On weighted Toeplitz, big Hankel operators and Carleson measures (Q989589)

Let \(B\) be the unit ball of the complex \(n\) space, \(n\geq 1\). Let \(H(B)\) be the space of all holomorphic functions on \(B\). For \(1<p<\infty\), a nonnegative integer \(k\) and \(M>0\), let \(L^p_{k, M}\) be the \(L^p\)-Sobolev space of order \(k\) with respect to the weighted measure \((1-|z|^2)^{M-1}d\nu(z)\), where \(\nu\) is the normalized Lebesgue measure on \(B\). Also, for \(1< p< \infty\) and \(s\) real, let \(B_s^p\) be the holomorphic Besov space consisting of all \(f\in H(B)\) such that \((1-|z|^2)^{k-s}( I + R)^k f(z)\in L^p((1-|z|^2)^{-1}d\nu)\) for some nonnegative \(k>s\). Here, \(I\) is the identity operator and \(R=\sum_{j=1}^n z_j{{\partial}\over{\partial z_j}}\). It is known that \(B_s^p=H(B)\cap L^p_{k, (k-s)p}\) with equivalent norms for each \(k>s\). For \(N>0\), let \(P^N\) be the integral operator on \(L^1_{0,N}\) given by \[ P^N\varphi(z)= c_N\int_B {{\varphi(w)}\over {(1-z\bar w})^{n+N}} (1-|w|^2)^{N-1} \,d\nu(w), \] where \(C_N\) is the normalizing constant. For \(g\in H(B)\), the authors consider operators (i) \(M_g: B^p_s\to H(B)\) and (ii) \(M_{\bar g}: B^p_s\to C^\infty(B)\). Here, \(M_\psi\) denotes the pointwise multiplication operator by \(\psi\). With the restriction \(g\in B^1_{-N}\), they also consider (iii) the densely-defined Toeplitz operator \(T_{\bar g}^N=P^NM_{\bar g}: B^p_s\to H(B)\) and (iv) the densely-defined Hankel operator \(H_{\bar g}^N=(I-P^N)M_{\bar g}: B^p_s\to C^\infty(B)\). The main results are characterizations of the symbol function \(g\) to assure the boundedness of each operator in (i)--(iv) from \(B_s^p\) to \(L^p_{k, (k-s)p}\) for arbitrary \(N>0\) and \(s\) real. The characterizations depend on the ranges of parameters \(p\), \(n\), \(N\), and \(s\). Moreover, optimal operator norm estimates are obtained in terms of the symbol function \(g\). These results extend well-known and classical results for the standard weighted Bergman spaces.