No Arabic abstract
Volterra integral operators ${cal A}=sum_{k=0}^m {cal A}_k$, $({cal A}_k f)(x)= a_k (x)int_0^x t^k f(t) ,dt$, are studied acting between weighted $L_2$ spaces on $(0,+infty)$. Under certain conditions on the weights and functions $a_k$, it is shown that $cal A$ is bounded if and only if each ${cal A}_k$ is bounded. This result is then applied to describe spaces of pointwise multipliers in weighted Sobolev spaces on $(0,+infty)$.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<infty$ of the Hankel operators $H_ f$ and $H_{bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
Let $mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $mathbb{D}$. We characterize bounded and compact Volterra type integration operators [ J_{g}(f)(z)=int_{0}^{z}f(lambda)g(lambda)dlambda ] between weighted Bergman spaces $L_{a}^{p}(omega )$ induced by $mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<infty$.
A full description of the membership in the Schatten ideal $S_ p(A^2_{omega})$ for $0<p<infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,eta)$ are elements of $C^{r}_{*}S^{m}_{1,delta}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $mathcal{H}^{s,p}_{FIO}(mathbb{R}^{n})$ and $mathcal{H}^{t,p}_{FIO}(mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$. Our main result implies that for $m=0$, $delta=1/2$ and $r>n-1$, $a(x,D)$ acts boundedly on $mathcal{H}^{p}_{FIO}(mathbb{R}^{n})$ for all $pin(1,infty)$.