Strongly nonlinear Gagliardo–Nirenberg inequality in Orlicz spaces and Boyd indices

Abstract

Given a N-function $A$ and a continuous function $h$ satisfying certain assumptions, we derive the inequality

with constants $C_1,C_2$ independent of $f$, where $f\ge 0$ belongs locally to the Sobolev space $W^{2,1}(\mathbb{R})$, $f^{{}^{\prime }}$ has compact support, $p>1$ is smaller than the lower Boyd index of $A$, ${\mathcal{T}}_{h,p}(\cdot )$ is certain nonlinear transform depending of $h$ but not of $A$ and $\mathcal{M}$ denotes the Hardy–Littlewood maximal function. Moreover, we show that when $h\equiv 1$, then $\mathcal{M}{f}^{{}^{\prime \prime }}$ can be improved by $f^{{}^{\prime \prime }}$. This inequality generalizes a previous result by the third author and Peszek, which was dealing with $p=2$.