Elliptic equations with discontinuous coefficients in generalized Morrey spaces

2017 ◽  
Vol 3 (3) ◽  
pp. 728-762 ◽  
Author(s):  
Giuseppe Di Fazio ◽  
Denny Ivanal Hakim ◽  
Yoshihiro Sawano
Keyword(s):  
Elliptic Equations ◽  
Discontinuous Coefficients ◽  
Morrey Spaces ◽  
Generalized Morrey Spaces

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

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
V. S. Guliyev ◽  
M. N. Omarova ◽  
M. A. Ragusa ◽  
A. Scapellato
Keyword(s):  
Elliptic Equations ◽  
Representation Formula ◽  
Morrey Spaces ◽  
Divergence Form ◽  
Order Equation ◽  
Regularity Of Solutions ◽  
Generalized Morrey Spaces ◽  
Regularity Results ◽  
Almost All ◽  
Derivatives Of

AbstractAim 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.

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

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

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.

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

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract 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.

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