the weighted ${L^p}$-integrability of the invariant Green’s function for the unit ball of ${\mathbb {C}^n}$ is obtained.

References

D. H. Armitage, Further results on the global integrability of superharmonic functions, J. London Math. Soc. (2) 6 (1972), 109–121. MR 313524, DOI 10.1112/jlms/s2-6.1.109
On the global integrability of superharmonic functions in balls, J. London Math. Soc. (2) 4 (1971), 365–373. MR 291484, DOI 10.1112/jlms/s2-4.2.365
J. A. Cima and C. S. Stanton, Admissible limits of $M$-subharmonic functions, Michigan Math. J. 32 (1985), no. 2, 211–220. MR 783575, DOI 10.1307/mmj/1029003188
E. B. Fabes and D. W. Stroock, The $L^p$-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), no. 4, 997–1016. MR 771392, DOI 10.1215/S0012-7094-84-05145-7
K. T. Hahn and David Singman, Boundary behavior of invariant Green’s potentials on the unit ball in $\textbf {C}^n$, Trans. Amer. Math. Soc. 309 (1988), no. 1, 339–354. MR 957075, DOI 10.1090/S0002-9947-1988-0957075-X
L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1969. MR 0261018
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
M. Stoll, Rate of growth of $p$th means of invariant potentials in the unit ball of $\textbf {C}^n$, J. Math. Anal. Appl. 143 (1989), no. 2, 480–499. MR 1022549, DOI 10.1016/0022-247X(89)90054-1
David Ullrich, Radial limits of $M$-subharmonic functions, Trans. Amer. Math. Soc. 292 (1985), no. 2, 501–518. MR 808734, DOI 10.1090/S0002-9947-1985-0808734-8