Interpolation on Convex-set Valued Lebesgue Spaces and its Applications
Yuxun Zhang, Jiang Zhou
TL;DR
This work develops a twofold interpolation theory for convex-set valued Lebesgue spaces $L_{\\mathcal{K}}^p(\Omega,\rho)$, generalizing both Marcinkiewicz and Riesz-Thorin theorems to set-valued contexts. It proves Marcinkiewicz-type interpolation for convex-set valued operators and applies it to convex-set valued fractional averaging and maximal operators, establishing their boundedness on $L_{\\mathcal{K}}^p(\Omega,\rho)$. It also establishes a Riesz-Thorin-type interpolation framework and uses it to derive a reverse factorization property for matrix weights via norm-function and convex-set techniques. The results extend set-valued harmonic analysis and matrix weight theory, providing new tools for weighted inequalities in matrix-valued and set-valued settings.
Abstract
In this paper, we obtain two interpolation theorems on convex-set valued Lebesgue spaces, which generalize the Marcinkiewicz interpolation theorem and Riesz-Thorin interpolation theorem on classical Lebesgue spaces, respectively. As applications, we obtain the boundedness of convex-set valued fractional averaging operators and fractional maximal operators, and prove the reverse factorization property of matrix weights.