We consider the weighted Bergman spaces HL(B, μλ), where we set dμλ(z) = cλ(1−|z| 2) dτ (z), with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. S...

Here #n is the standard Gaussian measure in R, of density d#n(x)=>k=1 ,(xk) dxk , x=(x1 , . . ., xn) # R , ,(xk)=1 2? exp(&xk 2), 8 is the inverse of the distribution function 8 of #1 , and A=[x # R: |x&a|<h for some a # A] denotes the open h-neighborhood of A. (1) becomes identity for all half-spaces A of measure p. In these notes we suggest an equivalent analytic form for (1) involving a rela...

We investigate the problem of approximating function f in power-type weighted variable exponent Sobolev space Wr,p(.) ?(.) (0,1), (r = 1, 2, ...), by Hardy averaging operator A (f) (x) 1/x ?x0 f(t)dt. If lies ?(.)(0, 1), it is shown that A||(f)?f|| p(.),?(.)?rp(.) ? C ||f(r) p(.),?(.) , where a positive constant. Moreover, we consider boundedness grand Lebesgue spaces Lp(.),? ?(.)(0,1). The suf...

Our aim in this paper is to establish Hardy-Sobolev inequalities for Sobolev functions and generalized Riesz potentials central Herz-Morrey spaces on the unit ball. As an application, we obtain norm Green potentials.

This paper is concerned with the problem of nonlinear best simultaneous approximations in K?the Bochner function spaces respect to Minkowski? norms Euclidean spaces. Characterization results generalized approximation are established. These considered a generalization concerning Lebesgue and Orlicz

In this paper we study approximations of functions Sobolev spaces W p , loc 2 ( Ω ) ⊂ R n by Lipschitz continuous functions. We prove that if f ∈ 1 ≤ < ∞ then there exists a sequence closed sets { A k } = + such the restrictions | are and cap S 0 ∖ ⋃ . Using these change variables formula in Lebesgue integral for mappings ; with Luzin capacity-measure N -property.

We introduce the notion of a generalized diierential on an abstract measure space and construct corresponding strong and weak Sobolev spaces. We show that the validity of a \weak equals strong"{theorem, i. e. the coincidence of the weak and strong Sobolev space, implies Markov uniqueness for the diiusion operator corresponding to the generalized diierential. Applications include a necessary and...