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

Given a N-function A and a continuous function h satisfying certain assumptions, we derive the inequality integral(R) A(vertical bar f' (x)vertical bar h(f(x))) dx <= C-1 integral(R) A(C-2 (p) root vertical bar Mf"(x)T-h,T-p (f,x)vertical bar. h(f(x))dx, with constants C-1, C-2 independent of f, where f >= 0 belongs locally to the Sobolev space W-2,W-1 (R), f' has compact support, p > 1 is smaller than the lower Boyd index of A, T-h,(p)(.) is certain nonlinear transform depending of h but not of A and M denotes the Hardy- Littlewood maximal function. Moreover, we show that when h=1, then Mf" can be improved by f". This inequality generalizes a previous result by the third author and Peszek, which was dealing with p = 2.