Several Problems In Nonlinear Geometry Of Banach Space

In this paper,We study regularity problem of nonlinear embedddings of Banach space and Lipschitz extension problem.In 1988,M.Lovblom proved that,If 1<p ? q ? 2,B(Lp)can be a-H(?)lder embedded into Lq,then ??p/q,Motivated by this result,we future give the corresponding results for all 1? p,q<+?,and show the estimates are optimal.And then for the case of lp,we prove that if 1 ? p ? q<+?,and B(lp)can be ?-Holder embedded into lq,then?? p/q.The results are different from the corresponding results for Lp,and we also show the estimates are sharp.Classical results tell us:If<p,q<? and ?>min(?),then there is a subset E of Lp,and a ?-Holder map:f:E?Lq,while the map couldn't be extended on the whole space,such that the extension map is also a a-Holder map with the same Holder constat as f.Motivated by Y.Benyamini and J.Lindenstrauss,we give a new simple proof for the above result.And then we give the corresponding results for the case of L1 for the first time.