JournalsrlmVol. 28, No. 1pp. 119–141

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

  • Claudia Capone

    Consiglio Nazionale delle Ricerche, Napoli, Italy
  • Alberto Fiorenza

    Università di Napoli Federico II, Italy and Consiglio Nazionale delle Ricerche, Napoli, Italy
  • Agnieszka Kałamajska

    University of Warsaw, Poland and Polish Academy of Sciences, Warsaw, Poland
Strongly nonlinear Gagliardo–Nirenberg inequality in Orlicz spaces and Boyd indices cover
Download PDF

A subscription is required to access this article.

Abstract

Given a N-function AA and a continuous function hh satisfying certain assumptions, we derive the inequality

RA(f(x)h(f(x)))dxC1RA(C2Mf(x)Th,p(f,x)ph(f(x)))dx,\int_\mathbb R A(|f^{'}(x)|h(f(x)))dx\leq C_1\int_\mathbb R A\left(C_2 \sqrt[p]{ |{\cal M}f^{''}(x){\cal T}_{h,p}(f,x)| }\cdot h(f(x)) \right)dx,

with constants C1,C2C_1,C_2 independent of ff, where f0f\ge 0 belongs locally to the Sobolev space W2,1(R)W^{2,1}(\mathbb R), ff^{'} has compact support, p>1p > 1 is smaller than the lower Boyd index of AA, Th,p(){\mathcal T}_{h,p}(\cdot) is certain nonlinear transform depending of hh but not of AA and M{\mathcal M} denotes the Hardy–Littlewood maximal function. Moreover, we show that when h1h\equiv 1, then Mf{\mathcal M}f^{''} can be improved by ff^{''}. This inequality generalizes a previous result by the third author and Peszek, which was dealing with p=2p=2.

Cite this article

Claudia Capone, Alberto Fiorenza, Agnieszka Kałamajska, Strongly nonlinear Gagliardo–Nirenberg inequality in Orlicz spaces and Boyd indices. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 28 (2017), no. 1, pp. 119–141

DOI 10.4171/RLM/755