Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces

We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space 𝓛 p,φ (X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space 𝓛 q,ψ (X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent.

Some estimates for the commutators of multilinear maximal function on Morrey-type space

In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

Parameter θ -Type Marcinkiewicz Integral on Nonhomogeneous Weighted Generalized Morrey Spaces

Let X , d , μ be a nonhomogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. In this setting, the author proves that parameter θ -type Marcinkiewicz integral M θ ρ is bounded on the weighted generalized Morrey space L p , ϕ , τ ω for p ∈ 1 , ∞ . Furthermore, the boudedness of M θ ρ on weak weighted generalized Morrey space W L p , ϕ , τ ω is also obtained.

Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form $$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$ L u : = ∑ i , j = 1 n a ij ( x ) u x i x j = ∇ · f , for almost all x ∈ Ω where the coefficients $$a_{ij}$$ a ij belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces $$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$ LM ~ { x 0 } p , φ . Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.

Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces

In this paper, we give a boundedness criterion for the potential operator { {\mathcal I} }^{\alpha } in the local generalized Morrey space L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the generalized Morrey space {M}_{p,\varphi }(\text{Γ}) defined on Carleson curves \text{Γ} , respectively. For the operator { {\mathcal I} }^{\alpha } , we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness on L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the strong and weak Adams-type boundedness on {M}_{p,\varphi }(\text{Γ}) .