Note on Affine Gagliardo–Nirenberg Inequalities

Abstract

Abstract. This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine L p −Sobolev inequalities. The logarithmic version of affine L p −Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical Pólys-Szegö principle.

Sharp affine Sobolev type inequalities via the L Busemann–Petty centroid inequality

“…The sharp affine L p log-Sobolev inequality, proved independently by Haberl, Schuster, Xiao [24] and Zhai [46] for 1 < p < n, states that Theorem 1.1. Let n ≥ 1 and p > 1.…”

New approach to the affine Pólya–Szegö principle and the stability version of the affine Sobolev inequality

“…(i) If α > 1, then there exists a constant G(n, α, p) such that for any function f ∈ [64] for λ = 1/2, and recently in [34] by a different proof. The case λ = 1/2 was proved in [31].…”

“…In particular, on the L p Petty projection inequality [44] (see [9] for an alternative proof) and on the solution of the normalized L p Minkowski problem [46].The main aim of the present paper is to give a new proof for these affine Pólya-Szegötype principles. Our approach is based on a recent work of Haddad, Jiménez and Montenegro (see [34]) where they give a new proof of some sharp (symmetric) affine Sobolev-type inequalities (like Sobolev, Gagliardo-Nirenberg and logarithmic-Sobolev inequalities [45,64]) by using the L p Busemann-Petty centroid inequality [44]. We show that their method also can be applied to general cases considered in [32,33,62], and hence gives an alternative proof for the results in [13,33,62].…”

Fractional Besov Trace/Extension Type Inequalities via the Caffarelli-Silvestre extension

“…The L p affine energy has been applied to establish the affine Sobolev inequalities (cf. [18,32,31]) and the affine version of the Pólya-Szegö principle (cf. [7,11]).…”