Weighted Morrey Spaces Research Articles

Vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey space

In this paper, we prove weighted norm inequalities for vector-valued multilinear singular integrals with nonsmooth kernels and commutators on generalized weighted Morrey space.

Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights

We define a scale of weighted Morrey spaces which contains different versions of weighted spaces appearing in the literature. This allows us to obtain weighted estimates for operators in a unified way. In general, we obtain results for weights of the form |x|αw(x) with w∈Ap and nonnegative α. We study particularly some properties of power-weighted spaces and in the case of the Hardy-Littlewood maximal operator our results for such spaces are sharp. By using extrapolation techniques the results are given in abstract form in such a way that they are automatically obtained for many operators.

Some inequalities for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces

The aim of this paper is to prove the boundedness of the oscillation and variation operators for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces.

Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces

The goal of this paper is to characterize the local sharp estimate $$(I_{\rho } f)^{\#}(x) \le C \, M_{\rho } f(x)$$ and by using this inequality to get necessary and sufficient conditions on the triple functions $$(\varphi , \rho , \omega )$$ which satisfy the equivalence of norms of the generalized fractional integral operator $$I_{\rho }$$ and the generalized fractional maximal operator $$M_{\rho }$$ on the generalized weighted Morrey spaces $${\mathcal {M}}_{p,\varphi }({\mathbb {R}}^{n},\omega )$$ and generalized weighted central Morrey spaces $$\dot{{\mathcal {M}}}_{p,\varphi }({{\mathbb {R}}}^n,\omega )$$ , when $$\omega \in A_{\infty }$$ -Muckenhoupt class.

Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

Let ℒ=−Δ+V be a Schrödinger operator on ℝd, d≥3, where Δ is the Laplacian operator on ℝd, and the nonnegative potential V belongs to the reverse Hölder class RHs with s≥d/2. For given 0<α<d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐα=ℒ−α/2. Suppose that b is a locally integrable function on ℝd and the commutator generated by b and ℐα is defined by b.ℐαfx=bx⋅ℐαfx−ℐαbfx. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHs with s≥d/2. Then, we will establish the boundedness properties of the fractional integrals ℐα on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator b,ℐα in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than Ap,q, BMOℝd, and Lp,κμ,ν corresponding to the classical case (that is V≡0).

An extension of the Muckenhoupt–Wheeden theorem to generalized weighted Morrey spaces

Abstract In this paper, we find the condition on a function ω and a weight v which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces ℳ p , ω ⁢ ( ℝ n , v ) {{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} and generalized weighted central Morrey spaces ℳ ˙ p , ω ⁢ ( ℝ n , v ) {\dot{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} , when v belongs to the Muckenhoupt A ∞ {A_{\infty}} -class.

Characterizations for the fractional maximal operator and its commutators in generalized weighted Morrey spaces on Carnot groups

In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional maximal operator $$M_{\alpha }$$ , $$0\le \alpha <Q$$ on Carnot group $${{\mathbb {G}}}$$ on generalized weighted Morrey spaces $$M_{p,\varphi }({{\mathbb {G}}},w)$$ , where Q is the homogeneous dimension of $${{\mathbb {G}}}$$ . Also we give a characterization for the Spanne type boundedness of the fractional maximal commutator operator $$M_{b,\alpha }$$ on generalized weighted Morrey spaces.

Two‐weight norm inequalities for fractional maximal functions and fractional integral operators on weighted Morrey spaces

AbstractIn this paper, we consider weighted norm inequalities for fractional maximal operators and fractional integral operators. For suitable weights, we prove the two‐weight norm inequalities for both operators on weighted Morrey spaces.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Regularity estimates in weighted Morrey spaces for quasilinear elliptic equations

We study regularity for solutions of quasilinear elliptic equations of the form \mathrm {div}\mathbf{A}(x,u,\nabla u)=\mathrm {div}\mathbf{F} in bounded domains in \mathbb{R}^n . The vector field \mathbf{A} is assumed to be continuous in u , and its growth in \nabla u is like that of the p -Laplace operator. We establish interior gradient estimates in weighted Morrey spaces for weak solutions u to the equation under a small BMO condition in x for \mathbf{A} . As a consequence, we obtain that \nabla u is in the classical Morrey space \mathcal{M}^{q,\lambda} or weighted space L^q_w whenever |\mathbf{F}|^{1/(p-1)} is respectively in \mathcal{M}^{q,\lambda} or L^q_w , where q is any number greater than p and w is any weight in the Muckenhoupt class A_{q/p} . In addition, our two-weight estimate allows the possibility to acquire the regularity for \nabla u in a weighted Morrey space that is different from the functional space that the data |\mathbf{F}|^{1/(p-1)} belongs to.

