Bounded function on which infinite iterations of weighted Berezin transform exist
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초록
We exhibit some properties of the weighted Berezin transform $T_{\h}f$ on $L^{\infty}(B_{n})$ and on $L^{1}(B_{n})$. As the main result, we prove that if $f \in L^{\infty}(B_{n})$ with $\lim_{k\rightarrow\infty} T_{\h}^{k}f$ exists, then there exist unique $\mathcal{M}$-harmonic function $g$ and $h \in \overline{(I-T_{\h})L^{\infty}(B_{n})}$ such that $f=g+h$. We also show that of the norm of weighted Berezin operator $T_{\h}$ on $L^{1}(B_n, \nu)$ converges to 1 as $\h$ tends to infinity, where $\nu$ is an ordinary Lebesgue measure.
Abstract
We exhibit some properties of the weighted Berezin transform $T_{\h}f$ on $L^{\infty}(B_{n})$ and on $L^{1}(B_{n})$. As the main result, we prove that if $f \in L^{\infty}(B_{n})$ with $\lim_{k\rightarrow\infty} T_{\h}^{k}f$ exists, then there exist unique $\mathcal{M}$-harmonic function $g$ and $h \in \overline{(I-T_{\h})L^{\infty}(B_{n})}$ such that $f=g+h$. We also show that of the norm of weighted Berezin operator $T_{\h}$ on $L^{1}(B_n, \nu)$ converges to 1 as $\h$ tends to infinity, where $\nu$ is an ordinary Lebesgue measure.
참고문헌 (8)
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