Caltech/UCLA Joint Analysis Seminar

I will report some recent progress regarding the boundedness of the map f↦|∇M β f| f↦|∇Mβf| f \mapsto |\nabla M_\beta f| from the endpoint space W 1,1 (R d ) W1,1(Rd) W^{1,1}(\mathbb{R}^d) to L d/(d−β) (R d ) Ld/(d−β)(Rd) L^{d/(d-\beta)}(\mathbb{R}^d) , where M β Mβ M_\beta denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a pointwise relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.

This is joint work with Jose Madrid.