Beurling–Ahlfors Commutators on Weighted Morrey Spaces and Applications to Beltrami Equations

Let $p\in (1, \infty )$, ? ? (0,1) and $w\in A_{p}({\mathbb C}).$ In this article, the authors obtain a boundedness (resp., compactness) characterization of the Beurling–Ahlfors commutator $[\mathcal B, b]$ on the weighted Morrey space $L_{w}^{p, \kappa }(\mathbb C)$ via $\text {BMO}({\mathbb C})$ [resp., $\text {CMO}({\mathbb C})$], where $\mathcal B$ denotes the Beurling–Ahlfors transform and $b\in \text {BMO}({\mathbb C})$ [resp., $\text {CMO}({\mathbb C})$]. Moreover, an application to the Beltrami equation is also given.

Approximation by trigonometric polynomials in weighted Morrey spaces

In this paper we investigate the best approximation by trigonometric polynomials in weighted Morrey spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$, where the weight function $w$ is in the Muckenhoupt class $A_{p}(I_{0})$ with $1 < p < \infty$ and $I_{0}=[0, 2\pi]$. We prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces $\mathcal{\widetilde{M}}_{p,\lambda}(I_{0},w)$ the closure of $C^{\infty}(I_{0})$ in $\mathcal{M}_{p,\lambda}(I_{0},w)$. We give the characterization of $K-$functionals in terms of the modulus of smoothness and obtain the Bernstein type inequality for trigonometric polynomials in the spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$.

Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type

Abstract In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space L ω p , k ( X ) L_\omega ^{p,k}\left( X \right) with κ ∈ (0, 1) and ω ∈ Ap (X), 1 &lt; p &lt; ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T] is compact on the same weighted Morrey space if and only if b belongs to the VMO space. We note that there is no extra assumptions on the quasimetric d and the doubling measure µ.

Fractional maximal operator and its higher order commutators in generalized weighted Morrey spaces on Heisenberg group

Fractional maximal operator and its higher order commutators in generalized weighted Morrey spaces on Heisenberg group

Maximal operator with rough kernel and its commutators in generalized weighted Morrey spaces

Maximal operator with rough kernel and its commutators in generalized weighted Morrey spaces

Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators

Let L = −Δ + V(x) be a Schrodinger operator, where Δ is the Laplacian on ℝn, while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators. Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schroodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schroodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood (H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.

Variation Inequalities for One-Sided Singular Integrals and Related Commutators

We establish one-sided weighted endpoint estimates for the ϱ -variation ( ϱ &gt; 2 ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the ϱ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the λ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones.

Variation inequalities related to Schrödinger operators on weighted Morrey spaces

Abstract This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

Note on the integral operators in weighted Morrey spaces

We investigate the boundedness of the maximal operator, the fractional maximal operator, and the fractional integral operator within the framework of weighted Morrey spaces. In particular, we consider the endpoint cases. This result can be recognized as the endpoint case of two weight multilinear norm inequality [4, Theorem 3.3] and, as a special case, recovers the Olsen inequality with small parameters [9, Examples 4.7].

Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

LetL=-Δ+Vbe a Schrödinger operator, whereΔis the Laplacian onRdand the nonnegative potentialVbelongs to the reverse Hölder classRHqforq≥d. The Riesz transform associated with the operatorL=-Δ+Vis denoted byR=∇(-Δ+V)-1/2and the dual Riesz transform is denoted byR⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classRHqforq≥d. Then we will establish the mapping properties of the operatorRand its adjointR⁎on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[b,R]and[b,R⁎]are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanAp,BMO(Rd), andLp,κ(w)corresponding to the classical Riesz transforms (V≡0).