Words of analytic paraproducts on Bergman spaces
Carmen Cascante, Joan Fábrega, Daniel Pascuas, José Ángel Peláez
TL;DR
This work delivers a complete quantitative characterization of bounded N-letter g-words on weighted Bergman spaces A^p_{ω_p} with smooth rapidly decreasing radial weights ω=e^{-2φ}. It shows a sharp dichotomy: if no T_g appears (n=0), boundedness is equivalent to g∈H^∞ with ∥L_g∥ ≍ ∥g∥_{H^∞}^N; if at least one T_g appears (n≥1), boundedness is equivalent to g belonging to the fractional Bloch-type class 𝔅^s_φ with s=N/n, and ∥L_g∥ ≍ ∥g∥^{N}_{𝔅^s_φ}. The authors develop φ-Bloch spaces, establish radicality, and introduce intermediate operators Q_g^{σ,ℓ} to obtain sharp upper and lower norm estimates, aided by Bergman kernel bounds and dilations g_r. The results extend to a rich family of weights SW, including weights that decrease to zero faster than any polynomial, and provide structural decompositions of g-words that underpin the boundedness criteria. This yields precise criteria for complex operator compositions in analytic function spaces, with implications for cancellation phenomena in paraproducts on weighted Bergman spaces.
Abstract
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 $|z|\to 1^-$.