Search

In this paper we study some estimates of norms in variable exponent Lebesgue spaces for singular integral operators that are imaginary powers of the Laplace operator in ℝ n . Using the Mellin transform argument, fromthese estimates we obtain the boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.

Inthisnoteweconsider “generalizedRieszpoints”forcom-pactandquasinilpotentperturbationsofToeplitzoperatorsactingontheHardyspaceoftheunitcircle. 1. IntroductionIf M is a subset of C, write isoM, accM, and ∂M for the isolated points,the accumulation points, and the boundary of M, respectively. Let X be aninfinite dimensional complex Banach space and write B(X) for the set of allbounded linear operators acting on X. We recall ([1], [5], [6]) that an operatorT ∈ B(X) is Fredholm if T(X) is closed and both T −1 (0) and X/T(X) arefinite dimensional. If T ∈ B(X) is Fredholm we can define the indexof T byindex(T) = dimT −1 (0)−dimX/T(X). An operator T ∈ B(X) is called Weylif it is Fredholm of index zero. The essential spectrum σ e (T) and the Weylspectrum ω(T) of T ∈ B(X) are defined by(1) σ e (T) = {λ ∈ C: T −λI is not Fredholm}and(2) ω(T) = {λ ∈ C: T −λI is not Weyl}.If T ∈ B(X) we write(3) π left (T) = {λ ∈ C: (T −λI) −1 (0) 6= {0}}for the set of all eigenvalues of T,(4) π left0 (T) = {λ ∈ iso σ(T) : 0 < dim(T −λI)

We introduce families of weighted grand Lebesgue spaces which generalize weighted grand Lebesgue spaces (known also as Iwaniec-Sbordone spaces). The generalization admits a possibility of expanding usual (weighted) Lebesgue spaces to grand spaces by various ways by means of additional functional parameter. For such generalized grand spaces we prove a theorem on the boundedness of linear operators under the information of their boundedness in ordinary weighted Lebesgue spaces. By means of this theorem we prove boundedness of the Hardy-Littlewood maximal operator and the Calderon-Zygmund singular operators in the weighted grand spaces.

In this paper the author extends the mapping properties of some singular integral operators in Zygmund spaces equipped with uniform norm. As a by-product quadrature methods for solving CSIE having indices 0 and 1 are proposed. Their stability and convergence are proved and error estimates in Zygmund norm are given. Some numerical tests are also shown.

We show that many harmonic analysis operators in the Bessel setting, including maximal operators, Littlewood-Paley-Stein type square functions, multipliers of Laplace or Laplace-Stieltjes transform type and Riesz transforms are, or can be viewed as, Calderon-Zygmund operators for all possible values of type parameter λ in this context. This extends results existing in the literature, but being justified only for a restricted range of λ.

We consider a potential-type integral operator with oscillating kernel in the space of functions representable as Riesz potentials. We obtain estimates of its norm in the Lebesgue spaces which depend on the oscillation parameter to a negative power.

We illustrate how limiting variants of Krasnoselʼskiĭʼs compact interpolation theorem may be obtained. As a special case of our results it follows that compactness is preserved on certain Lorentz–Zygmund spaces close to the end-point Lebesgue spaces.

We characterize the class of weights related to the boundedness of maximal op- erators associated to a Young function η in the context of variable Lebesgue spaces. Fractional versions of these results are also obtained by means of a weighted Hedberg type inequality. These results are new even in the classical Lebesgue spaces. We also deal with Wiener's type inequalities for the mentioned operators in the variable context. As applications of the strong type results for the maximal operators, we derive weighted boundedness properties for a large class of operators controlled by them.

In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted Holder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted Holder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

We prove the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces. The operators include Calder?n-Zygmund singular integral operator, Littlewood-Paley operator and Marcinkiewicz operator.

In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are proved. The integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

In approximation theory, the Lebesgue constant of an interpolation operator plays an important role. The Lebesgue constant of Berrut's interpolation operator has been studied recently. In the present work, an improved upper bound for equidistant nodes is given.

The norm of the grand Lebesgue spaces is defined through the supremum of Lebesgue norms, balanced by an infinitesimal factor. In this paper we consider the spaces defined by a norm with an analogous expression, where Lebesgue norms are replaced by grand Lebesgue norms. Without the use of interpolation theory, we prove an iteration-type theorem, and we establish that the new norm is again equivalent to the norm of grand Lebesgue spaces. We prove that the expression involved satisfy the axioms of Banach Function Spaces, and we find explicit values of the constants of the equivalence. Analogous results are proved for small Lebesgue spaces.

