Elena de la Rosa

Let $$\mathbb {D}$$D denote the unit disc in $$\mathbb {C}$$C . We define the generalized Cesàro operator as follows: $$\begin{aligned} C_{\omega }(f)(z)=\int _0^1 f(tz)\left( \frac{1}{z}\int _0^z B^{\omega }_t(u)\,\textrm{d}u\right) \,\omega (t)\textrm{d}t, \end{aligned}$$C ω ( f ) ( z ) = ∫ 0 1 f ( t z ) 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 study the action of the operator $$C_{\omega }$$C ω on weighted Hardy spaces of analytic functions $$\mathcal {H}_{\gamma }$$H γ , $$\gamma >0$$γ > 0 and on general weighted Bergman spaces $$A^{2}_{\mu }$$A μ 2 .

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Cesàro operator as follows $$ C_ω(f)(z)=\int_0^1 f(tz)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We study the action of the operator $C_ω$ on weighted Hardy spaces of analytic functions $\mathcal{H}_γ$, $γ>0$ and on general weighted Bergman spaces $A^2_μ$.

We establish new characterizations of the Bloch space $$\mathcal {B}$$B which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function $$f(z)=\sum _{n=0}^\infty \widehat{f}(n) z^n$$f ( z ) = ∑ n = 0 ∞ f ^ ( n ) z n in the unit disc $$\mathbb {D}$$D , we define the fractional derivative $$ D^{\mu }(f)(z)=\sum \limits _{n=0}^{\infty } \frac{\widehat{f}(n)}{\mu _{2n+1}} z^n $$D μ ( f ) ( z ) = ∑ n = 0 ∞ f ^ ( n ) μ 2 n + 1 z n induced by a radial weight $$\mu $$μ , where $$\mu _{2n+1}=\int _0^1 r^{2n+1}\mu (r)\,dr$$μ 2 n + 1 = ∫ 0 1 r 2 n + 1 μ ( r ) d r are the odd moments of $$\mu $$μ . Then, we consider the space $$ \mathcal {B}^\mu $$B μ of analytic functions f in $$\mathbb {D}$$D such that $$\Vert f\Vert _{\mathcal {B}^\mu }=\sup _{z\in \mathbb {D}} \widehat{\mu }(z)|D^\mu (f)(z)|<\infty $$‖ f ‖ B μ = sup z ∈ D μ ^ ( z ) | D μ ( f ) ( z ) | < ∞ , where $$\widehat{\mu }(z)=\int _{|z|}^1 \mu (s)\,ds$$μ ^ ( z ) = ∫ | z | 1 μ ( s ) d s . We prove that $$\mathcal {B}^\mu $$B μ is continously embedded in $$\mathcal {B}$$B for any radial weight $$\mu $$μ , and $$\mathcal {B}=\mathcal {B}^\mu $$B = B μ if and only if $$\mu \in \mathcal {D}=\widehat{\mathcal {D}}\cap \check{\mathcal {D}}$$μ ∈ D = D ^ ∩ D ˇ . A radial weight $$\mu \in \widehat{\mathcal {D}}$$μ ∈ D ^ if $$\sup _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( \frac{1+r}{2}\right) }<\infty $$sup 0 ≤ r < 1 μ ^ ( r ) μ ^ 1 + r 2 < ∞ and a radial weight $$\mu \in \check{\mathcal {D}}$$μ ∈ D ˇ if there exist $$K=K(\mu )>1$$K = K ( μ ) > 1 such that $$\inf _{0\le r1.$$inf 0 ≤ r < 1 μ ^ ( r ) μ ^ 1 - 1 - r K > 1 .

