Operators In Lebesgue Spaces Research Articles

A class of Hausdorff–Berezin operators on the unit ball

Abstract The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

Approximation results by multivariate Kantorovich-type neural network sampling operators in Lebesgue spaces with variable exponents

Approximation results by multivariate Kantorovich-type neural network sampling operators in Lebesgue spaces with variable exponents

HARDY-CESARO MAXIMAL OPERATOR IN LEBESGUE-BMO SPACES

In this paper defining the Lebesgue-BMO spaces. We prove the boundedness of the Hardy-Cesaro maximal operators in such spaces. Also prove necessary and sufficient condition the boundedness of the Hardy-Cesaro maximal operators in Lebesgue spaces.

Hausdorff and Dunkl–Hausdorff operators in Lebesgue spaces for monotone functions and monotone weights

Hausdorff and Dunkl–Hausdorff operators in Lebesgue spaces for monotone functions and monotone weights

Model for Aqueous Polymer Solutions with Damping Term: Solvability and Vanishing Relaxation Limit.

The main aim of this paper is to investigate the solvability of the steady-state flow model for low-concentrated aqueous polymer solutions with a damping term in a bounded domain under the no-slip boundary condition. Mathematically, the model under consideration is a boundary value problem for the system of strongly nonlinear partial differential equations of third order with the zero Dirichlet boundary condition. We propose the concept of a full weak solution (velocity–pressure pair) in the distributions sense. Using the method of introduction of auxiliary viscosity, the acute angle theorem for generalized monotone nonlinear operators, and the Krasnoselskii theorem on the continuity of the superposition operator in Lebesgue spaces, we obtain sufficient conditions for the existence of a full weak solution satisfying some energy inequality. Moreover, it is shown that the obtained solutions of the original problem converge to a solution of the steady-state damped Navier–Stokes system as the relaxation viscosity tends to zero.

Hausdorff operators associated with the Opdam–Cherednik transform in Lebesgue spaces

In this paper, we introduce the Hausdorff operator associated with the Opdam--Cherednik transform and study the boundedness of this operator in various Lebesgue spaces. In particular, we prove the boundedness of the Hausdorff operator in Lebesgue spaces, in grand Lebesgue spaces, and in quasi-Banach spaces that are associated with the Opdam--Cherednik transform. Also, we give necessary and sufficient conditions for the boundedness of the Hausdorff operator in these spaces.

Some properties of solutions of Itô equations with drift in [formula omitted

This paper is a natural continuation of Krylov (2020), where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in Ld+1(Rd+1). Here we study some properties of these processes such as higher summability of Green’s functions, a maximum principle for related transport equations with irregular coefficients, boundedness of resolvent operators in Lebesgue spaces, establish Itô’s formula, and so on.

Boundedness of the Dunkl–Hausdorff operator in Lebesgue spaces

In this paper, the Lvp(ℝ)-boundedness of the Dunkl–Hausdorff operator Hα,ϕ f(x)=∫ℝϕ(t)|t|2α+2fxtdt has been characterized and for a certain type of weight v, the operator norm ∥Hα,ϕ∥Lvp(ℝ)→Lvp(ℝ) has been obtained. This covers several of the existing results. Analogous results in two dimensions have also been proved.

Alternative criteria for boundedness of one class of integral operators in Lebesgue spaces

The paper discusses weighted Hardy type inequalities for a certain class of Volterra type integral operators with kernels. Criteria for the validity of the presented inequalities are found, which are different from the criteria obtained earlier.

Convergence Almost Everywhere of Non-convolutional Integral Operators in Lebesgue Spaces

The case of one-dimensional and multidimensional non-convolutional integral operators in Lebesgue spaces is considered in this paper. The convergence in the norm and almost everywhere of non-convolution integral operators in Lebesgue spaces was insufficiently studied. The kernels <img src=image/13420650_01.gif> of non-convolutional integral operators do not need to have a monotone majorant, so the well-known results on the convergence almost everywhere of convolutional averages are not applicable here. The kernels <img src=image/13420650_01.gif> of nonconvolutional integral operators take into account different behaviors at <img src=image/13420650_02.gif> and <img src=image/13420650_03.gif> depending on <img src=image/13420650_04.gif> (which is important in applications) and cover the situation in the particular case of convolutional and non-convolutional integral operators. We are interested in the behavior of function <img src=image/13420650_05.gif> as <img src=image/13420650_06.gif>. Theorems on convergence almost everywhere in the case of one-dimensional and multidimensional nonconvolution integral operators in Lebesgue spaces are proved. The theorems proved are more general ones (including for convolutional integral operators) and cover a wide class of kernels.

