Gagliardo-Nirenberg inequalities in weighted Orlicz spaces equipped with a nonnecessarily doubling measure

Abstract

We obtain Gagliardo-Nirenberg interpolation inequalities of the form $\| \nabla u\|_{X}\leq C_1\sqrt{\| u|\|_{Y}\|\nabla^{(2)}u\|_Z}+C_2\| u\|_Y$, where $X,Y,Z$ are Orlicz spaces related to a single measure which may not satisfy the doubling condition. Some examples among homogeneous, logarithmic and exponential spaces are given.