José Ángel Peláez

For $0<p,q<\infty$ and $ω$ a radial weight, the space $L^{p,q}_ω$ consists of complex-valued measurable functions $f$ on the unit disk such that $$ \| f\|_{L^{p,q}_ω}^q = \int_0^1 \left (\frac{1}{2π}\int_0^{2π}|f(re^{iθ})|^pdθ\right )^{\frac{q}{p}}rω(r)\,dr, $$ and the mixed norm space $A^{p,q}_ω$ is the subset of $L^{p,q}_ω$ consisting of analytic functions. We say that a radial weight $ω$ belongs to $\widehat{\mathcal{D}}$ if there exists $C=C(ω)>0$ such that $$\int_r^1ω(s)ds \leq C \int_{\frac{1+r}{2}}^1ω(s)\,ds \,\, \text{for every}\,\, 0\leq r <1.$$ We describe the dual space of $A^{p,q}_ω$ for every $0<p,q<\infty$ and $ω\in\widehat{\mathcal{D}}$. Later on, we apply the obtained description of the dual space of $A^{p,q}_ω$ to prove that the Bergman projection induced by $ω$, $P_ω$, is bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and $ω\in \widehat{\mathcal{D}}$. Besides, we also prove that $P_ω$ and the corresponding maximal Bergman projection $P_ω^+$ are not simultaneously bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and $ω\in \widehat{\mathcal{D}}$.

Given a radial doubling weight $μ$ on the unit disc $\mathbb{D}$ of the complex plane and its odd moments $μ_{2n+1}=\int_0^1 s^{2n+1}μ(s)\, ds$, we consider the fractional derivative $$ D^μ(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}}z^n, $$ of a function $ f(z)=\sum_{n=0}^{\infty}\widehat{f}(n)z^n$ analytic in $\mathbb{D}$. We also consider the fractional integral operator $I^μ(f)(z)=\sum_{n=0}^{\infty} μ_{2n+1}\widehat{f}(n)z^n$, and the fractional Volterra-type operator $$ V_{μ,g}(f)(z)= I^μ(f\cdot D^μ(g))(z),\quad f\in\mathcal{H}(\mathbb{D}), $$ for any fixed $g\in\mathcal{H}(\mathbb{D})$. We prove that $V_{μ,g}$ is bounded (compact) on a Hardy space $H^p$, $0<p<\infty$, if and only if $g$ belongs to $\text{BMOA}$ ($\text{VMOA}$). Moreover, if $\int_0^1 \frac{\left(\int_r^1 μ(s)\, ds\right)^p}{(1-r)^2}\,dr=+\infty$, we prove that $V_{μ,g}$ belongs to the Schatten class $S_p(H^2)$ if and only if $g=0$. On the other hand, if $\frac{\left(\int_r^1 μ(s)\, ds\right)^p}{(1-r)^2}$ is a radial doubling weight it is proved that $V_{μ,g} \in S_p(H^2)$ if and only if $g$ belongs to the Besov space $B_p$. En route, we obtain descriptions of $H^p$, $\text{BMOA}$, $\text{VMOA}$ and $B_p$ in terms of the fractional derivative $D^μ$.

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by $T_gf(z)=\int_0^zf(ζ)g'(ζ)dζ$, $S_gf(z)=\int_0^zf'(ζ)g(ζ)dζ$, and $M_gf(z)=g(z)f(z)$. An $N$-letter $g$-word is an operator of the form $L=L_1\cdots L_N$, where each $L_j$ is either $M_g$, $S_g$ or $T_g$. It has been recently proved, in a recent paper by A. Aleman and the authors of this paper, that understanding the boundedness of a $g$-word on classical Hardy and Bergman spaces is a challenging problem due to the potential cancellations involved. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary $g$-word on a weighted Bergman space $A^p_{ω^{p/2}}$, where $ω=e^{-2\varphi}$ is a smooth rapidly decreasing weight. In particular, it states that any $N$-letter $g$-word such that $\#\{j:L_j=T_g\}=n\ge 1$ is bounded on $A^p_{ω^{p/2}}$ if and only if $g$ satisfies the "fractional" Bloch-type condition \[ \|g\|_{\mathcal{B}^s_\varphi}^s= \sup_{z\in\mathbb{D}}\frac{s|g(z)|^{s-1}|g'(z)|}{1+\varphi'(|z|)} <\infty, \] where $s=\frac{N}{n}$, and $\|L\|_{A^p_{ω^{p/2}}}\simeq \|g\|_{\mathcal{B}^s_\varphi}^N$. The class of smooth rapidly decreasing weights contains the radial weights \begin{equation*} ω_n(z)=e^{-2\exp_{n}(g_{α,c}(|z|))}, \quad\mbox{where}\quad g_{α,c}(r)=\tfrac{c}{(1-r^2)^α}, \quad\mbox{for $c,α>0$,} \end{equation*} $\exp_0(x)=x$ and $\exp_n(x)=e^{\exp_{n-1}(x)}$, for $n\in\mathbb{N}$. Therefore it contains weights which decrease arbitrarily rapidly to zero as $|z|\to 1^-$.

Let $$\omega $$ ω be a radial weight on the unit disc of the complex plane $$\mathbb {D}$$ D and denote by $$\widehat{\omega }(r)=\int _r^1 \omega (s)\,ds$$ ω ^ ( r ) = ∫ r 1 ω ( s ) d s the tail integrals. A radial weight $$\omega $$ ω belongs to the class $$\widehat{\mathcal {D}}$$ D ^ if satisfies the upper doubling condition $$\begin{aligned} \sup _{0<r<1}\frac{\widehat{\omega }(r)}{\widehat{\omega }\left( \frac{1+r}{2}\right) }<\infty . \end{aligned}$$ sup 0 < r < 1 ω ^ ( r ) ω ^ 1 + r 2 < ∞ . If $$\nu $$ ν or $$\omega $$ ω belongs to $$\widehat{\mathcal {D}}$$ D ^ , we describe the boundedness of the Bergman projection $$P_\omega $$ P ω induced by $$\omega $$ ω on the growth space $$L^\infty _{\widehat{\nu }} =\{ f: \Vert f\Vert _{\infty ,v}=\text {ess sup}_{z\in \mathbb {D}} |f(z)|\widehat{\nu }(z)<\infty \}$$ L ν ^ ∞ = { f : ‖ f ‖ ∞ , v = ess sup z ∈ D | f ( z ) | ν ^ ( z ) < ∞ } in terms of neat conditions on the moments and/or the tail integrals of $$\omega $$ ω and $$\nu $$ ν . Moreover, we solve the analogous problem for $$P_\omega $$ P ω from $$L^\infty _{\widehat{\nu }}$$ L ν ^ ∞ to the Bloch type space $$\mathcal {B}^\infty _{\widehat{\nu }} = \{f\, \text {analytic in} \mathbb {D}: \Vert f\Vert _{\mathcal {B}^\infty _{\widehat{\nu }}} $$ B ν ^ ∞ = { f analytic in D : ‖ f ‖ B ν ^ ∞ $$= \sup _{z\in \mathbb {D}}(1-|z|)\widehat{\nu }(z)|f'(z)|<\infty \}.$$ = sup z ∈ D ( 1 - | z | ) ν ^ ( z ) | f ′ ( z ) | < ∞ } . Similar questions for exponentially decreasing radial weights will also be studied.

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.