ON HARMONICITY IN A DISC AND n-HARMONICITY
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let τ ≠ δ0 be either a power bounded radial measure with compact support on the unit disc D with τ(D) = 1 such that there is a δ > 0 so that |ˆτ(s)| ≠ 1 for every s ∈ ∑(δ) ＼ {0, 1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {h ∈ L∞(D) | limn→∞ h*τn exists}. Let τ1, . . . , τn be measures on D with above mentioned properties. Here, we prove that if f ∈ L∞(Dn) satisfies an invariant volume mean value property with respect to τ1, . . . , τn, then f is n-harmonic.
Abstract
Let τ ≠ δ0 be either a power bounded radial measure with compact support on the unit disc D with τ(D) = 1 such that there is a δ > 0 so that |ˆτ(s)| ≠ 1 for every s ∈ ∑(δ) ＼ {0, 1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {h ∈ L∞(D) | limn→∞ h*τn exists}. Let τ1, . . . , τn be measures on D with above mentioned properties. Here, we prove that if f ∈ L∞(Dn) satisfies an invariant volume mean value property with respect to τ1, . . . , τn, then f is n-harmonic.
참고문헌 (9)
- P. Ahern, "An invariant volume-mean-value property", J. Funct. Anal., 111(2), pp.380~397 (1993)
- J. Arazy, "Iterates and the boundary behavior of the Berezin transform", Ann. Inst. Fourier (Grenoble), 51(4), pp.1101~1133 (2001)
- Y. Benyamini, "Harmonic analysis of spherical functions on SU(1, 1)", Ann. Inst. Fourier (Grenoble), 42(3), pp.671~694 (1992)
- M. Engliš, "Functions invariant under the Berezin transform", J. Funct. Anal., 121(1), pp.233~254 (1994)
- H. Furstenberg, "A Poisson formula for semi-simple Lie groups", Ann. of Math., 77, pp.335~386 (1963)
- H. Furstenberg, "Boundaries of Riemannian symmetric spaces. in: Symmetric spaces, Pure and Appl. Math., Vol. 8", pp.359~377 (1972), Dekker
- Y. Katznelson, "On power bounded operators", J. Funct. Anal., 68(3), pp.313~328 (1986)
- J. Lee, "Weighted Berezin transform in the polydisc", J. Math. Anal. Appl., 338(2), pp.1489~1493 (2008)
- W. Rudin, "Function Theory in the Unit Ball of Cn" (1980), Springer-Verlag, New York Inc.