Jouni Rättyä

For $$0<p<\infty $$ 0 < p < ∞ , $$\Psi :[0,\infty )\rightarrow (0,\infty )$$ Ψ : [ 0 , ∞ ) → ( 0 , ∞ ) and a finite positive Borel measure $$\mu $$ μ on the unit disc $$\mathbb {D}$$ D , the Lebesgue–Zygmund space $$L^p_{\mu ,\Psi }$$ L μ , Ψ p consists of all measurable functions f such that $$\Vert f \Vert _{L_{\mu , \Psi }^{p}}^p =\int _{\mathbb {D}}|f|^p\Psi (|f|)\,d\mu < \infty $$ ‖ f ‖ L μ , Ψ p p = ∫ D | f | p Ψ ( | f | ) d μ < ∞ . For an integrable radial function $$\omega $$ ω on $$\mathbb {D}$$ D , the corresponding weighted Bergman-Zygmund space $$A_{\omega , \Psi }^{p}$$ A ω , Ψ p is the set of all analytic functions in $$L_{\mu , \Psi }^{p}$$ L μ , Ψ p with $$d\mu =\omega \,dA$$ d μ = ω d A . The purpose of the paper is to characterize bounded (and compact) embeddings $$A_{\omega ,\Psi }^{p}\subset L_{\mu , \Phi }^{q}$$ A ω , Ψ p ⊂ L μ , Φ q , when $$0<p\le q<\infty $$ 0 < p ≤ q < ∞ , the functions $$\Psi $$ Ψ and $$\Phi $$ Φ are essentially monotonic, and $$\Psi ,\Phi ,\omega $$ Ψ , Φ , ω satisfy certain doubling properties. The tools developed on the way to the main results are applied to characterize bounded and compact integral operators acting from $$A^p_{\omega ,\Psi }$$ A ω , Ψ p to $$A^q_{\nu ,\Phi }$$ A ν , Φ q , provided $$\nu $$ ν admits the same doubling property as $$\omega $$ ω .

Let $ω$ be a radial weight, $0<p,q<\infty$ and $Γ(ξ)=\left\{z\in\mathbb{D}:|\arg z-\argξ|<(|ξ|-|z|)\right\}$ for $ξ\in\overline{\mathbb{D}}$ . The average radial integrability space $L^q_p(ω)$ consists of complex-valued measurable functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|^q_{L^q_p(ω)}=\frac{1}{2π}\int_{0}^{2π}\left(\int_{0}^{1}|f(re^{iθ})|^pω(r)r\,dr\right)^{\frac{q}{p}}dθ<\infty,$$ and the tent space $T^q_p(ω)$ is the set of those $f$ for which $$\|f\|^q_{T_{p}^{q}(ω)}=\frac{1}{2π}\int_{\partial{\mathbb{D}}}\left(\int_{Γ(ξ)}|f(z)|^pω(z)\frac{dA(z)}{1-|z|}\right)^{\frac{q}{p}}\,|dξ|<\infty.$$ Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in $\mathbb{D}$. It is shown that the non-tangential maximal operator $$f\mapsto N(f)(ξ)=\sup_{z\inΓ(ξ)}|f(z)|,\quad ξ\in \mathbb{D},$$ is bounded from $AL^q_p(ω)=L^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ and $AT^q_p(ω)=T^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ to $L^q_p(ω)$ and $T^q_p(ω)$, respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in $AL^q_p(ω)$ and $AT^q_p(ω)$, and the identity $AL^q_p(ω)=AT^q_p(ω)$ for weights admitting a one-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection $P_γ$, induced by the standard weight $(γ+1)(1-|z|^2)^γ$, on $L^q_p(ω)$ and $T^q_p(ω)$ with $1<q,p<\infty$ is independent of $q$, and is described by a Bekollé-Bonami type condition.

The boundedness of the small Hankel operator $h^ω_{f}(g)=\overline{P_ω}(fg)$ induced by a measurable symbol $f$ and the Bergman projection $P_ω$ associated to a radial weight $ω$ acting from the weighted Bergman space $A^p_ω$ to its conjugate analytic counterpart $\overline{A^p_ω}$ is characterized on the range $1<p<\infty$ when $ω$ belongs to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. On the way to the proof a sharp integral estimate for certain modified Bergman kernels is obtained.

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 ( ...

