Bounded and compact multipliers between Bergman and Hardy spaces
Let \(D\) denote the open unit disc \(\{z\in\mathbb{C}: |z|<1\}\) of the complex plane \(\mathbb{C}\). If \(0< p\leq\infty\), \(0<q,s<\infty\), then \(H(p,q,s)\) is associated with the weighted Hardy space \(H^p_s= H(p,\infty,s)\), and contains elements of the form \(f\) in the space of holomorphic functions \(H(D)\), with norm \[ \|f\|_{p,q,s}= \Biggl(\int^1_0 M_p(r, f)^q(1- r)^{sq-1} dr\Biggr)^{1/q};\quad M_p(r, f)= \Biggl((2\pi)^{-1} \int^1_0|f(re^{i\theta})|^p d\theta\Biggr)^{1/p}. \] The weighted Bergman space \(A^p_\alpha\) is identifiable as \(H(p,p,(1+\alpha)/p)\), \(\alpha>-1\). Let \(X\), \(Y\) be function spaces consisting of functions of the form \(f\) with \(f(z)= \sum^\infty_{j=0} a_nz^n\). If \(\gamma= \{\gamma_n, n= 0,1,2,\dots\}\) is a preassigned sequence, and the operator \(T_\gamma\) is defined by \(T_\gamma(f)(z)= \sum^\infty_{n=0} \gamma_na_nz^n\), then \(\gamma\) is said to be a multiplier in \((X,Y)\) if \(T_\gamma: X\to Y\), and \(T_\gamma\) is a bounded operator. The fractional operator \(D^t\), \(t>0\), is defined as a multiplier operator with \(\gamma_j= j^t\), \(j= 0,1,2,\dots\)\ . Let \(\ell_{p,q}\) denote the space of sequences of the form \(a= (a_j)\) for which \(\{\left(\sum^{2^k- 1}_{j= 2^{k-1}}|a_j|^p\right)^{1/p}\), \(k= 1,2,\dots\}\) has finite \(\ell_q\)-norm, and let \(K(X,Y)\) denote the class of compact operators from \(X\) into \(Y\). The main results of this paper identify and indicate inclusions involving the spaces \((H^p,H^q)\) and \((A^p,A^q)\). In particular, it is shown in Theorem 5 that if \(1<p<2\), \(1/r= (1/p)-{1\over 2}= (2-p)/2p\), then \(D^{-1/r}\ell^{r,\infty,\infty}\subseteq (A^p,A^2)\subseteq D^{-1/r}\ell^\infty\). It is also shown in Theorem 14 that if \(p>1\), \(q\geq 1\), then \({\mathcal L}(A^p,A^q)= K(A^p,A^q)\) if and only if \(q<p\), where \({\mathcal L}(X,Y)\) is the space of bounded linear operators from \(X\) into \(Y\). The paper contains references to papers which include related results.
scientific article; zbMATH DE number 1346928
scientific article