Boundedness Of Operators Research Articles

Extrapolation of Compactness on Banach Function Spaces

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.

Existence and uniqueness of fractal-fractional equations generated by a new fractal-fractional operator utilizing the advanced gamma function

Numerous special functions, including the beta function, hypergeometric functions, and other orthogonal polynomials, are closely connected to the gamma function. Recently, gamma function has been enhanced by adding a new parameter. As a consequence, this gamma is called the parametric gamma function or b-gamma function. By utilizing the enhanced gamma function (or the parametric gamma function), we have present a generalization for the special function Rabotnov function. Consequently, new fractal-fractional operators (derivative and integral) involving the generalized Rabotnov function are defined. Analysis is introduced to discover the main properties of the suggested operators. Since the main challenge in the calculus of fractal-fractional operators is to design examples, we illustrate a set of examples including power series. We explore the boundedness of the recommended operators. There are many gains for checking the boundedness of fractal operators in general and fractal-fractional operators in particular. Moreover, as an application, we establish the existence and uniqueness solution of abstract fractal-fractional equation. Examples are presented at the end of the effort. Graphics and computations are observed using MATHEMATICA 13.3 software.•By using the parametric gamma function, the Rabotnov special function is generalized. New fractal-fractional operators are presented using the generalized Rabotnov function with examples;•Boundedness of these operators is investigated, where bounded operators give a strong foundation for understanding linear transformations across normed spaces, with many applications in science and mathematics.•Conditions of the existence and uniqueness of solutions of fractal-fractional differential abstract equation are established.

Boundedness of Operators on Weighted Morrey–Campanato Spaces in the Bessel Setting

Boundedness of Operators on Weighted Morrey–Campanato Spaces in the Bessel Setting

Littlewood–Paley–Rubio de Francia inequality for multi‐parameter Vilenkin systems

AbstractA version of Littlewood–Paley–Rubio de Francia inequality for bounded multi‐parameter Vilenkin systems is proved: For any family of disjoint sets such that are intervals in and a family of functions with Vilenkin–Fourier spectrum inside the following holds: where C does not depend on the choice of rectangles or functions . This result belongs to the line of studying of (multi‐parameter) generalizations of Rubio de Francia inequality to locally compact abelian groups. The arguments are mainly based on the atomic theory of multi‐parameter martingale Hardy spaces and, as a byproduct, yield an easy‐to‐use multi‐parameter version of Gundy's theorem on the boundedness of operators taking martingales to measurable functions. Additionally, some extensions and corollaries of the main result are obtained, including a weaker version of the inequality for exponents and an example of a one‐parameter inequality for an exotic notion of interval.

Preduals of variable Morrey–Campanato spaces and boundedness of operators

Preduals of variable Morrey–Campanato spaces and boundedness of operators

Boundedness of Operators on Campanato Spaces Related with Schrödinger Operators on Heisenberg Groups

Boundedness of Operators on Campanato Spaces Related with Schrödinger Operators on Heisenberg Groups

Boundedness of Operators and Inequalities on Morrey–Banach Spaces

This paper establishes the boundedness of the spherical maximal function, Rubio de Francia operators, Bochner–Riesz operators, Fourier integral operators, geometric maximal operator, minimal operator and strongly singular integral operators on Morrey spaces built on Banach function spaces. We also establish the Coifman–Fefferman inequalities, Gundy–Wheeden inequalities and Sobolev–Lieb–Thirring inequalities on Morrey–Banach spaces. In particular, our results include the boundedness of the above operators and inequalities on classical Morrey spaces, generalized Morrey spaces, Orlicz–Morrey spaces, Morrey–Lorentz spaces and Morrey spaces with variable exponents.

Littlewood–Paley–Rubio De Francia Inequality for the Two-Parameter Walsh System

A version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $$ {I}_k={I}_k^1\times {I}_k^2 $$ in ℤ+ × ℤ+ and a family of functions fk with Walsh spectrum inside Ik the following is true: $$ {\left\Vert \sum_k{f}_k\right\Vert}_{L^p}\le {C}_p{\left\Vert {\left(\sum_k{\left|{f}_k\right|}^2\right)}^{1/2}\right\Vert}_{L^p},\kern2em 1<p\le 2, $$ where Cp does not depend on the choice of rectangles {Ik} or functions {fk}. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, a two-parameter version of Gundy’s theorem on the boundedness of operators taking martingales to measurable functions is formulated, which might be of independent interest.

Integral operators on local Orlicz-Morrey spaces

We establish a general principle on the boundedness of operators on local Orlicz-Morrey spaces. As applications of this principle, we obtain the boundedness of the Calder?n-Zygmund operators, the nonlinear commutators of the Calder?n-Zygmund operators, the oscillatory singular integral operators, the singular integral operators with rough kernels and the Marcinkiewicz integrals on the local Orlicz-Morrey spaces.