Nonlocal operators are chaotic

We characterize for the first time the chaotic behavior of nonlocal operators that come from a broad class of time-stepping schemes of approximation for fractional differential operators. For that purpose, we use criteria for chaos of Toeplitz operators in Lebesgue spaces of sequences. Surprisingly, this characterization is proved to be-in some cases-dependent of the fractional order of the operator and the step size of the scheme.

Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spaces Lp⋅ with a variable exponent. By extending the boundedness of singular integral operators in variable Lebesgue spaces from scalar type to abstract-valued type, the maximal Lp⋅−regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space E01,p⋅I, together with the imbedding property of E01,p⋅I into the range-varying function space C−I,X1−1/p⋅,p⋅. Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.

The Clifford Hilbert transform and Riemann–Hilbert problems associated to the Dirac operator in Lebesgue spaces

In this paper, we establish some properties of the Hilbert transform in Clifford analysis setting, and mainly show the weak type (1,1) inequality for the Clifford Hilbert transform in Euclidean space Rm (m ≥ 3) using the Clifford algebra and analysis generalizations of the one dimensional proofs. Furthermore, we prove Kolmogorov’s inequality for Clifford Hilbert transform. With the help of the properties for the Hilbert transform and Clifford analytic techniques, we establish the Riemann–Hilbert problems in Clifford value Lebesgue p-integrable spaces. Based on the Newton embedding method, we prove an existence and uniqueness for the nonlinear Riemann–Hilbert problem and also give an error estimation for the approximate solutions in the Newton embedding procedure in Lebesgue spaces.

Hausdorff operator in Lebesgue spaces

Hausdorff operator in Lebesgue spaces

Hardy–Steklov Operators and Duality Principle in Weighted Sobolev Spaces of the First Order

Boundedness criteria for the Hardy–Steklov operator in Lebesgue spaces on the real axis are presented. As applications, two-sided estimates for the norms of spaces associated with weighted Sobolev spaces of the first order with various weight functions and summation parameters are established.

Multi-dimensional Hardy type inequalities in Hölder spaces

Most Hardy type inequalities concern boundedness of the Hardy type operators in Lebesgue spaces. In this paper we prove some new multi-dimensional Hardy type inequalities in Holder spaces.

On the Boundedness of Singular Integrals in Morrey Spaces and its Preduals

We reduce the boundedness of operators in Morrey spaces $$L_p^r\left( {\mathbb R}^n\right) $$ , its preduals, $$H^{\varrho }L_p ({\mathbb R}^n)$$ , and their preduals $$\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) $$ to the boundedness of the appropriate operators in Lebesgue spaces, $$L_p({\mathbb R}^n)$$ . Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications in harmonic analysis.

The Bijectivity of the Tight Frame Operators in Lebesgue Spaces

 Abstract—The motivation of this dissertation mainly is which affine frame wavelet systems span Lebesgue spaces have been investigated less. The technique we used is analogous to technique of Calderon-Zygmund operators, but we rely on Calderon-Zygmund decomposition theorem. We prove our main results without smoothness assumption on frame wavelets. We prove that affine tight frame wavelets span Lebesgue spaces under the condition, we also show that the affine tight frame operator extends from 2 () L  to a bounded, linear and bijective operator on () p L  , for 1. p    Under such condition, the

Boundedness and compactness of a supremum-involving integral operator

We obtain criteria of boundedness and compactness of a supremum-involving integral operator in Lebesgue spaces on a half-axis.

A generalization of Bihari’s lemma to the case of Volterra operators in Lebesgue spaces

For operators acting in the Lebesgue space Lq(Π), 1 < q < ∞, an abstract analog of Bihari’s lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat-Darboux problem.