Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWeighted estimates of the Bergman projection with matrix weights
183
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of $P$: [|P|_{L^2(Omega,W)}leq C(mathcal B_2(W))^{{2}}.] Here $mathcal B_2(W)$ is the Bekolle-Bonami constant for the matrix weight $W$ and $C$ is a constant that is independent of the weight $W$ but depends upon the dimension and the domain.
rate research
Read More
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational power of the absolute value of a special defining function $rho$ of the domain: we prove (weighted) Sobolev-$L^{p}$ and Lipchitz estimates for domains in $mathbb{C}^{2}$ (or, more generally, for domains having a Levi form of rank $geq n-2$ and for decoupled domains) and for convex domains. In particular, for these defining functions, we generalize results obtained by A. Bonami & S. Grellier and D. C. Chang & B. Q. Li. We also obtain a general (weighted) Sobolev-$L^{2}$ estimate.
In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}left(Omega,dmu_{0}right)$ where $Omega$ is a smoothly bounded pseudoconvex domain of finite type in $mathbb{C}^{n}$ and $mu_{0}=left(-rho_{0}right)^{r}dlambda$, $lambda$ being the Lebesgue measure, $rinmathbb{Q}_{+}$ and $rho_{0}$ a special defining function of $Omega$, are still valid for the Bergman projection of $L^{2}left(Omega,dmuright)$ where $mu=left(-rhoright)^{r}dlambda$, $rho$ being any defining function of $Omega$. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations are obtained for weighted $L^{p}$-Sobolev and lipschitz estimates in the case of pseudoconvex domain of finite type in $mathbb{C}^{2}$ and for some convex domains of finite type.
A radial weight $omega$ belongs to the class $widehat{mathcal{D}}$ if there exists $C=C(omega)ge 1$ such that $int_r^1 omega(s),dsle Cint_{frac{1+r}{2}}^1omega(s),ds$ for all $0le r<1$. Write $omegaincheck{mathcal{D}}$ if there exist constants $K=K(omega)>1$ and $C=C(omega)>1$ such that $widehat{omega}(r)ge Cwidehat{omega}left(1-frac{1-r}{K}right)$ for all $0le r<1$. In a recent paper, we have recently prove that these classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights. Classical results by Hardy and Littlewood, and Shields and Williams, show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $omegainwidehat{mathcal{D}}setminus check{mathcal{D}}$ and $0<ple 1$. In this paper we establish sharp estimates for the norm of the analytic Bergman space $A^p_omega$, with $omegainwidehat{mathcal{D}}setminus check{mathcal{D}}$ and $0<p<infty$, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.
We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $L^1(mathbb{C})$ whose associated Bergman kernels have infinitely many zeroes is exhibited.
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
