A non-doubling Trudinger inequality
Амиран Гогатишвили, Pekka Koskela · Studia Mathematica · 2005
We establish a Trudinger inequality for functions that satisfy a suitable Poincaré inequality in a Euclidean space equipped with a Borel measure that need not be doubling.1. Introduction.It is by now well understood that a Poincaré inequality improves itself to a Sobolev-type inequality when we consider a doubling measure.To be more precise, suppose that a pair u, g of measurable functions with g ≥ 0 satises the inequalityHere and in what follows, 4 A refers to µ(A) -1 T A , u A is the average of u over a set A, and we assume that u is integrable on each ball B. Assume then that µ is doubling: