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For $c>-1$, let $\nu_{c}$ denote a weighted radial measure on $\mathbb{C}$ normalized so that $\nu_{c}(D)=1.$ For $c_1, c_2 >-1$ and $f \in L^{1}(D^{2},\ \nu_{c_1} \times \nu_{c_2})$, we define the weighted Berezin transform $B_{c_1,c_2}f$ on $D^2$ by $$\big(B_{c_1, c_2}\big)f(z,w) = \int_{D}\int_{D} f\big( \varphi_{z}(x) , \varphi_{w}(y) \big)\ d\nu_{c_1}(x) d\nu_{c_2}(y).$$ This paper is about the space $M_{c_1, c_2}^{p}$ of function $f \in L^{p}(D^{2},\ \nu_{c_1} \times \nu_{c_2})$ satisfying $B_{c_1,c_2}f=f$ for $1 \le p< \infty$. We find the identity operator on $M_{c_1, c_2}^{p}$ by using invariant Laplacians and we characterize some special type of functions in $M_{c_1, c_2}^{p}$.
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참고문헌 (7)
참고문헌 신청
P. Ahern / 1993 / An invariant volume-mean-value property / J. Funct. Anal 111(2):380~397
J. P Ferrier / 1973 / Spectral Theory and Complex Analysis / North-Holland
H. Furstenberg / 1972 / Pure and Appl. Math. Vol. 8 / Dekker :359~377
J. Lee / 2008 / Weighted Berezin transform in the polydisc / J. Math. Anal. Appl 338(2):1489~1493
이재성 / 2017 / Some properties of the Berezin transform in the bidisc / 대한수학회논문집 32(3):779~787
J. P Ferrier / 1973 / Spectral Theory and Complex Analysis / North-Holland
H. Furstenberg / 1972 / Pure and Appl. Math. Vol. 8 / Dekker :359~377
J. Lee / 2008 / Weighted Berezin transform in the polydisc / J. Math. Anal. Appl 338(2):1489~1493
이재성 / 2017 / Some properties of the Berezin transform in the bidisc / 대한수학회논문집 32(3):779~787
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