Weighted composition operators on weighted Bergman spaces of bounded symmetric domains

In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator W _φψ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.     

Essential norm of the difference of weighted composition operators

We study differences of weighted composition operators between weighted Banach spaces H _ν ^∞ of analytic functions with weighted sup-norms and give an expression for the essential norm of these differences. We apply our result to estimate the essential norm of differences of composition operators acting on Bloch-type spaces. Authors’ addresses: Mikael Lindstr?m, Department of Mathematics, Abo Akademi University, FIN 20500 Abo, Finland; Elke Wolf, Mathematical Institute, University of Paderborn, D-33095 Paderborn, Germany     

Boundedness and compactness of multidimensional integral operators with homogeneous kernels

We obtain sufficient conditions for the boundedness and compactness of multidimensional integral operators with homogeneous kernels acting from a weighted L _p -space to a weighted L _q -space.     

On interpolation of bilinear operators

In this paper we study interpolation of bilinear operators between products of Banach spaces generated by abstract methods of interpolation in the sense of Aronszajn and Gagliardo. A variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c₀ spaces into Banach spaces of non-trivial the periodic Fourier cotype. This result is then extended to the spaces generated by the well-known minimal and maximal methods of interpolation determined by quasi-concave functions. In the case when a maximal construction is generated by Hilbert spaces, we obtain a general variant of bilinear interpolation theorem. Combining this result with the abstract Grothendieck theorem of Pisier yields further results. The results are applied in deriving a bilinear interpolation theorem for Calderón-Lozanovsky, for Orlicz spaces and an embedding interpolation formula for (E,p)-summing operators.     

Weighted approximation by Baskakov-type operators

The uniform weighted approximation errors of Baskakov-type operators are characterized for weights of the form $\left(\dfrac{x}{1+x}\right)^{\gamma_{0}}{(1+x)}^{\gamma_{\infty}}$ for ?? ₀,?? _????[?1,0]. Direct and strong converse theorems are proved in terms of the weighted K-functional.     

Schatten class weighted composition operators on weighted Fock spaces

We describe the Schatten class weighted composition operators on Fock–Sobolev spaces and a large class of weighted Fock spaces, where the weights defining such spaces are radial, decay at least as fast as the classical Gaussian weight, and satisfy certain mild smoothness condition. To prove our main results, we characterize the Schatten class membership of the Toeplitz operators T _μ induced by nonnegative measures μ on the complex space ${\mathbb{C}^n}$ .     

TheL p-saturation results weighted simultaneous approximation by the method of Bernstein-Durrmeyer operators

The aim of the present paper is to prove new equivalence results and L_p-stauration results on weighted simultaneous approximation by the method of Bernstein-Durrmeyer operators (including results in [7]). One of the main tools and crucial estimates managing the converse results is given by a direct modified Vornorskaja theorem which uses the third order weighted modulus of smoothness. Supported by NSF of Hebei Province.     

Rates of -statistical convergence of positive linear operators

In this paper we study the rates of A-statistical convergence of sequences of positive linear operators mapping the weighted space C_ρ1 into the weighted space B_ρ2.     

Gårding’s inequality for elliptic operators with degeneracy

In this paper, we prove Gårding’s weighted inequality for degenerate elliptic operators in an arbitrary (bounded or unbounded) domain of n-dimensional Euclidean space ? ⁿ and use this inequality to study the unique solvability of a specific variational problem. It is assumed that the lower coefficients of the operators under consideration belong to some weighted L _p -spaces.     

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy s = (s_y,s_ù){\sigma=(\sigma_\psi,\sigma_\wedge)} , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol s_ù{\sigma_\wedge} which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.     

Spectra of weighted composition operators on weighted banach spaces of analytic functions

We determine the spectra of weighted composition operators acting on the weighted Banach spaces of analytic functionsH _{ν p} ^∞ when the symbolφ has a fixed point in the open unit disk. Further, we apply this result to give the spectra of composition operators on Bloch type spaces. In particular, we answer in the affirmative a conjecture by MacCluer and Saxe. The research of the second author was partially supported by the Academy of Finland Project No. 51906; the research of this paper was carried out while this author was visiting Kent State University, whose hospitality is acknowledged with thanks.     

On discrete Picard operators

The present paper is concerned with the approximation properties of discrete version of Picard operators. We first give exact equalities for the moments of the operators. In calculations of these moments, Eulerian numbers play a crucial role. We discuss convergence of these operators in weighted spaces and give Voronovskaya‐type asymptotic formula. The weighted approximation of the operators in quantitative mean in terms of different modulus of continuities is also considered. We emphasize that the rate of convergence of the operators is better than the one obtained in 1 . Copyright © 2016 John Wiley & Sons, Ltd.     

Weighted estimates for multilinear pseudodifferential operators

In this paper,we study the weighted estimates for multilinear pseudodifferential operators.We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions.Our result generalizes similar ones from the classical Ap weights to multiple weights.     

Essential norm of composition operators between weighted Hardy spaces

An asymptotic formula for the essential norm of composition operators acting between two weighted Hardy spaces H_w1 and H_w2, where w₁ and w₂ are two admissible weight functions, is given. The boundedness of the operators is also characterized.     

On properties of the generalized elliptic pseudo-differential operators on closed manifolds

It is shown that two classes of generalized elliptic pseudo-differential operators, GEL(X) and REL(X), selected by the author from the class of classical linear pseudo-differential operators coincide. It is also shown that for any operators A, B ∈GEL(X) their composition AB and their global parametrices P_A, P_B belong to GEL(X). An operator A belongs to GEL(X) independently of the choice of a basis in E and of the weighted order of A. Some properties of the classes EFL(U) and REL(U) arising in microalocal analysis of generalized elliptic operators are studied. Bibliography: 12 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 215–269. Translated by N. A. Karazeeva.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

Weak type estimates for some maximal operators on the weighted Hardy spaces

Weighted weak type estimates are proved for some maximal operators on the weighted Hardy spacesH _ω ^p (0 <p < 1, ω ∈A ₁) (0<p<1, ω∞A₁); in particular, weighted weak type endpoint estimates are obtained for the maximal operators arising from the Bochner-Riesz means and the spherical means onH _ω ^p .     

Spectrum of operators in ideal spaces

One considers “weighted translation” operators in ideal Banach spaces. It is proved that if the translation is aperiodic (the set of periodic points has measure zero), then the spectrum of such an operator is rotationinvariant. This result can be extended (under certain additional restrictions) to “weighted translation” operators acting in regular subspaces of ideal spaces, in particular, to operators in Hardy spaces. In this note we prove the rotation-invariance of the spectrum of aperiodic operators of “weighted translation” in ideal spaces and uniform B-algebras. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 196–198, 1976.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach

The pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ₊, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ₊ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.