54, 1986. EXTENSIONS OF LIPSCHITZ MAPS. 131. (a). (b) and. (c). PROOF. ..... J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis, ...
ISRAEL JOURNAL OF MATHEMATICS, Vol. 54, No. 2, 1986
EXTENSIONS OF LIPSCHITZ MAPS INTO BANACH SPACES
BY
WILLIAM B. JOHNSON,a'b'* JORAM LINDENSTRAUSS°'''*~ AND GIDEON SCHECHTMANd'"'t "Ohio State University, Columbus, 0H43210, USA; bTexas A & M University, College Station, Texas, USA ; ° The Hebrew University of Jerusalem, Jerusalem, Israel; and The Weizmann Institute of Science, Rehovot 76100, Israel ABSTRACT It is p r o v e d t h a t if Y C X a r e m e t r i c s p a c e s w i t h Y h a v i n g n > 2 p o i n t s t h e n a n y
map f from Y into a Banach space Z can be extended to a map f from X into Z so that Ilfll,~,,--- c Iogn Ilflh. where c is an absolute constant. A related result is obtained for the ease where X is assumed to be a finite-dimensional normed space and Y is an arbitrary subset of X. C o n s i d e r the following extension p r o b l e m : given three metric spaces X, Y, Z with Y _C X, what is the smallest constant L such that any Lipschitz function f : Y---~ Z admits an extension f ' X
~ Z with
Ilfll,io-- 6(log k / l o g log k) ~/2 in the second (cf. [2]). In this note we c o n c e r n ourselves with a m o r e general situation: Z is a general B a n a c h space and we put no geometrical restrictions on X. W e p r o v e the following two t h e o r e m s : t Supported in part by US-Israel Binational Science Foundation and by NSF MCS-7903042. " Supported in part by NSF MCS-8102714. Received April 18, 1985 129
