Let ? be a positive finite Borel measure on the unit circle. The associated Dirichlet space D(?) consists of holomorphic functions disc whose derivatives are square integrable when weighted against Poisson integral ?. We give sufficient condition subset E circle which ensures that is uniqueness set for D(?). also some examples measures and sets

We solve the Dirichlet problem for a nonlinear degenerate elliptic equation that arises in modeling weak shock reeection at a wedge. The equation exhibits a nonlinear version of a degeneracy rst studied by Keldysh. Using monotone operator techniques, we prove existence of a weak solution in a weighted Sobolev space. For negative boundary data, the solution is smooth up to the degenerate boundar...

This paper is concerned with the existence of positive solutions for a class of quasilinear singular elliptic systems with Dirichlet boundary condition. By studying the competition between theCaffarelli–Kohn–Nirenberg exponents, the sign-changing potentials and the nonlinear terms, we establish an interval on the range of multiple parameters over which solutions exist in an appropriate weighted...

We study inhomogeneous Dirichlet boundary value problems associated to a linear parabolic equation dudt=Au with strongly elliptic operator A on bounded and unbounded domains white noise data. Our main assumption is that the heat kernel of corresponding homogeneous problem enjoys Gaussian type estimates taking into account distance boundary. Under mild assumptions about domain, we show generates...

in this paper, we considered composition operators on weighted hilbert spaces of analytic functions and  observed that a formula for the  essential norm, gives a hilbert-schmidt characterization and characterizes the membership in schatten-class for these operators. also, closed range composition operators  are investigated.

We show that if u is a compactly supported distribution on the complex plane such that, for every pair of entire functions f, g, $$\begin{aligned} \langle u,f{\overline{g}}\rangle =\langle u,f\rangle u,{\overline{g}}\rangle , \end{aligned}$$ then at single point. As an application, we complete classification all weighted Dirichlet spaces unit disk are de Branges–Rovnyak by showing spaces, weigh...

Multipliers of reproducing kernel Hilbert spaces can be characterized in terms positivity $n \times n$ matrices analogous to the classical Pick matrix. We study for which it suffices consider bounded size $n$. connect this problem notion subhomogeneity non-selfadjoint operator algebras. Our main results show that multiplier algebras many analytic functions, such as Dirichlet space and Drury-Arv...