Weighted Spaces Research Articles

Hadamard–Bergman Operators on Weighted Spaces

Hadamard–Bergman Operators on Weighted Spaces

Best Co-Approximation In Weighted Space

In this paper, we present best co- approximation in weighted space. The results considered are these of existence of functions of best co- approximation, specifications of co-a proximal and specification co- Chebyshev subspaces.

A Priori Estimate of Solutions of One Boundary-Value Problem in a Strip for a Higher-Order Degenerate Elliptic Equation

In this paper, we prove coercive a priori estimates of solutions of a Dirichlet-type boundary-value problem in a strip for a certain higher-order degenerate elliptic equation containing weighted derivatives of a special form up to the order 2m and ordinary partial derivatives up to the order 2k−1 under the condition 2m > 2k −1. At the boundary of the strip, Dirichlet-type conditions are imposed. A coercive a priori estimate for solutions of the problem considered in special weighted Sobolev-type spaces is obtained.

Piecewise Monotone Approximation of Unbounded Functions In Weighted Space LP,w([-,

In this paper, investigate the approximation of unbounded functions in weighted space, by using trigonometric polynomials considered. We introduced type of polynomials piecewise monotone having same local monotonicity as unbounded functions without affecting the order of huge error have a finite number of max. and min. unbounded functions that amount. In addition, we established not included any of extreme points of this functions, of and closed subset γ on closed intervals then there exist class of polynomials such that the best of approximation has high or order of and such that for sufficiently great of the polynomials and functions have the same monotonicity at each of γ.

Limiting Dynamics for Stochastic FitzHugh–Nagumo Lattice Systems in Weighted Spaces

Limiting Dynamics for Stochastic FitzHugh–Nagumo Lattice Systems in Weighted Spaces

$${\mathcal{L}}$$-Wiener–Hopf Operators in Weighted Spaces in Case of Reflectionless Potential

$${\mathcal{L}}$$-Wiener–Hopf Operators in Weighted Spaces in Case of Reflectionless Potential

Volterra operators between limits of Bergman-type weighted spaces of analytic functions

We characterize continuity and compactness of the Volterra integral operator $T_g$ with the non-constant analytic symbol $g$ between certain weighted Fréchet or (LB)-spaces of analytic functions on the open unit disc, which arise as projective (resp. inductive) limits of intersections (resp. unions) of Bergman spaces of order $1<p<\infty$ induced by the standard radial weight $(1-|z|^2)^\alpha$ for $0<\alpha<\infty$. Motivated from the earlier results obtained for weighted Bergman spaces of standard weight, we also establish several results concerning the spectrum of the Volterra operators acting on the weighted Bergman Fréchet space $A^p_{\alpha+}$, and acting on the weighted Bergman (LB)-space $A^p_{\alpha-}$.

Heat kernel of fractional Laplacian with Hardy drift via desingularizing weights

We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.

Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.

Banach algebras generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols

Let $$D_3$$ be the three-dimensional Siegel domain and $${\mathcal {A}}_\lambda ^2(D_3)$$ the weight-ed Bergman space with weight parameter $$\lambda >-1$$ . In the present paper, we analyse the commutative (not $$C^*$$ ) Banach algebra $${\mathcal {T}}(\lambda )$$ generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on $${\mathcal {A}}_\lambda ^2(D_3)$$ . We remark that $${\mathcal {T}}(\lambda )$$ is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed.

CLOSURE OF FINITE FUNCTIONS IN ONE WEIGHT SOBOLEV TYPE SPACE

The description of the closure of finite or smooth finite functions in functional spaces are classical tasks of functional space theory. This task is important in smooth functional spaces such as those of Sobolev, Nikolski, Besov and in their various generalizations. Usually, in a weightless space of smooth functions, the set of compactly finite functions, generally speaking, is not dense. But in the weighted space of smooth functions, for example, in the Sobolev weighted space, with strong degeneracy of the weight, many compactly finite functions can be dense. Therefore, an important issue is the problem of characterizing the closure of compactly finite functions in the weight space under consideration. Here we consider a weighted space of Sobolev type of the second order with three weights and it describes the closure of the set of functions with compact supports.

Hilbert space valued Gabor frames in weighted amalgam spaces

AbstractLet{\mathbb{H}}be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the{\mathbb{H}}-valued Gabor frame operator on{\mathbb{H}}-valued weighted amalgam spaces{W_{\mathbb{H}}(L^{p},L^{q}_{v})},{1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on{W_{\mathbb{H}}(L^{p},L^{q}_{v})},{1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space{W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on{W_{\mathbb{H}}(L^{p},L^{q}_{v})},{1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space{\mathbb{H}}.

On the Separation Property of the Sturm–Liouville Operator in Weighted Spaces of Multiplicators

In this paper, we prove the separation theorem for the Sturm–Liouville operator in terms of point multiplicators in weighted Sobolev spaces. The research method is based on local estimates on intervals of characteristic length.

Higher Summability and Discrete Weighted Muckenhoupt and Gehring Type Inequalities

AbstractIn this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space $l_{w}^{p}({\open N})$ form summability on the weighted space $l_{w}^{q}({\open N})$ when p>q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration.

The sufficient condition for inclusion properties of discrete weighted lebesgue spaces

In this paper, we define the discrete weighted Lebesgue spaces as generalization of discrete Lebesgue spaces, and have proven sufficient condition for inclusion properties on those spaces. To get the result, we will compare some parameters on discrete weighted Lebesgue spaces. In addition, the weak type of the discrete Lebesgue spaces is discussed.

Nonlinear harmonic analysis of integral operators in weighted grand Lebesgue spaces and applications

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderon–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

On the existence of solutions to a class of boundary value problems in a strip for high-order degenerate elliptic equations

Problems in a strip for high-order elliptic equations that degenerate into an odd-order equation on the strip boundary are studied. Coercive a priori estimates and existence and uniqueness theorems for solutions of such problems are obtained in special weighted Sobolev-type spaces. The norms in these spaces are defined with the help of a special integral transform.

On degenerate elliptic equations of high order and pseudodifferential operators with degeneration

Problems for high-order degenerate elliptic equations in a half-space are studied. Coercive a priori estimates and existence theorems for solutions of such problems in special weighted Sobolev-type spaces are obtained. The norms in these spaces are defined with the help of a special integral transform. Pseudodifferential operators with degeneration constructed using a special integral transform are studied. Pseudodifferential operators with degeneration are used to factorize the symbol of a high-order degenerate elliptic operator and to construct a separating operator of the Leray–Sakamoto type.

A q-analogue of the weighted Bergman space on the disk and associated second q-Bargmann transform

We provide a q-analogue of the classical weighted Bergman space on the complex unit disk and we give the explicit expression of the corresponding reproducing kernel function. Moreover, we give a q-analogue of the second Bargmann integral transform introduced by V. Bargmann in [4, p. 203]. We show that it defines a unitary integral transform from the Hilbert space on the nonnegative real half line spanned by the q-Laguerre polynomials with respect to the weight function xαexpq⁡(−(1−q)x), onto the considered weighted q-Bergman Hilbert space.

Non-analytic superposition results on modulation spaces with subexponential weights

Motivated by classical results for Gevrey spaces and their applications to nonlinear partial differential equations we define so-called Gevrey-modulation spaces. We establish analytic as well as non-analytic superposition results on Gevrey-modulation spaces. These results are extended to a special weighted modulation space where the weight increases stronger than any polynomial but less than as in the Gevrey case.