In this paper, we introduce a class of singular integral operators which generalize Calderon-Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a general hyper curve. We show that such operators have similar properties as ordinary Calderon-Zygmund operators. In particular, we prove that they are of weak-type (1, 1) and strong type (p,p) for 1

The aim of this paper is to prove new uncertainty principles for an integral operator with a bounded kernel. To do so we prove a Nash-type inequality and a Carlson-type inequality for this transformation. From this we deduce a variation on Heisenberg’s uncertainty inequality and Faris’s local uncertainty principle. We also prove a variation on Donoho-Stark’s uncertainty principle. Our results can be applied to a wide variety of integral operators, including the Fourier transform, the Fourier-Bessel transform, the generalized Fourier transform and the-transform.

For maximal function and Riesz potential on p-adic linear space ℚpn we give sufficient conditions of its boundedness in generalized Morrey spaces. For radial weights of special kind this condition for Riesz potential is sharp. Also we prove that if Riesz potential Iα(f) exists at point b, then b is Lq Lebesgue point for some q.

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

In the present paper, we introduce the Baskakov-Durrmeyer-Stancu operators based on q-integers. First we obtain moment estimates for these operators. We study the direct theorem and the rate of convergence in weighted space for the operators Dn,q(α,β)(f, x). Also Voronovskaja type theorem is studied.

In this work, we consider a generalized system of partial differential operators, we define the related Fourier transform and establish some harmonic analysis results. We also investigate a wide class of integral transforms of Riemann–Liouville type. In particular we give a good estimate of these integrals kernels, inversion formula and a Plancherel theorem for the dual. Keywords: Fourier transform, Convolution product, Integral operators, Mehler representation, Dual operator, Inversion formula

We investigate the problem of pointwise convergence of the family of non-linear integral operators: L_{\lambda}(f,x)=\int \limits_{a}^{b}\sum \limits_{m=1}^{N}f^{m}(t)K_{\lambda ,m}(x,t)dt, x\in \left( a,b\right), where N\geq 1 is a finite natural number, \lambda is a non-negative real parameter, K_{\lambda ,m}(x,t) is a non-negative kernel and f is the function in L_{1}(a,b). We consider two cases such that (a,b) denotes finite interval of $\mathbb{R}$ and $(a,b)$ denotes the whole real axis.

In the present paper we develop a theory of pseudo-linear operators. A generalization of the classical L^p spaces to the setting of pseudo-analysis is constructed. We obtain a Riesz representation theorem together with a Lax-Milgram type lemma.

We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.

In this paper, we continue a previous study concerning different types of pseudo-convergences of sequences of measurable functions with respect to set-valued nonadditive monotonic set functions and we establish some pseudo-versions of Lebesgue and Riesz theorems in the set-valued case. We also characterize some important structural properties of fuzzy multimeasures.

We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.Keywords: Vector integral, integral operators, operator kernel

The aim of this paper is to establish a sufficient condition for certain weighted norm inequalities for singular integral operators with non-smooth kernels and for the commutators of these singular integrals with BMO functions. Our condition is applicable to various singular integral operators, such as the second derivatives of Green operators associated with Dirichlet and Neumann problems on convex domains, the spectral multipliers of non-negative self-adjoint operators with Gaussian upper bounds, and the Riesz transforms associated with magnetic Schrodinger operators.

In this paper, we study the approximation of the Lebesgue integrable functions at fixed characteristic points with the use of statistical convergence and matrix summability of the sequence of convolution type integral operators. (C) 2012 Elsevier Inc. All rights reserved.

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyen, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.

This paper gives a criterion on the weighted norm estimates of the oscillatory and variation operators for the commutators of Calderón–Zygmund singular integrals in dimension 1. As applications, the weighted Lp -boundedness of the oscillation operators and the ρ-variation operators for commutators of the Hilbert transform and the Hermitian Riesz transform are obtained. In addition, the corresponding results for the λ-jump operators and the numbers of up-crossings associated with these operators are also given.

Boundedness results for bilinear square functions and vector-valued operators on products of Lebesgue, Sobolev, and other spaces of smooth functions are presented. Bilinear vector-valued Calderon-Zygmund operators are introduced and used to obtain bounds for the optimal range of estimates in target Lebesgue spaces including exponents smaller than one.

We introduce a class of double-complex integral linear operators. Some geometric properties are investigated and a statistical approximation theorem is obtained. In a particular case, our operators turn into the complex Picard operators.

The paper deals with the monotonicity of singular integral operators of the form where is the Cauchy singular integral operator on the interval (0,1) of the real axis and q is a power or logarithmic function. Under suitable assumptions, such singular integral operators are proved to be monotone and maximal monotone in spaces with power weights. Moreover, two related integral equations with weakly singular kernels of logarithmic type are studied. Copyright © 2012 John Wiley & Sons, Ltd.

