Interpolating Sequences in Harmonically Weighted Dirichlet Spaces

In this article we show that interpolating sequences on certain harmonically weighted Dirichlet spaces can be characterized in terms of a separation condition and a Carleson-measure condition. This is the first example of a space with Nevanlinna–Pick kernel with non-radially symmetric weights in which this characterization remains true.     

Interpolating sequences in harmonically weighted Dirichlet spaces

We describe the interpolating sequences and weak interpolating sequences for the multiplier algebras of harmonically weighted Dirichlet spaces when is a finitely atomic measure.

     

Composition operators on weighted Dirichlet spaces

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

     

Cantor sets and cyclicity in weighted Dirichlet spaces

We treat the problem of characterizing the cyclic vectors in the weighted Dirichlet spaces, extending some of our earlier results in the classical Dirichlet space. The absence of a Carleson-type formula for weighted Dirichlet integrals necessitates the introduction of new techniques.     

Weighted composition operators between Dirichlet spaces

In this article, we study the boundedness of weighted composition operators between different vector-valued Dirichlet spaces. Some sufficient and necessary conditions for such operators to be bounded are obtained exactly, which are different completely from the scalar-valued case. As applications, we show that these vector-valued Dirichlet spaces are different counterparts of the classical scalar-valued Dirichlet space and characterize the boundedness of multiplication operators between these different spaces.     

De Branges-Rovnyak spaces and Dirichlet spaces

Sarason has shown that the local Dirichlet spaces Dλ may be considered as manifestations of de Branges-Rovnyak spaces H(b), and has used this identification to give a new proof that the spaces Dλ are star-shaped. We investigate which other Dirichlet spaces D(μ) arise as de Branges-Rovnyak spaces, and which other de Branges-Rovnyak spaces H(b) are star-shaped. We also prove a transfer principle which represents H(b)-spaces inside Dλ.     

A priori bounds in weighted spaces

In unbounded domains we state some a priori bounds for solutions of the Dirichlet problem for linear second order elliptic differential equations in nondivergence form with discontinuous coefficients in weighted spaces. The weight function is related to the distance function from a fixed subset S of ∂Ω.     

Hilbert spaces of Dirichlet series and their multipliers

We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted norms of the Dirichlet coefficients. We describe the multiplier algebras of these spaces. The functions in the multiplier algebra may or may not extend to be analytic on a larger half-plane than the functions in the Hilbert space.

     

Compactness of the Hardy-Littlewood operator on some spaces of harmonic functions

We study the compactness of the Hardy-Littlewood operator on several spaces of harmonic functions on the unit ball in ? ⁿ such as: a-Bloch, weighted Hardy, weighted Bergman, Besov, BMO _p , and Dirichlet spaces.     

Compact differences of composition operators on weighted Dirichlet spaces

Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S ², the space of analytic functions whose first derivative is in H ², and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.     

On order bounded weighted composition operators between Dirichlet spaces

Positivity - Recently, Gao et al. (Chin Ann Math 37B:585–594, 2016) proved a sufficient condition for order boundedness of a weighted composition operator acting between Dirichlet spaces. In...     

Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces

The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted Lp's theory with 1<p<∞. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1) (1997) 55-81] with Dirichlet and Neumann conditions.     

Adjoints of rationally induced weighted composition operators on the Hardy,Bergman and Dirichlet spaces

We determine the adjoint of a multiplication operator with rational symbol u acting on various spaces of analytic functions, in which the denominator of u is a product of distinct linear factors. We use the results to represent the adjoints of weighted composition operators with rational symbols on the Hardy, Bergman and Dirichlet spaces.     

Gradient estimates for parabolic equations in generalized weighted Morrey spaces

We consider the Cauchy–Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain.The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others.We obtain Calderón–Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Jumping nonlinearities and weighted Sobolev spaces

Working in a weighted Sobolev space, a new result involving jumping nonlinearities for a semilinear elliptic boundary value problem in a bounded domain in R^N is established. The nonlinear part of the equation is assumed to grow at most linearly and to be at resonance with the first eigenvalue of the linear part on the right. On the left, the nonlinearity crosses over (or jumps over) several higher eigenvalues. Existence is obtained through the use of infinite-dimensional critical point theory in the context of weighted Sobolev spaces and appears to be new even for the standard Dirichlet problem for the Laplacian.     

The Dirichlet problem for elliptic equations in weighted Sobolev spaces on unbounded domains of the plane

This paper deals with the Dirichlet problem for second order linear elliptic equations in unbounded domains of the plane in weighted Sobolev spaces. We prove an a priori bound and an existence and uniqueness result.     

Non-homogeneous Dirichlet boundary value problems in weighted Sobolev spaces

The paper presents existence and multiplicity results for non-linear boundary value problems on possibly non-smooth and unbounded domains under possibly non-homogeneous Dirichlet boundary conditions. We develop here an appropriate functional setting based on weighted Sobolev spaces. Our results are obtained by using global minimization and a minimax approach using a non-smooth critical point theory.     

Closed ideals in Dirichlet spaces

It is shown that the closed lattice ideals of Dirichlet spaces and of the Sobolev spacesW ^1,p are those subspaces which consist of all functions which vanish on a prescribed set.     

The dirichlet problem in weighted spaces and some uniqueness theorems for harmonic functions

This paper considers the Dirichlet problem in weighted spaces L ¹(??) in the half-plane and in the disk. The obtained results are applied to studying the uniqueness questions of harmonic functions in the half-plane and in the half-space. Also, the uniqueness theorem of harmonic functions in the unit disk is proved.