Let $\omega$ and $\nu$ be radial weights on the unit disc of the complex plane such that $\omega$ admits the doubling property $\sup _{0\le r<1}\frac {\int _r^1 \omega (s)\,ds}{\int _{\frac {1+r}{2}}^1 \omega (s)\,ds}<\infty$. Consider the one weight inequality \begin{equation} \|P_\omega (f)\|_{L^p_\nu }\le C\|f\|_{L^p_\nu },\quad 1<p<\infty ,\tag {\dag } \end{equation} for the Bergman projection $P_\omega$ induced by $\omega$. It is shown that the Muckenhoupt-type condition \begin{equation*} A_p(\omega ,\nu )=\sup _{0\le r<1}\frac {\left (\int _r^1 s\nu (s)\,ds \right )^{\frac {1}{p}}\left (\int _r^1 s\left (\frac {\omega (s)}{\nu (s)^{\frac 1p}}\right )^{p’}\,ds \right )^{\frac {1}{p’}}}{\int _r^1 s\omega (s)\,ds}<\infty \end{equation*} is necessary for ($\dag$) to hold, and sufficient if $\nu$ is of the form \begin{align*} \nu (s)=\omega (s)\left (\int _r^1 s\omega (s)\,ds \right )^\alpha \end{align*} for some $-1<\alpha <\infty$. This result extends the classical theorem due to Forelli and Rudin for a much larger class of weights. In addition, it is shown that for any pair $(\omega ,\nu )$ of radial weights the Calderón operator \begin{equation*} H^\star _\omega (f)(z)+H_\omega (f)(z) =\int _{0}^{|z|} f\left (s\frac {z}{|z|}\right )\frac {s\omega (s)\,ds}{\int _s^1 t\omega (t)\,dt} +\frac {\int _{|z|}^1f\left (s\frac {z}{|z|}\right ) s\omega (s)\,ds}{\int _{|z|}^1 s\omega (s)\,ds} \end{equation*} is bounded on $L^p_\nu$ if and only if $A_p(\omega ,\nu )<\infty$.

The Bergman projection $$P_\alpha $$P α , induced by a standard radial weight, is bounded and onto from $$L^\infty $$L ∞ to the Bloch space $$\mathcal {B}$$B . However, $$P_\alpha : L^\infty \rightarrow \mathcal {B}$$P α : L ∞ → B is not a projection. This fact can be emended via the boundedness of the operator $$P_\alpha :\textrm{BMO}_2(\Delta )\rightarrow \mathcal {B}$$P α : BMO 2 ( Δ ) → B , where $$\textrm{BMO}_2(\Delta )$$BMO 2 ( Δ ) is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection $$P_\omega $$P ω and the space $$\textrm{BMO}_{\omega ,p}(\Delta )$$BMO ω , p ( Δ ) of functions of bounded mean oscillation induced by $$1<p<\infty $$1 < p < ∞ and a radial weight $$\omega \in \mathcal {M}$$ω ∈ M . Here $$\mathcal {M}$$M is a wide class of radial weights defined by means of moments of the weight, and it contains the standard and the exponential-type weights. We describe the weights such that $$P_\omega :\textrm{BMO}_{\omega ,p}(\Delta )\rightarrow \mathcal {B}$$P ω : BMO ω , p ( Δ ) → B is bounded. They coincide with the weights for which $$P_\omega : L^\infty \rightarrow \mathcal {B}$$P ω : L ∞ → B is bounded and onto. This result seems to be new even for the standard radial weights when $$p\ne 2$$p ≠ 2 .

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 $0<β, \, \widetildeβ\le 1$.

The problem of describing the analytic functions g on the unit disc such that the integral operator $$T_g(f)(z)=\int _0^zf(\zeta )g'(\zeta )\,d\zeta $$T g ( f ) ( z ) = ∫ 0 z f ( ζ ) g ′ ( ζ ) d ζ is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space $$H^\infty $$H ∞ is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of $$T_g$$T g acting from a weighted Dirichlet space $$D^p_\omega $$D ω p , induced by an upper doubling weight $$\omega $$ω , to $$H^\infty $$H ∞ . We also characterize, in terms of neat conditions on $$\omega $$ω , the upper doubling weights for which $$T_g: D^p_\omega \rightarrow H^\infty $$T g : D ω p → H ∞ is bounded (or compact) only if g is constant.