Toeplitz operators on Bergman spaces of polygonal domains

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

Whitney Decomposition, Embedding Theorems and Interpolation Questions in Weighted Spaces of Analytic Functions

По классической теореме Уитни каждое открытое множество на плоскости можно представить в виде объединения специальных квадратов, внутренности которых не пересекаются. В статье, используя эти свойства квадратов Уитни, вводится новое понятие: для каждого центра $a_k$ квадрата Уитни существует точка $a_k^*\in C/G$ такая, что расстояние до границы открытого множества $G$ заключается между двумя константами независимо от $k$. Используя свойства Уитни в~статье, в частности, устанавливается необходимое и достаточное условие на ${z_k }_1^{\infty}\subset G$, при котором оператор $R(f)=(f(z_1),f(z_2),\ldots,f(z_n),\ldots)$ отображает обобщенные плоские классы Неванлинны по множеству $G$ в $l^p.$

ON THE VARIATIONAL CONSTANT ASSOCIATED TO THE -HARDY INEQUALITY

Let$\unicode[STIX]{x1D6FA}$be a domain in$\mathbb{R}^{m}$with nonempty boundary. In Ward [‘On essential self-adjointness, confining potentials and the$L_{p}$-Hardy inequality’, PhD Thesis, NZIAS Massey University, New Zealand, 2014] and [‘The essential self-adjointness of Schrödinger operators on domains with non-empty boundary’,Manuscripta Math.150(3) (2016), 357–370] it was shown that the Schrödinger operator$H=-\unicode[STIX]{x1D6E5}+V$, with domain of definition$D(H)=C_{0}^{\infty }(\unicode[STIX]{x1D6FA})$and$V\in L_{\infty }^{\text{loc}}(\unicode[STIX]{x1D6FA})$, is essentially self-adjoint provided that$V(x)\geq (1-\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA}))/d(x)^{2}$. Here$d(x)$is the Euclidean distance to the boundary and$\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$is the nonnegative constant associated to the$L_{2}$-Hardy inequality. The conditions required for a domain to admit an$L_{2}$-Hardy inequality are well known and depend intimately on the Hausdorff or Aikawa/Assouad dimension of the boundary. However, there are only a handful of domains where the value of$\unicode[STIX]{x1D707}_{2}(\unicode[STIX]{x1D6FA})$is known explicitly. By obtaining upper and lower bounds on the number of cubes appearing in the$k\text{th}$generation of the Whitney decomposition of$\unicode[STIX]{x1D6FA}$, we derive an upper bound on$\unicode[STIX]{x1D707}_{p}(\unicode[STIX]{x1D6FA})$, for$p>1$, in terms of the inner Minkowski dimension of the boundary.

On minimal thinness, reduced functions and Green potentials

Let Ω be an open connected subset of the unit disc U, let E = U\Ω and let {Ωk} be a Whitney decomposition of U. If z(Q) is the centre of the “square” Q, if T is the unit circle and t = dist.(Q, T), we considerwhere Ek = E ∩ Qk and c(Ek) is the capacity of Ek. We prove that the set E is minimally thin at τ ∈ T in U if and only if W(τ)< ∞. We study functions of type W and discuss the relation between certain results of Naim on minimal thinness [15], a minimum principle of Beurling [3], related results due to Dahlberg [7] and Sjögren [16] and recent work of Hayman-Lyons [15] (cf. also Bonsall [4]) and Volberg [19]. For simplicity, we discuss our problems in the unit disc U in the plane. However, the same techniques work for analogous problems in higher dimensions and in more complicated regions.