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Register a new userBoundedness of operators generated by fractional semigroups associated with Schrodinger operators on Campanato type spaces via $T1$ theorem
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Let $mathcal{L}=-Delta+V$ be a Schr{o}dinger operator, where the nonnegative potential $V$ belongs to the reverse H{o}lder class $B_{q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, ${e^{-tmathcal{L}^{alpha}}}_{t>0},$ associated with $mathcal{L}$. As an application, we obtain the $BMO^{gamma}_{mathcal{L}}$-boundedness of the maximal function, and the Littlewood-Paley $g$-functions associated with $mathcal{L}$ via $T1$ theorem, respectively.
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In this paper we introduce a class of generalized Morrey spaces associated with Schrodinger operator $L=-Delta+V$. Via a pointwise estimate, we obtain the boundedness of the operators $V^{beta_{2}}(-Delta+V)^{-beta_{1}}$ and their dual operators on these Morrey spaces.
Schrodinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically verifiable sufficient condition for their spectrum to be singular continuous and supported on a Cantor set of zero Lebesgue measure. Applications to specific examples are discussed.
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.
We provide a boundedness criterion for the integral operator $S_{varphi}$ on the fractional Fock-Sobolev space $F^{s,2}(mathbb C^n)$, $sgeq 0$, where $S_{varphi}$ (introduced by Kehe Zhu) is given by begin{eqnarray*} S_{varphi}F(z):= int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z- bar{w}) dlambda(w) end{eqnarray*} with $varphi$ in the Fock space $F^2(mathbb C^n)$ and $dlambda(w): = pi^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(mathbb R^n)$.
We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_alpha(mathbb{B}_ n)$ to the Lebesgue spaces $L^q(mathbb{S}_ n)$ for all $0<p,q<infty$. For the case $n=1$, some partial results were previously obtained by Wu. Especially, in the case $q<p$ and $q<s$, we obtain the new characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.
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