We prove the existence of pairs of unitary (or self-adjoint) operators with singular spectral measure whose difference is a rank-two operator for which the Abel wave operators fail to exist. Also, we discuss the closely related problem of constructing the Hilbert transform with respect to a singular measure on the unit circle.

The paper is devoted to singular integral operators acting on weighted variable exponent Lebesgue spaces on certain composed Carleson curves. Necessary and sufficient conditions for Fredholmness and an index formula are obtained.

In this paper we estimate various norms of some integral operators related to the Poisson equation defined in a bounded domain in the complex plane with vanishing boundary data.

Let (X,d,μ) be a space of homogeneous type. Under the assumption μ({x})=0 for all x∈X, we prove a decomposition theorem for singular integral operators on (X,d,μ). Isotropic Haar expansion gives a representation of the integral operator as a series of simple shifts and rearrangements plus two paraproducts. This yields a UMD-valued T(1) theorem on spaces of homogeneous type.

We prove a dyadic representation theorem for bi-parameter singular integrals. That is, we represent certain bi-parameter operators as rapidly decaying averages of what we call bi-parameter shifts. A new version of the product space T1 theorem is established as a consequence., 26 pages

‎A singular value inequality for sums and products of Hilbert space operators‎ ‎is given‎. ‎This inequality generalizes several recent singular value‎ ‎inequalities‎, ‎and includes that if $A$‎, ‎$B$‎, ‎and $X$ are positive operators‎ ‎on a complex Hilbert space $H$‎, ‎then ‎\begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XB^{^{1/2}}\right) \leq \frac{1}{2}\left\Vert‎ ‎X\right\Vert \text{ }s_{j}\left( A+B\right) \text{, ‎\‎ ‎}j=1,2,\cdots\text{,}‎ ‎\end{equation*} ‎which is equivalent to‎ ‎ \begin{equation*}‎ ‎s_{j}\left( A^{^{1/2}}XA^{^{1/2}}-B^{^{1/2}}XB^{^{1/2}}\right) \leq‎ ‎\left\Vert X\right\Vert s_{j}\left( A\oplus B\right) \text{, ‎\ }j=1,2,\cdots ‎\text{.}‎ ‎\end{equation*}‎ ‎ Other singular value inequalities for sums and products of operators are‎ ‎presented‎. ‎Related arithmetic-geometric mean inequalities are also‎ ‎discussed‎.

An asymmetric Fuglede-Putnam’s Theorem for w -hyponormal operators and dominant operators is proved, as a consequence of this result, we obtain that the range of the generalized derivation induced by the above classes of operators is orthogonal to its kernel. Mathematics subject classification (2010): 47B47, 47A30, 47B20.

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.

We study several fundamental operators in harmonic analysis related to Bessel operators, including maximal operators related to heat and Poisson semigroups, Littlewood–Paley–Stein square functions, multipliers of Laplace transform type and Riesz transforms. We show that these are (vector-valued) Calderon–Zygmund operators in the sense of the associated space of homogeneous type, and hence their mapping properties follow from the general theory.

Explicit forms of Lebesgue functions are not described in the mathematical literature by now. This issue is related to the problem of reducing sums of modules of fundamental polynomials or the corresponding Dirichlet kernels. That is why the complete study of graphs of Lebesgue functions remains a complicated topical problem in the theory of approximation. In this paper we solve the mentioned problems both for odd and even numbers of interpolation nodes. We find explicit forms of Lebesgue functions and study them by means of the differential calculus. All mentioned forms are new.

The purpose of this paper is to investigate the mapping properties of the strongly singular convolution operators on general weighted modulation spaces [Formula: see text] for 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ ℝ. Our results show that modulation spaces are good substitutions for Lebesgue spaces.

The uniqueness theorem is proved for the solution of the inverse spec- tral problem for second-order integro-di®erential operators on a ¯nite interval. These operators are perturbations of the Sturm-Liouville operator with convolution and one- dimensional operators. The main tool is an integral transform connected with solutions of integro-di®erential operators.

We obtain the necessary and sufficient conditions for the boundedness of the weighted singular integral operator with power weights in grand Lebesgue spaces. Because of applications to singular integral equations, the underlying set on which the functions are defined is a Carleson curve in the complex plane. Note that weighted boundedness of an operator in grand Lebesgue space is not the same as the boundedness in weighted grand Lebesgue space.

We study a two-weight problem for the Hardy–Littlewood maximal operator in variable exponent Lebesgue spaces L p (·). The exponential function satisfies some logarithmic type continuity conditions. Bibliography: 25 titles.

Various type of two–weight criteria for fractional maximal operators (involving the Hardy–Littlewood maximal functions) and singular integrals in variable exponent Lebesgue spaces defined on the real line are established. Bibliography: 30 titles.