Extension of results about p-summing operators to Lipschitz p-summing maps and their respective relatives

Abstract

In this dissertation, we study about the extension of results of psumming operators to Lipschitz p-summing maps and their respective relatives for 1 ≤ p < ∞ . Lipschitz p-summing and Lipschitz p-integral maps are the nonlinear version of (absolutely) p-summing and p-integral operators respectively. The p-summing operators were first introduced in the paper [13] by Pietsch in 1967 for 1 < p < ∞ and for p = 1 go back to Grothendieck which he introduced in his paper [9] in 1956. They were subsequently taken on with applications in 1968 by Lindenstrauss and Pelczynski as contained in [12] and these early developments of the subject are meticulously presented in [6] by Diestel et al. While the absolutely summing operators (and their relatives, the integral operators) constitute important ideals of operators used in the study of the geometric structure theory of Banach spaces and their applications to other areas such as Harmonic analysis, their confinement to linear theory has been found to be too limiting. The paper [8] by Farmer and Johnson is an attempt by the authors to extend known useful results to the non-linear theory and their first interface in this case has appealed to the uniform theory, and in particular to the theory of Lipschitz functions between Banach spaces. We find analogues for p-summing and p-integral operators for 1 ≤ p < ∞. This then divides the dissertation into two parts. In the first part, we consider results on Lipschitz p-summing maps. An application of Bourgain’s result as found in [2] proves that a map from a metric space X into ℓ2X 1 with |X| = n is Lipschitz 1-summing. We also apply the non-linear form of Grothendieck’s Theorem to prove that a map from the space of continuous real-valued functions on [0, 1] into a Hilbert space is Lipschitz p-summing for some 1 ≤ p < ∞. We also prove an analogue of the 2-Summing Extension Theorem in the non-linear setting as found in [6] by showing that every Lipschiz 2-summing map admits a Lipschiz 2-summing extension. When X is a separable Banach space which has a subspace isomorphic to ℓ1, we show that there is a Lipschitz p-summing map from X into R2 for 2 ≤ p < ∞ whose range contains a closed set with empty interior. Finally, we prove that if a finite metric space X of cardinality 2k is of supremal metric type 1, then every Lipschitz map from X into a Hilbert space is Lipschitz p-summing for some 1 ≤ p < ∞. In the second part, we look at results on Lipschitz p-integral maps. The main result is that the natural inclusion map from ℓ1 into ℓ2 is Lipschitz 1-summing but not Lipschitz 1-integral.