SOME PROPERTIES OF THE BEREZIN TRANSFORM IN THE BIDISC

Let m be the Lebesgue measure on ${\mathbb{C}}$ normalized to $m(D)=1,{\mu}$ be an invariant measure on D defined by $d_{\mu}(z)=(1-{\mid}z{\mid}^2)^{-2}dm(z)$. For $f{\in}L^1(D^n,m{\times}{\cdots}{\times}m)$, Bf the Berezin transform of f is defined by, $$(Bf)(z_1,{\ldots},z_n)={\displaystyle\smashmargin{2}{\int\nolimits_D}{\cdots}{\int\nolimits_D}}f({\varphi}_{z_1}(x_1),{\ldots},{\varphi}_{z_n}(x_n))dm(x_1){\cdots}dm(x_n)$$. We prove that if $f{\in}L^1(D^2,{\mu}{\times}{\mu})$ is radial and satisfies ${\int}{\int_{D^2}}fd{\mu}{\times}d{\mu}=0$, then for every bounded radial function ${\ell}$ on $D^2$ we have $$\lim_{n{\rightarrow}{\infty}}{\displaystyle\smashmargin{2}{\int\int\nolimits_{D^2}}}(B^nf)(z,w){\ell}(z,w)d{\mu}(z)d{\mu}(w)=0$$. 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.