Bounded operators on weighted spaces of holomorphic functions on the upper half-plane (Q2891266)

A standard weight \(v\) on the upper half-plane \(\mathbb{G}\) is a positive continuous function constant on horizontal lines increasing with respect to the height and such that \(\lim_{t\to 0^+}v(it)=0.\)NEWLINENEWLINEThe space \(H_v(\mathbb{G})=\{f:\mathbb{G}\to\mathbb{C}\text{ analytic }: \|f\|_v=\sup_{z\in \mathbb{G}}v(z)|f(z)|<\infty\}\) endowed with the \(\|\cdot\|_v\)-norm is considered. The present paper studies the differentiation operator \(D:H_v(\mathbb{G})\to H_{v_1}(\mathbb{G})\), where \(v_1(z)=\text{im} z v(z)\). The main results areNEWLINENEWLINETheorem 1: NEWLINE\[NEWLINE\text{ran} (D)\subset H_{v_1}(\mathbb{G})\,\mathrm{iff}\, D \text{ is a bounded operator }\mathrm{iff}NEWLINE\]NEWLINENEWLINENEWLINE\[NEWLINE \text{ there are constants } c,\beta >0 \text{ such that }\frac{v(it)}{v(is)}\leq c \left(\frac{t}{s}\right)^\beta\text{ for } 0<s\leq t, \text{ and} NEWLINE\]NEWLINE Theorem 2: NEWLINE\[NEWLINE\text{ran} (D)= H_{v_1}(\mathbb{G}) \,\mathrm{iff}\, D \text{ is an isomorphism }\mathrm{iff}NEWLINE\]NEWLINE NEWLINE\[NEWLINE \text{ there are constants } c,\beta, d, \gamma >0 \text{ such that } d \left(\frac{t}{s}\right)^\gamma \leq \frac{v(it)}{v(is)}\leq c \left(\frac{t}{s}\right)^\beta \text{ for } 0<s\leq t.NEWLINE\]NEWLINE The proofs of the results are rather technical and, along the way, the authors show that the boundedness of \(D\) implies that the weight has to be essential, that is, for the associated weight \(\tilde{v}(z)=\inf\{\frac{1}{|h(z)|}: h\in H_v(\mathbb{G}),\, \|h\|_v\leq 1 \}\), there is a constant \(C>0\) such that \(C v(z)\geq \tilde{v}(z)\), \(z\in \mathbb{G}\).