Extrapolation to mixed Herz spaces and its applications

Mingquan Wei

Mathematische Nachrichten, 2024, vol. 297, issue 6, 2067-2091

Abstract: In this paper, we extend the extrapolation theory to mixed Herz spaces K̇q⃗α,p(Rn)$\dot{K}^{\alpha,p}_{\vec{q}}(\mathbb {R}^n)$ and Kq⃗α,p(Rn)$K^{\alpha,p}_{\vec{q}}(\mathbb {R}^n)$. To prove the main result, we first study the dual spaces of mixed Herz spaces, and then give the boundedness of the Hardy–Littlewood maximal operator on mixed Herz spaces. By using the extrapolation theorems, we obtain the boundedness of many integral operators on mixed Herz spaces. We also give a new characterization of boundedmeanoscillationspace(BMO)(Rn)${\rm{bounded\ mean\ oscillation\ space}}\ ({\rm BMO})(\mathbb {R}^n)$ via the boundedness of commutators of some operators on mixed Herz spaces.

Date: 2024
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