Title of Thesis

Limiting Reiteration For Real Interpolation And Optimal Sobolev Embeddings

Author(s)

Irshaad Ahmed

Abstract
Firstly, sharp reiteration theorems for the K􀀀interpolation method in limiting cases are proved using two-sided estimates of the K􀀀functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case. Secondly, we prove optimal embeddings of the homogeneous Sobolev spaces built-up over function spaces in Rn with K􀀀monotone and rearrangement invariant norm into another rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f.

1

1.1 Introduction
1.2 Estimates of the K􀀀functional
1.3 Auxiliary results and embeddings
1.4 Reiteration theorems
1.5 An application

2.1 Introduction
2.2 Pointwise estimates involving the decreasing rearrangements
2.3 The generalized homogeneous Sobolev spaces
2.4 Admissible couples
2.5 Optimal norms