JournalsrlmVol. 30, No. 4pp. 847–864

Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces

  • Alberto Fiorenza

    Università degli Studi di Napoli Federico II, Italy and Consiglio Nazionale delle Ricerche, Napoli, Italy
  • Maria Rosaria Formica

    Università degli Studi di Napoli Parthenope, Italy
  • Tomáš G. Roskovec

    University of South Bohemia, České Budějovice, Czech Republic and Czech Technical University in Prague, Czech Republic
  • Filip Soudský

    University of South Bohemia, České Budějovice, Czechia
Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces cover
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Abstract

The classical Gagliardo–Nirenberg interpolation inequality is a well-known estimate which gives, in particular, an estimate for the Lebesgue norm of intermediate derivatives of functions in Sobolev spaces. We present an extension of this estimate into the scale of the general rearrangement-invariant Banach function spaces with the proof based on the Maz’ya’s pointwise estimates. As corollaries, we present the Gagliardo–Nirenberg inequality for intermediate derivatives in the case of triples of Orlicz spaces and triples of Lorentz spaces. Finally, we promote the scaling argument to validate the optimality of the Gagliardo–Nirenberg inequality and show that the presented estimate in Orlicz scale is optimal.

Cite this article

Alberto Fiorenza, Maria Rosaria Formica, Tomáš G. Roskovec, Filip Soudský, Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 4, pp. 847–864

DOI 10.4171/RLM/872