Leibenzon's backward shift and composition operators (Q2750842)

Let \(n\) be a positive integer, and let \(B= B_n\) denote the open unit ball of the complex Euclidean space \(C_n\). If \(\varphi:B\to B\) is holomorphic, then the composition operator \({\mathcal C}_\varphi\) is defined by \({\mathcal C}_\varphi(f)(z)= f(\varphi(z))\), \(z\in B\), where \(f: B\to C\) is holomorphic. If \(1< p< \infty\), then the norm \(\|.\|_{Hp}= |.|_p\) on Hardy space \(H^p(B)\) is defined by \(|f|^p_p= \sup\{\int_S|f(r\zeta)^p d\sigma(\zeta)\), \(0< r< 1\}\), where \(\sigma\) is the normalized Lebesgue measure on \(S= \partial B\). The weighted Bergman space \(A^p_q(B)\) is defined for \(0< p<\infty\), \(q>-1\), by NEWLINE\[NEWLINEA^p_q(B)= \Biggl\{f:f\text{ is holomorphic on \(B\) and }\int_B|f(z)|^p (1-|z|^2)^q dv(z)< \infty\Biggr\},NEWLINE\]NEWLINE where \(v\) is Lebesgue measure on \(B\).NEWLINENEWLINENEWLINEThe holomorphic function \(f(z)= \sum\{c_jz^j, j= 0,\pm 1,\pm 2,\dots\}\), \(z\in B\), is said to be in the weighted Hardy space \(H^2_q(B)\) if NEWLINE\[NEWLINE\|f\|^2_q= \sum\Biggl\{{|j|+ q\choose q}|c_j|^2\updownarrow z \updownarrow^2_2, j= 0,\pm 1,\pm 2,\dots\Biggr\},NEWLINE\]NEWLINE where \({a\choose b}= \Gamma(a+ 1)/(\Gamma(a+ b+ 1)\Gamma(b+ 1))\), and \(\updownarrow z \updownarrow^2_2=\|z\|_{L^2(S)}\).NEWLINENEWLINENEWLINEIn the main theorem of this paper, the author shows that:NEWLINENEWLINENEWLINEif \(\varphi: B\to B\) is holomorphic and \(\varphi(0)= 0\), then NEWLINE\[NEWLINE\|{\mathcal C}_\varphi(f)\|_0\leq\|f\|_{1-n},\quad f\in H^2_{1-n}(B).NEWLINE\]NEWLINE Earlier results referred to in the introduction of the paper include statements of results due to \textit{B. D. MacCluer} and \textit{P. R. Mercer} [Proc. Am. Math. Soc. 123, No. 7, 2093-2102 (1995; Zbl 0831.47023)], indicating that \({\mathcal C}_\varphi\) maps \(H^p(B)\) into \(A^p_{n-2}(B)\).