A short proof of the Gagliardo-Nirenberg inequality with BMO term

Miyazaki, Yoichi

doi:10.1090/proc/15048

A short proof of the Gagliardo-Nirenberg inequality with BMO term
HTML articles powered by AMS MathViewer

by Yoichi Miyazaki

Proc. Amer. Math. Soc. 148 (2020), 4257-4261

DOI: https://doi.org/10.1090/proc/15048

Published electronically: May 27, 2020

PDF | Request permission

Abstract:

We give a short proof of the Gagliardo-Nirenberg inequality with BMO term as well as the classical Gagliardo-Nirenberg inequality, applying Hedberg’s method, which was used for the Riesz potential, to Muramatu’s integral formula. Compared with the proof given by Strzelecki [Bull. London Math. Soc. 38 (2006), pp. 294–300], we do not need the duality of the Hardy space and the BMO space.

References

A. Fiorenza, M. R. Formica, T. Roskovec, and F. Soudský, Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks, (2018), 1–14, arXiv:1812.04281v1.
Emilio Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 8 (1959), 24–51 (Italian). MR 109295
Lars Inge Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510. MR 312232, DOI 10.1090/S0002-9939-1972-0312232-4
Giovanni Leoni, A first course in Sobolev spaces, 2nd ed., Graduate Studies in Mathematics, vol. 181, American Mathematical Society, Providence, RI, 2017. MR 3726909, DOI 10.1090/gsm/181
E. È. Lokharu, The Gagliardo-Nirenberg inequality for maximal functions that measure smoothness, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 389 (2011), no. Issledovaniya po Lineĭnym Operatoram i Teorii Funktsiĭ. 38, 143–161, 287 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 182 (2012), no. 5, 663–673. MR 2822534, DOI 10.1007/s10958-012-0771-x
Vladimir Maz′ya and Tatyana Shaposhnikova, On pointwise interpolation inequalities for derivatives, Math. Bohem. 124 (1999), no. 2-3, 131–148. MR 1780687
Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530, DOI 10.1007/978-3-642-15564-2
David S. McCormick, James C. Robinson, and Jose L. Rodrigo, Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO, Milan J. Math. 81 (2013), no. 2, 265–289. MR 3129786, DOI 10.1007/s00032-013-0202-6
Yves Meyer and Tristan Rivière, A partial regularity result for a class of stationary Yang-Mills fields in high dimension, Rev. Mat. Iberoamericana 19 (2003), no. 1, 195–219. MR 1993420, DOI 10.4171/RMI/343
Yoichi Miyazaki, Introduction to $L_p$ Sobolev spaces via Muramatu’s integral formula, Milan J. Math. 85 (2017), no. 1, 103–148. MR 3654906, DOI 10.1007/s00032-017-0267-8
Yoichi Miyazaki, Sobolev inequalities via Muramatu’s integral formula, Matematiche (Catania) 74 (2019), no. 2, 289–299. MR 4041941, DOI 10.4418/2019.74.2.5
Tosinobu Muramatu, On Besov spaces and Sobolev spaces of generalized functions definded on a general region, Publ. Res. Inst. Math. Sci. 9 (1973/1974), 325–396. MR 0341063, DOI 10.2977/prims/1195192564
L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162. MR 109940
P. Strzelecki, Gagliardo-Nirenberg inequalities with a BMO term, Bull. London Math. Soc. 38 (2006), no. 2, 294–300. MR 2215922, DOI 10.1112/S0024609306018169