Weighted Hardy Operators Research Articles

ON SOME INEQUALITIES FOR EXPONENTIALLY WEIGHTED FRACTIONAL HARDY OPERATORS WITH ∆-INTEGRAL CALCULUS

Dynamic equations, inequalities and operators are the indispensable cornerstones of harmonic analysis and time scale calculus. We know that the topics in question have many applications in different scientific fields. In this paper, some inequalities will be proved for variable exponentially weighted Hardy operators with ∆-integral calculus.

Mixed radial-angular bounds for Hardy-type operators on Heisenberg group

<abstract><p>In this paper, we study $ n $-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg group and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $ n $-dimensional weighted Hardy operator and weighted Cesàro operator are also obtained.</p></abstract>

GENERALIZED WEIGHTED HARDY OPERATORS AND THEIR COMMUTATORS IN THE LOCAL ”COMPLEMENTARY” GENERALIZED VARIABLE EXPONENT WEIGHTED MORREY SPACES

In this paper we study the boundedness of weighted Hardy operators Hα u and Hα u in local ”complementary” generalized variable exponent weighted Morrey spaces {M p(·),ω,ϕ {0} , characterized by variable exponent p(x), a general function ω(r) and a weight ϕ. We also study the boundedness of the commutators of Hα u and Hα u in the spaces {M p(·),ω,ϕ {0} .

Weighted Boundedness of Certain Sublinear Operators in Generalized Morrey Spaces on Quasi-Metric Measure Spaces Under the Growth Condition

We prove weighted boundedness of Calderón–Zygmund and maximal singular operators in generalized Morrey spaces on quasi-metric measure spaces, in general non-homogeneous, only under the growth condition on the measure, for a certain class of weights. Weights and characteristic of the spaces are independent of each other. Weighted boundedness of the maximal operator is also proved in the case when lower and upper Ahlfors exponents coincide with each other. Our approach is based on two important steps. The first is a certain transference theorem, where without use homogeneity of the space, we provide a condition which insures that every sublinear operator with the size condition, bounded in Lebesgue space, is also bounded in generalized Morrey space. The second is a reduction theorem which reduces weighted boundedness of the considered sublinear operators to that of weighted Hardy operators and non-weighted boundedness of some special operators.

Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesàro function spaces

The boundedness of the weighted iterated Hardy-type operators \(T_{u,b}\) and \(T_{u,b}^*\) involving suprema from weighted Lebesgue space \(L_p(v)\) into weighted Cesàro function spaces \({\operatorname{Ces}}_{q}(w,a)\) are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator \(R_u\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator \(P_{u,b }\) from \(L^p(v)\) into \({\operatorname{Ces}}_{q}(w,a)\) on the cone of monotone non-increasing functions. Under additional condition on \(u\) and \(b\), we are able to characterize the boundedness of weighted iterated Hardy-type operator \(T_{u,b}\) involving suprema from \(L^p(v)\) into \({\operatorname{Ces}}_q(w,a)\) on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function \(M_{\gamma}\) from \(\Lambda^p(v)\) into \(\Gamma^q(w)\).

Estimate for the Entropy Numbers of the Weighted Hardy Operators that Act from Banach Space to q-Banach Space

We study weighted Hardy operators from the Banach space Lp to the q-Banach space Lq and obtain estimates for the entropy numbers of one- and two-weighted Hardy operators with nonnegative measurable weight functions.

Estimate for the entropy numbers of the weighted Hardy operators that act from Banach space to $q$-Banach space

Estimate for the entropy numbers of the weighted Hardy operators that act from Banach space to $q$-Banach space

Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces

We study one‐ and multi‐dimensional weighted Hardy operators on functions with Holder‐type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which ...

Commutators with Coefficients in CMO of Weighted Hardy Operators in Generalized Local Morrey Spaces

We prove theorems on the boundedness of commutators [a,H_w^alpha ] of the weighted multidimensional Hardy operator H^alpha _w:= w H^alpha frac{1}{w} from a generalized local Morrey space mathcal {L}^{p,varphi ;0}({mathbb {R}^n}) to local or global space mathcal {L}^{q,psi }({mathbb {R}^n}). The main impacts of these theorems arethe use of CMO_s-class of coefficients a for the commutators;the general setting when the function varphi defining the Morrey space and the weight w are independent of one another and the weight w is not assumed to be in A_p;recovering the Sobolev–Adams exponent q instead of Sobolev–Spanne type exponent in the case of classical Morrey spacesboundedness from local to global Morrey spaces;the obtained estimates contain the parameter s > 1 which may be arbitrarily chosen. Its choice regulates in fact an equilibrium between assumptions on the coefficient a and the characteristics of the space. The obtained results are new also in non-weighted case.

Compactness for the weighted Hardy operator in variable exponent spaces

In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operatorHυ,ωf(x)=υ(x)∫0xf(t)ω(t)dt to be compactly acting from Lp(⋅)(0,∞) to Lq(⋅)(0,∞).

L p $L^{p}$ and BMO bounds for weighted Hardy operators on the Heisenberg group

In the setting of the Heisenberg group $\mathbb{H}^{n}$ , we characterize those nonnegative functions w defined on $[0,1]$ for which the weighted Hardy operator $\mathsf{H}_{w}$ is bounded on $L^{p}(\mathbb{H}^{n})$ , $1\leq p\leq\infty$ , and on $\operatorname{BMO}(\mathbb{H}^{n})$ . Meanwhile, the corresponding operator norm in each case is derived. Furthermore, we introduce a type of weighted multilinear Hardy operators and obtain the characterizations of their weights for which the weighted multilinear Hardy operators are bounded on the product of Lebesgue spaces in terms of Heisenberg group. In addition, the corresponding norms are worked out.