Let $$\omega $$ω and $$\nu $$ν be radial weights on the unit disc of the complex plane, and denote $$\sigma =\omega ^{p'} \nu ^{-\frac{p'}{p}}$$σ = ω p ′ ν - p ′ p and $$\omega _x =\int _0^1\,s^x \omega (s)\,ds$$ω x = ∫ 0 1 s x ω ( s ) d s for all $$1\le x<\infty $$1 ≤ x < ∞ . Consider the one-weight inequality $$\begin{aligned} \Vert P_\omega (f)\Vert _{L^p_\nu }\le C\Vert f\Vert _{L^p_\nu },\quad 1<p<\infty , \qquad \qquad (\dagger ) \end{aligned}$$‖ P ω ( f ) ‖ L ν p ≤ C ‖ f ‖ L ν p , 1 < p < ∞ , ( † ) for the Bergman projection $$P_\omega $$P ω induced by $$\omega $$ω . It is shown that the moment condition $$\begin{aligned} D_p(\omega ,\nu )=\sup _{n\in {\mathbb {N}}\cup \{0\}} \frac{\left( \nu _{np+1}\right) ^\frac{1}{p}\left( \sigma _{np'+1} \right) ^\frac{1}{p'}}{\omega _{2n+1}}<\infty \end{aligned}$$D p ( ω , ν ) = sup n ∈ N ∪ { 0 } ν n p + 1 1 p σ n p ′ + 1 1 p ′ ω 2 n + 1 < ∞ is necessary for ( $$\dagger $$† ) to hold. Further, $$D_p(\omega ,\nu )<\infty $$D p ( ω , ν ) < ∞ is also sufficient for ( $$\dagger $$† ) if $$\nu $$ν admits the doubling properties $$\sup _{0\le r<1}\frac{\int _r^1\nu (s) s\,ds}{\int _{\frac{1+r}{2}}^1\nu (s)s\,ds}<\infty $$sup 0 ≤ r < 1 ∫ r 1 ν ( s ) s d s ∫ 1 + r 2 1 ν ( s ) s d s < ∞ and $$\sup _{0\le r<1}\frac{\int _r^1\nu (s)s\,ds}{\int _r^{1-\frac{1-r}{K}} \nu (s)s\,ds}<\infty $$sup 0 ≤ r < 1 ∫ r 1 ν ( s ) s d s ∫ r 1 - 1 - r K ν ( s ) s d s < ∞ for some $$K>1$$K > 1 . In addition, an analogous result for the one weight inequality $$\Vert P_\omega (f)\Vert _{D^p_{\nu ,k}} \le C\Vert f\Vert _{L^p_\nu }$$‖ P ω ( f ) ‖ D ν , k p ≤ C ‖ f ‖ L ν p , where $$\begin{aligned} \Vert f \Vert _{D^p_{\nu , k}}^p =\sum \limits _{j=0}^{k-1}\vert f^{(j)}(0)\vert ^p +\int _{{\mathbb {D}}} \vert f^{(k)}(z)\vert ^p (1-\vert z \vert )^{kp} \nu (z)\,dA(z)<\infty , \quad k\in {\mathbb {N}}, \end{aligned}$$‖ f ‖ D ν , k p p = ∑ j = 0 k - 1 | f ( j ) ( 0 ) | p + ∫ D | f ( ...

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 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.

We establish new characterizations of the Bloch space $\mathcal{B}$ which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty \widehat{f}(n) z^n$ in the unit disc $\mathbb{D}$, we define the fractional derivative $ D^μ(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}} z^n $ induced by a radial weight $μ$, where $μ_{2n+1}=\int_0^1 r^{2n+1}μ(r)\,dr$ are the odd moments of $μ$. Then, we consider the space $ \mathcal{B}^μ$ of analytic functions $f$ in $\mathbb{D}$ such that $\|f\|_{\mathcal{B}^μ}=\sup_{z\in \mathbb{D}} \widehatμ(z)|D^μ(f)(z)|<\infty$, where $\widehatμ(z)=\int_{|z|}^1 μ(s)\,ds$. We prove that $\mathcal{B}^μ$ is continously embedded in $\mathcal{B}$ for any radial weight $μ$, and $\mathcal{B}=\mathcal{B}^μ$ if and only if $μ\in \mathcal{D}=\widehat{\mathcal{D}}\cap\check{\mathcal{D}}$. A radial weight $μ\in \widehat{\mathcal{D}}$ if $\sup_{0\le r <1}\frac{\widehatμ(r)}{\widehatμ\left(\frac{1+r}{2}\right)}<\infty$ and a radial weight $μ\in \check{\mathcal{D}}$ if there exist $K=K(μ)>1$ such that $\inf_{0\le r1.$

