Abstract
In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.
Highlights
The fractional power of the Laplacian operators 4 are defined by Citation: Zhang, H.; Zhou, J.Mixed-Norm Amalgam Spaces andTheir Predual
This paper investigates the generalization of the Hardy–Littlewood–Sobolev theorem on mixed-norm amalgam spaces
We point out that the boundedness of fractional integral operators and their commutators have been studied in classical amalgam spaces
Summary
The fractional power of the Laplacian operators 4 are defined by Citation: Zhang, H.; Zhou, J. Let. The mixed Morrey spaces M~p0 (Rn ) were defined to be the set of all measurable functions f such that their quasi-norms k f kM p0 = sup sup | Q( x, r )| p0. We point out that the boundedness of fractional integral operators and their commutators have been studied in classical amalgam spaces. We study the necessary condition of the boundedness of [b, Iγ ] from ( L~p , L~s )α (Rn ) to ( L~q , L~s ) β (Rn ), which is a new result even for the classical amalgam spaces.
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