Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn regularity and irregularity of certain holomorphic singular integral operators
127
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We survey recent work and announce new results concerning two singular integral operators whose kernels are holomorphic functions of the output variable, specifically the Cauchy-Leray integral and the Cauchy-SzegH o projection associated to various classes of bounded domains in $mathbb C^n$ with $ngeq 2$.
rate research
Read More
We show that for an entire function $varphi$ belonging to the Fock space ${mathscr F}^2(mathbb{C}^n)$ on the complex Euclidean space $mathbb{C}^n$, the integral operator begin{eqnarray*} S_{varphi}F(z)=int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z- bar{w}),dlambda(w), zin mathbb{C}^n, end{eqnarray*} is bounded on ${mathscr F}^2(mathbb{C}^n)$ if and only if there exists a function $min L^{infty}(mathbb{R}^n)$ such that $$ varphi(z)=int_{mathbb{R}^n} m(x)e^{-2left(x-frac{i}{2} z right)cdot left(x-frac{i}{2} z right)} dx, zin mathbb{C}^n. $$ Here $dlambda(w)= pi^{-n}e^{-leftvert wrightvert^2}dw$ is the Gaussian measure on $mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_varphi$. Moreover, we obtain the reducing subspaces of $S_{varphi}$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${mathscr F}^2(mathbb{C})$ on the complex plane ${mathbb C}$ (Integr. Equ. Oper. Theory {bf 81} (2015), 451--454).
Let $f:{mathbb B}^n to {mathbb B}^N$ be a holomorphic map. We study subgroups $Gamma_f subseteq {rm Aut}({mathbb B}^n)$ and $T_f subseteq {rm Aut}({mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When $Gamma_f$ contains the center of ${bf U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $Phi:Gamma_f to T_f$ such that $f$ is equivariant with respect to $Phi$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of ${rm Ker}(Phi)$ to older results on invariant proper maps between balls. When $f$ is proper but completely non-rational, we show that either both $Gamma_f$ and $T_f$ are finite or both are noncompact.
In this note, we study the boundedness of integral operators $I_{g}$ and $T_{g}$ on analytic Morrey spaces. Furthermore, the norm and essential norm of those operators are given.
This article develops a novel approach to the representation of singular integral operators of Calderon-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderon-Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of $T(1)$ theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the $A_2$ theorem; that is, sharp dependence of the Sobolev norm of $T$ on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative $A_p$ estimates which are best known, and sharp in the range $max{p,p}geq 3$ for the fully cancellative case.
We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 le p < infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $ell^2$ when $p e 2.$
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