Let $ω$ and $ν$ be radial weights on the unit disc of the complex plane, and denote $σ=ω^{p'}ν^{-\frac{p'}p}$ and $ω_x =\int_0^1 s^x ω(s)\,ds$ for all $1\le x<\infty$. Consider the one-weight inequality \begin{equation}\label{ab1} \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad 1<p<\infty,\tag† \end{equation} for the Bergman projection $P_ω$ induced by $ω$. It is shown that the moment condition $$ D_p(ω,ν)=\sup_{n\in \mathbb{N}\cup\{0\}}\frac{\left(ν_{np+1}\right)^\frac1p\left(σ_{np'+1}\right)^\frac1{p'}}{ω_{2n+1}}<\infty $$ is necessary for \eqref{ab1} to hold. Further, $D_p(ω,ν)<\infty$ is also sufficient for \eqref{ab1} if $ν$ admits the doubling properties $\sup_{0\le r<1}\frac{\int_r^1 ω(s)s\,ds}{\int_{\frac{1+r}{2}}^1 ω(s)s\,ds}<\infty$ and $\sup_{0\le r<1}\frac{\int_r^1 ω(s)s\,ds}{\int_r^{1-\frac{1-r}{K}} ω(s)s\,ds}<\infty$ for some $K>1$. In addition, an analogous result for the one weight inequality $ \|P_ω(f)\|_{D^p_{ν,k}}\le C\|f\|_{L^p_ν}, $ where $$ \Vert f \Vert_{D^p_{ν, k}}^p =\sum\limits_{j=0}^{k-1}| f^{(j)}(0)|^p+ \int_{\mathbb{D}} \vert f^{(k)}(z)\vert^p (1-|z| )^{kp}ν(z)\,dA(z)<\infty, \quad k\in \mathbb{N}, $$ is established. The inequality \eqref{ab1} is further studied by using the necessary condition $D_p(ω,ν)<\infty$ in the case of the exponential type weights $ν(r)=\exp \left(-\fracα{(1-r^l)^β} \right)$ and $ω(r)= \exp \left(-\frac{\widetildeα}{(1-r^{\widetilde{l}})^{\widetildeβ}} \right)$, where $0<α, \, \widetildeα, \, l, \, \widetilde{l}<\infty$ and $0<β, \, \widetildeβ\le 1$.

We consider the Hilbert-type operator defined by $$ H_{\omega}(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$ where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the unit disc $\mathbb{D}$. We prove that $H_{\omega}$ is bounded on the Hardy space $H^p$, $1<p<\infty$, if and only if \begin{equation} \label{abs1} \sup_{0\le r<1} \frac{\widehat{\omega}(r)}{\widehat{\omega}\left( \frac{1+r}{2}\right)}<\infty, \tag{\dag} \end{equation} and \begin{equation*} \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{equation*} where $\widehat{\omega}(r)=\int_r^1 \omega(s)\,ds$. We also prove that $H_\omega: H^1\to H^1$ is bounded if and only if \eqref{abs1} holds and $$ \sup\limits_{r \in [0,1)} \frac{\widehat{\omega}(r)}{1-r} \left(\int_0^r \frac{ds}{\widehat{\omega}(s)}\right)<\infty.$$ As for the case $p=\infty$, $H_\omega$ is bounded from $H^\infty$ to $BMOA$, or to the Bloch space, if and only if \eqref{abs1} holds. In addition, we prove that there does not exist radial weights $\omega$ such that $H_{\omega}: H^p \to H^p $, $1\le p<\infty$, is compact and we consider the action of $H_{\omega}$ on some spaces of analytic functions closely related to Hardy spaces.