Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the $p$-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains
206
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This is a significant improvement of the results of the first author and K. Wang, where the same result is shown to hold for polynomial-generated principal submodules of the Bergman module on the unit ball $mathbb{B}_n$ of $mathbb{C}^n$. As a consequence of our main result, we prove that the submodule of $L_a^2(mathbb{B}_n)$ consisting of functions vanishing on a pure analytic subsets of codimension $1$ is $p$-essentially normal for all $p>n$.
rate research
Read More
We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.
Let $mathcal{H}_d^{(t)}$ ($t geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $mathbb{B}_d$ with kernel [ k(z,w) = frac{1}{(1-langle z, w rangle)^{d+t+1}} . ] We prove that if an ideal $I triangleleft mathbb{C}[z_1, ldots, z_d]$ (not necessarily homogeneous) has what we call the approximate stable division property, then the closure of $I$ in $mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$.
Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones.
We obtain sharp ranges of $L^p$-boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating $L^p$-boundedness on a domain and its quotient by a finite group. The range of $p$ for which the Bergman projection is $L^p$-bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases.
Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of $n$-fold tensor products of Drury-Arveson spaces.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
