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Haar Shifts, Commutators, and Hankel Operators

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 Added by Michael T. Lacey
 Publication date 2008
  fields
and research's language is English




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Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. An decomposition of the commutator [H,b] into paraproducts is presented.



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For $mathbb B^n$ the unit ball of $mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^Phi_alpha$, which are generalizations of classical Bergman spaces. We characterize the dual space of large Bergman-Orlicz space, and bounded Hankel operators between some Bergman-Orlicz spaces $A_alpha^{Phi_1}(mathbb B^n)$ and $A_alpha^{Phi_2}(mathbb B^n)$ where $Phi_1$ and $Phi_2$ are either convex or concave growth functions.
In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{omega _varphi}$, where $omega _varphi= e^{-varphi}$ and $varphi$ is a subharmonic function. We consider compact Hankel operators $H_{overline {phi}}$, with anti-analytic symbols ${overline {phi}}$, and give estimates of the trace of $h(|H_{overline phi}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{overline {phi}}))_n$ in terms of decreasing rearrangement of $|phi |/sqrt{Delta varphi}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{overline {phi}}))_n$ is achieved by $(s_n (H_{overline{z}}))_n$. Namely, we establish that if $s_n(H_{overline {phi}})= o (s_n(H_{overline {z}}))$, then $H_{overline {phi}} = 0$. Then, we show that if $Delta varphi (z) asymp frac{1}{(1-|z|^2)^{2+beta}}$ with $beta geq 0$, then $s_n(H_{overline {phi}}) = O(s_n(H_{overline {z}}))$ if and only if $phi $ belongs to the Hardy space $H^p$, where $p= frac{2(1+beta)}{2+beta}$. Finally, we compute the asymptotics of $s_n(H_{overline {phi}})$ whenever $ phi in H^{p }$.
Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{Omega}$ be the convolution singular integral operator with kernel $frac{Omega(x)}{|x|^n}$. For $bin{rm BMO}(mathbb{R}^n)$, let $T_{Omega,,b}$ be the commutator of $T_{Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $Llog L$ type for $T_{Omega,,b}$ when $Omegain L^q(S^{n-1})$ for some $qin (1,,infty]$.
92 - Guoen Hu , Xiangxing Tao 2020
Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_{Omega}$ be the homogeneous singular integral operator with kernel $frac{Omega(x)}{|x|^d}$ and $T_{Omega,,b}$ be the commutator of $T_{Omega}$ with symbol $b$. In this paper, we prove that if $Omegain L(log L)^2(S^{d-1})$, then for $bin {rm BMO}(mathbb{R}^d)$, $T_{Omega,,b}$ satisfies an endpoint estimate of $Llog L$ type.
We characterize bounded Toeplitz and Hankel operators from weighted Bergman spaces to weighted Besov spaces in tube domains over symmetric cones. We deduce weak factorization results for some Bergman spaces of this setting.
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