Non-smooth Atomic Decompositions for Generalized Orlicz–Morrey Spaces of the Third Kind

We deal with the generalized Orlicz–Morrey space ${\mathcal {M}}_{\phi,\varPhi}({\mathbb {R}}^{n})$ of the third kind and consider the decomposition method. Also we characterize its predual space. Some maximal estimates for generalized Orlicz–Morrey spaces of the third kind are also obtained by using the weighted Hardy operators. As an application, we consider the Olsen inequality, which is a bilinear estimate on the fractional integral operator. As an appendix, we consider a general form of the vector-valued boundedness of the Hardy–Littlewood maximal operator, where $\phi$ in the definition of ${\mathcal {M}}_{\phi,\varPhi}({\mathbb {R}}^{n})$ depends on $x$ as well. This paper contains a remedy for the mistake in the proof of the Olsen inequality of the 2014 paper by the second author (Iida et al. in Z. Anal. Anwend. 33(2):149–170, 2014).

Necessary and sufficient conditions for generalized Hausdorff operators and commutators

In this paper, we introduce a type of generalized Hausdorff operators and characterize the boundedness of these operators on Lebesgue spaces and central Morrey spaces. Moreover, we obtain the operator norms on these spaces. We also obtain sufficient and necessary conditions which ensure the boundedness of their commutators on Lebesgue spaces and central Morrey spaces with symbols in central BMO spaces. As applications, we give a new method to obtain sharp bounds for weighted Hardy operators and weighted Cesaro operators on Lebesgue spaces and central Morrey spaces.

Weighted Hardy operators in the local generalized vanishing Morrey spaces

In this paper we study \(p\rightarrow q\)-boundedness of the multi-dimensional Hardy type operators in the vanishing local generalized Morrey spaces \(V\mathcal L ^{p,\varphi }_\mathrm{{loc}}(\mathbb R ^n,w)\) defined by an almost increasing function \(\varphi (r)\) and radial type weight \(w(|x|)\). We obtain sufficient conditions, in terms of some integral inequalities imposed on \(\varphi \) and \(w\), for such a boundedness. In the case where the function \(\varphi (r)\) and the weight are power functions, these conditions are also necessary.

Weighted Hardy Operators in Complementary Morrey Spaces

We study the weightedp→q-boundedness of the multidimensional weighted Hardy-type operatorsHwαandℋwαwith radial type weightw=w(|x|), in the generalized complementary Morrey spacesℒ∁{0}p,ψ(ℝn)defined by an almost increasing functionψ=ψ(r). We prove a theorem which provides conditions, in terms of some integral inequalities imposed onψandw, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the functionψand the weightware power functions. We also prove that the spacesℒ∁{0}p,ψ(Ω)over bounded domains Ω are embedded between weighted Lebesgue spaceLpwith the weightψand such a space with the weightψ, perturbed by a logarithmic factor. Both the embeddings are sharp.

Boundedness of Weighted Hardy Operator and Its Adjoint on Triebel-Lizorkin-Type Spaces

Letp∈[1,∞],q∈[1,∞),τ∈(0,∞), andα∈(0,1)such thatτ>1/p-1/qandα≤n(1/p-τ), letUψbe the weighted Hardy operator andVψits adjoint operator with respect to the weight functionψ. In this paper, the authors establish a sufficient and necessary condition on weight functionψto ensure the boundedness ofUψandVψon the Triebel-Lizorkin-type spacesḞp,qα,τ(ℝn)and their predual spaces, Triebel-Lizorkin-Hausdorff spaces, which unify and generalize the known results onQ-type spaces.

A Weighted Variant of Riemann‐Liouville Fractional Integrals on ℝn

We introduce certain type of weighted variant of Riemann‐Liouville fractional integral on ℝn and obtain its sharp bounds on the central Morrey and λ‐central BMO spaces. Moreover, we establish a sufficient and necessary condition of the weight functions so that commutators of weighted Hardy operators (with symbols in λ‐central BMO space) are bounded on the central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Weyl and Cesàro.

Coincidence of strict s-numbers of weighted Hardy operators

We establish the equality of all the so-called strict s-numbers of the weighted Hardy operator T : L p ( I ) → L p ( I ) , where 1 < p < ∞ , I = ( a , b ) ⊂ R and T f ( x ) = v ( x ) ∫ a x u ( t ) f ( t ) d t , u and v being prescribed functions satisfying certain integrability conditions.

Commutators of weighted Hardy operators on Herz-type spaces

Commutators of weighted Hardy operators on Herz-type spaces

Generalized Poincaré embeddings and weighted Hardy operator on [formula omitted] spaces

The Poincaré embeddings W ˙ 1 , n ( R n ) ⊂ BMO ( R n ) and the John–Nirenberg inequality in BMO ( R n ) are classical results in modern analysis and partial differential equations. This paper studies the generalized Poincaré embeddings and the John–Nirenberg inequality in Q p α , q ( R n ) spaces. Furthermore, for the weighted Hardy operator U ψ and its adjoint V ψ , this paper gives sufficient and necessary conditions on weight function ψ to ensure the boundedness of them on Q p α , q ( R n ) spaces.