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

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,

Given a N-function A and a continuous function h satisfying certain assumptions, we derive the inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with constants [C.sub.1], [C.sub.2] independent of f, where f [greater than or equal to] 0 belongs locally to the Sobolev space [W.sup.2,1] (R), f' has compact support, p 1 is smaller than the lower Boyd index of A, [T.sub.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 [equivalent to] 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.