ABSTRACT
This chapter focuses on the sharp Sobolev inequalities known as the Moser–Trudinger and Adams inequalities and recalls some basic definition and standard results. It also focuses on Moser–Trudinger type inequalities in spaces of infinite measure in Euclidean space and other geometries. The inequality of Equation is known as the Moser–Trudinger inequality. After this seminal work there has been lot of interest on these types of inequalities including their validity in various other contexts, improvements of this inequality, and the question of the existence of extremal functions. Adams used the Riesz transform to convert it into a convolution type estimate and then use a lemma due to O’Neil for convolution operators of nonincreasing rearrangement of functions to prove the result.
