Hilbert-type operator induced by radial weight on Hardy spaces

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