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Let $m$ be the Lebesgue measure on $\mathbb C$ normalized to $m(D)=1$, $\mu$ be an invariant measure on $D$ defined by $d \mu(z) = (1-|z|^{2})^{-2} \ dm(z)$. For $f \in L^{1}(D^{n} , m \times \cdot \cdot \cdot \times m) , \ Bf$ the Berezin transform of $f$ is defined by, $$(Bf)(z_{1} , \ldots, z_{n})\! =\! \int_{D} \cdot \cdot \cdot \int_{D} f \big( \varphi_{z_{1}}(x_{1}), \ldots , \varphi_{z_{n}}(x_{n}) \big) dm(x_{1}) \cdots dm(x_{n}).$$ We prove that if $f \in L^{1}(D^{2},\mu\times\mu)$ is radial and satisfies $ \int \int_{D^{2}} f d\mu \times d\mu = 0, $ then for every bounded radial function $\ell$ on $D^2$ we have \[\lim_{n \rightarrow \infty} \int\!\!\int_{D^{2}} (B^{n} f) (z,w) \ell(z,w) d\mu(z) d\mu(w) = 0. \] Then, using the above property we prove $n$-harmonicity of bounded function which is invariant under the Berezin transform. And we show the same results for the weighted the Berezin transform in the polydisc.
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참고문헌 (6)
참고문헌 신청
P. Ahern / 1993 / An invariant volume-mean-value property / J. Funct. Anal 111(2):380~397
H. Furstenberg / 1963 / A Poisson formula for semi-simple Lie groups / Ann. of Math. (2) 77:335~386
H. Furstenberg / 1970 / Boundaries of Riemannian symmetric spaces, Symmetric spaces, Short Courses / Washington Univ.
J. Lee / 1998 / An invariant mean value property in the polydisc / Illinois J. Math 42(3):406~419
J. Lee / 2008 / Weighted Berezin transform in the polydisc / J. Math. Anal. Appl 338(2):1489~1493
H. Furstenberg / 1963 / A Poisson formula for semi-simple Lie groups / Ann. of Math. (2) 77:335~386
H. Furstenberg / 1970 / Boundaries of Riemannian symmetric spaces, Symmetric spaces, Short Courses / Washington Univ.
J. Lee / 1998 / An invariant mean value property in the polydisc / Illinois J. Math 42(3):406~419
J. Lee / 2008 / Weighted Berezin transform in the polydisc / J. Math. Anal. Appl 338(2):1489~1493
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