A Weyl pseudodifferential calculus associated with exponential weights on Rd

We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted Lp spaces over Rd with weights of the form exp(−ϕ(x)), for ϕ a C2 function, a setting in which the operator associated to the weighted Dirichlet form typically has only holomorphic functional calculus. A symbol class giving rise to bounded operators on Lp is determined, and its properties are analyzed. This theory is used to calculate an upper bounded on the H∞ angle of relevant operators and deduces known optimal results in some cases. Finally, the symbol class is enriched and studied under an algebraic viewpoint.

Mixed Hardy spaces and their applications

Multi-parameter mixed Hardy space $$H_{{\rm{mix}}}^p$$ is introduced by a new discrete Calderon’s identity. As an application, we obtain the $$H_{{\rm{mix}}}^p \to {L^p}\left({{ℝ^{{n_1} + {n_2}}}} \right)$$ boundedness of operators in the mixed Journe’s class.

The Boundedness on Mixed Hardy Spaces

The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journé class on mixed Hardy spaces by a direct method.

On the boundedness of fractional maximal and Riesz potential operators associated with Kontorovich–Lebedev transform

ABSTRACT We define the maximal and fractional maximal operators associated with the Kontorovich–Lebedev transform. We study the boundedness of these operators in the space . Using these results we establish the Hardy–Littlewood–Sobobev theorem for the Riesz potential operator associated with Kontorovich–Lebedev transform.

Boundedness of operators with partial integralswith the mixed norm. II

Boundedness of operators with partial integralswith the mixed norm. II

Singular integrals unsuitable for the curvature method whose L2-boundedness still implies rectifiability

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L2(μ)-boundedness of certain singular integral operators to the geometric properties of the support of measure μ, e.g., rectifiability. It can be applied, however, only if Menger curvature-like permutations directly associated with the kernel of the operator are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the L2(μ)-boundedness of the operator still implies that the support of μ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors–David regularity conditions.

New molecular characterizations of anisotropic Musielak–Orlicz Hardy spaces and their applications

Let φ:Rn×[0,∞)→[0,∞) be an anisotropic growth function and A a general expansive matrix on Rn. Let HAφ(Rn) be the anisotropic Musielak–Orlicz Hardy space associated with A defined via the non-tangential grand maximal function. In this article, the authors establish a new molecular characterization of HAφ(Rn) via its known atomic characterizations, which positively answer a question mentioned by Li et al. (2016) [34]. As an application, the authors obtain a criterion on the boundedness of linear operators on HAφ(Rn), which further implies the boundedness of integral anisotropic Calderón–Zygmund operators on HAφ(Rn). In addition, the boundedness of these operators from HAφ(Rn) to the Musielak–Orlicz space Lφ(Rn) is also presented.

Conical square functions for degenerate elliptic operators

Abstract The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let L w = - w - 1 ⁢ div ( w ⁢ A ⁢ ∇ ) {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)} , where w ∈ A 2 {w\in A_{2}} and A is an n × n {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the L 2 ⁢ ( w ) {L^{2}(w)} -Kato square root problem obtaining that L w {\sqrt{L_{w}}} is equivalent to the gradient on L 2 ⁢ ( w ) {L^{2}(w)} . The same authors in collaboration with the second named author of this paper studied the L p ⁢ ( w ) {L^{p}(w)} -boundedness of operators that are naturally associated with L w {L_{w}} , such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in L p ⁢ ( v ⁢ d ⁢ w ) {L^{p}(v\,dw)} for v ∈ A ∞ ⁢ ( w ) {v\in A_{\infty}(w)} ), and in particular a class of “degeneracy” weights w was found in such a way that the classical L 2 {L^{2}} -Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on L p ⁢ ( w ) {L^{p}(w)} and on L p ⁢ ( v ⁢ d ⁢ w ) {L^{p}(v\,dw)} , with v ∈ A ∞ ⁢ ( w ) {v\in A_{\infty}(w)} , of the conical square functions that one can construct using the heat or Poisson semigroup associated with L w {L_{w}} . As a consequence of our methods, we find a class of degeneracy weights w for which L 2 {L^{2}} -estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with L w {L_{w}} .

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt $$A_1$$ weighted $$L_p$$ spaces, which enable us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings (instead of using duality), one is neither restricted in the parameter range of the p’s (in particular $$p=1$$ is admissible and applies to weak-type inequalities) nor the operator has to be linear. Another remarkable consequence of our results is that vector-valued inequalities in Morrey spaces are automatically deduced. On the other hand, we also obtain $$A_\infty $$ -weighted inequalities with Morrey quasinorms.

Boundedness of operators arising from Schwarz BVP in modified local Morrey-type spaces

In this paper, we prove the boundedness of a class of operators arising from Schwarz BVP in modified local Morrey-type spaces in the unit disc of the complex plane.

The boundedness of operators in Muckenhoupt weighted Morrey spaces via extrapolation techniques and duality

The bidual of the closure of smooth functions with respect to the Morrey norm coincides with the Morrey space. This assertion is generalized to some Muckenhoupt weighted Morrey spaces. We combine this fact with basic extrapolation techniques due to Rubio de Francia adapted to weighted Morrey spaces. This leads to new results on the boundedness of operators even for the unweighted case.