On the Integrability of Eigenfunctions of the Laplace-Beltrami Operator in the Unit Ball of C
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Authors: | Manfred Stoll |
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Institution: | (1) Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA |
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Abstract: | Let B denote the unit ball in C
n
, n1, and let ,
, and denote the volume measure, gradient, and Laplacian respectively, with respect to the Bergman metric on B. For R and 0<p<, we denote by L
p
the set of real, or complex-valued measurable functions f on B for which
B
(1–|z|2)|f(z)|
p
d(z)<, and by D
p
the Dirichlet space of C
1 functions f on B for which |
f|L
p
. Also, for C, we denote by X
the set of C
2 real, or complex-valued functions f on B for which
f=f. The main result of the paper is as follows: Let 0<p< and suppose R with –n
2. Then L
p
X
={0}, and for 0, D
p
X
={0}(a) for all n+
when p1, and(b) for all
when 0<p<1.By example it is shown that the result is best possible for all values of p with pn/(n+
. |
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Keywords: | Dirichlet spaces eigenfunctions Laplace-Beltrami operator -harmonic functions" target="_blank">gif" alt="phmmat" align="MIDDLE" BORDER="0">-harmonic functions |
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