Hilbert-type operator induced by radial weight on Hardy spaces

Abstract

AbstractWe consider the Hilbert-type operator defined by $$\begin{aligned} H_{\omega }(f)(z)=\int _0^1 f(t)\left( \frac{1}{z}\int _0^z B^{\omega }_t(u)\,du\right) \,\omega (t)dt, \end{aligned}$$ H ω ( f ) ( z ) = ∫ 0 1 f ( t ) 1 z ∫ 0 z B t ω ( u ) d u ω ( t ) d t , where $$\{B^{\omega }_\zeta \}_{\zeta \in \mathbb {D}}$$ { B ζ ω } ζ ∈ D are the reproducing kernels of the Bergman space $$A^2_\omega $$ A ω 2 induced by a radial weight $$\omega $$ ω in the unit disc $$\mathbb {D}$$ D . We prove that $$H_{\omega }$$ H ω is bounded on the Hardy space $$H^p$$ H p , $$1<p<\infty $$ 1 < p < ∞ , if and only if $$\begin{aligned} \sup _{0\le r<1} \frac{\widehat{\omega }(r)}{\widehat{\omega }\left( \frac{1+r}{2}\right) }<\infty , (\dag )\end{aligned}$$ sup 0 ≤ r < 1 ω ^ ( r ) ω ^ 1 + r 2 < ∞ , ( † ) and $$\begin{aligned} \sup \limits _{0<r<1}\left( \int _0^r \frac{1}{\widehat{\omega }(t)^p} dt\right) ^{\frac{1}{p}} \left( \int _r^1 \left( \frac{\widehat{\omega }(t)}{1-t}\right) ^{p'}\,dt\right) ^{\frac{1}{p'}} <\infty , \end{aligned}$$ sup 0 < r < 1 ∫ 0 r 1 ω ^ ( t ) p d t 1 p ∫ r 1 ω ^ ( t ) 1 - t p ′ d t 1 p ′ < ∞ , where $$\widehat{\omega }(r)=\int _r^1 \omega (s)\,ds$$ ω ^ ( r ) = ∫ r 1 ω ( s ) d s . We also prove that $$H_\omega : H^1\rightarrow H^1$$ H ω : H 1 → H 1 is bounded if and only if ($$\dag $$ † ) holds and $$\begin{aligned} \sup \limits _{r \in [0,1)} \frac{\widehat{\omega }(r)}{1-r} \left( \int _0^r \frac{ds}{\widehat{\omega }(s)}\right) <\infty . \end{aligned}$$ sup r ∈ [ 0 , 1 ) ω ^ ( r ) 1 - r ∫ 0 r ds ω ^ ( s ) < ∞ . As for the case $$p=\infty $$ p = ∞ , $$H_\omega $$ H ω is bounded from $$H^\infty $$ H ∞ to $$\mathord \textrm{BMOA}$$ BMOA , or to the Bloch space, if and only if ($$\dag $$ † ) holds. In addition, we prove that there does not exist radial weights $$\omega $$ ω such that $$H_{\omega }: H^p \rightarrow H^p $$ H ω : H p → H p , $$1\le p<\infty $$ 1 ≤ p < ∞ , is compact and we consider the action of $$H_{\omega }$$ H ω on some spaces of analytic functions closely related to Hardy spaces.