Modified Morrey Spaces Research Articles

Two-weighted inequalities of maximal commutators in the modified Morrey spaces, associated with the Laplace-Bessel differential operator

We consider the generalized shift operator, associated with the LaplaceBessel differential operator ∆B =∑ni=1∂2∂x2i+∑ki=1γixi∂∂xi. The maximal commutators Mb,γ, associated with the generalized shift operator are investigated. At first, we prove that the maximal commutators is bounded from the modified Morrey space Lep,λ,φ1,γ(Rnk,+) to Lep,λ,φ1,γ(R n k,+) for all 1 < p < ∞, b ∈ BMOγ(Rnk,+), (φ1, φ2) ∈ Aep,γ(Rnk,+).

HARDY-LITTLEWOOD-STEIN-WEISS THEOREMS FOR RIESZ POTENTIALS IN MODIFIED MORREY SPACES

The aim of this paper is to prove the Hardy-Littlewood-Stein-Weiss theorems for Riesz potentials in modified Morrey spaces.

On embeddings into the Morrey and modified Morrey spaces in the Dunkl setting

On embeddings into the Morrey and modified Morrey spaces in the Dunkl setting

A remark on modified Morrey spaces on metric measure spaces

Morrey norms, which are originally endowed with two parameters, are considered on general metric measure spaces. Volberg, Nazarov and Treil showed that the modified Hardy-Littlewood maximal operator is bounded on Legesgue spaces. The modified Hardy-Littlewood maximal operator is known to be bounded on Morrey spaces on Euclidean spaces, if we introduce the third parameter instead of adopting a natural extension of Morrey spaces. When it comes to geometrically doubling, as long as an auxiliary parameter is introduced suitably, the Morrey norm does not depend on the third parameter and this norm extends naturally the original Morrey norm. If the underlying space has a rich geometric structure, there is still no need to introduce auxiliary parameters. However, an example shows that this is not the case in general metric measure spaces. In this paper, we present an example showing that Morrey spaces depend on the auxiliary parameters.

Potential operators in modified Morrey spaces defined on Carleson curves

In this paper we study the potential operator IΓα, 0<1 in the modified Morrey space L˜p,λ(Γ) and the spaces BMO(Γ) defined on Carleson curves Γ. We prove that for 1<p<(1−λ)∕α the potential operator IΓα is bounded from the modified Morrey space L˜p,λ(Γ) to L˜q,λ(Γ) if and in the case of infinite curve only if α≤1∕p−1∕q≤α∕(1−λ), and from the spaces L˜1,λ(Γ) to WL˜q,λ(Γ) if and in the case of infinite curve only if α≤1−1q≤α1−λ. Furthermore, for the limiting case (1−λ)∕α≤p≤1∕α we show that if Γ is an infinite Carleson curve, then the modified potential operator I˜Γα is bounded from L˜p,λ(Γ) to BMO(Γ), and if Γ is a finite Carleson curve, then the operator IΓα is bounded from L˜p,λ(Γ) to BMO(Γ).

Necessary and sufficient conditions for boundedness of multilinear fractional integrals with rough kernels on Morrey type spaces

In this article, we study necessary and sufficient conditions on the parameters of the boundedness on Morrey spaces and modified Morrey spaces for $T_{\Omega,\alpha}$ and $M_{\Omega,\alpha}$ , which are a multilinear fractional integral and a multilinear fractional maximal operator with rough kernel, respectively. Our results extend some known results significantly.

Necessary and sufficient conditions for the boundedness of rough multilinear fractional operators on Morrey-type spaces

In this paper, we study the necessary and sufficient conditions on the parameters for the boundedness of the multilinear fractional maximal operator $\mathcal{M}_{\Omega,\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\Omega,\alpha}$ with rough kernels on Morrey spaces and modified Morrey spaces, respectively. This extends some recent results of Guliyev, Hasnov and Zeren; the necessary and sufficient conditions for the boundedness of $M_{\alpha}$ and $I_{\alpha}$ on modified spaces are considered.

On the boundedness of the maximal operator and Riesz potential in the modified Morrey spaces

On the boundedness of the maximal operator and Riesz potential in the modified Morrey spaces

Morrey-Sobolev Spaces on Metric Measure Spaces

In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,\mu)$. The embedding of the Newton-Morrey-Sobolev space into the H\"older space is obtained if $\mathscr{X}$ supports a weak Poincar\'e inequality and the measure $\mu$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Haj{\l}asz gradient, the authors also introduce the Haj{\l}asz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Haj{\l}asz-Morrey-Sobolev space when $\mu$ is doubling and $\mathscr{X}$ supports a weak Poincar\'e inequality. In particular, on the Euclidean space ${\mathbb R}^n$, the authors obtain the coincidence among the Newton-Morrey-Sobolev space, the Haj{\l}asz-Morrey-Sobolev space and the classical Morrey-Sobolev space. Finally, when $(\mathscr{X},d)$ is geometrically doubling and $\mu$ a non-negative Radon measure, the boundedness of some modified (fractional) maximal operators on modified Morrey spaces is presented; as an application, when $\mu$ is doubling and satisfies some measure decay property, the authors further obtain the boundedness of some (fractional) maximal operators on Morrey spaces, Newton-Morrey-Sobolev spaces and Haj{\l}asz-Morrey-Sobolev spaces.

Necessary and sufficient conditions for the boundedness of the anisotropic Riesz potential in anisotropic modified Morrey spaces

Abstract We prove that the anisotropic fractional maximal operator Mα,σ and the anisotropic Riesz potential operator Iα,σ, 0 &lt; α &lt; ∣σ∣ are bounded from the anisotropic modified Morrey space L̃1,b,σ(Rn) to the weak anisotropic modified Morrey space WL̃q,b,σ(Rn) if and only if, α/|σ|≤1-1/q≤α/(|σ|(1-b)) and from L̃p,b,σ(Rn) to L̃q,b,σ(Rn) if and only if, α/|σ| ≤ 1/p-1/q≤α ((1-b) |σ|). In the limiting case we prove that the operator Mα,σ is bounded from L̃p,b,σ(Rn) to L∞ (Rn) and the modified anisotropic Riesz potential operator Ĩα,σ is bounded from L̃p,b,σ(Rn) to BMOσ(Rn).

Riesz potential on the Heisenberg group and modified Morrey spaces

Abstract In this paper we study the fractional maximal operator Mα, 0 ≤ α &lt; Q and the Riesz potential operator ℑα, 0 &lt; α &lt; Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).

Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces

Necessary and sufficient conditions for the boundedness of the Riesz potential in modified Morrey spaces

Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

We consider the generalized shift operator, associated with the Dunkl operator Λα(f)(x) = (d/dx)f(x)+((2α + 1)/x)((f(x) − f(−x))/2), α &gt; −1/2. We study some embeddings into the Morrey space (D‐Morrey space) Lp,λ,α, 0 ≤ λ &lt; 2α + 2 and modified Morrey space (modified D‐Morrey space) associated with the Dunkl operator on ℝ. As applications we get boundedness of the fractional maximal operator Mβ, 0 ≤ β &lt; 2α + 2, associated with the Dunkl operator (fractional D‐maximal operator) from the spaces Lp,λ,α to L∞(ℝ) for p = (2α + 2 − λ)/β and from the spaces to L∞(ℝ) for (2α + 2 − λ)/β ≤ p ≤ (2α + 2)/β.