Gagliardo–Nirenberg inequalities in logarithmic spaces

Kałamajska, Agnieszka; Pietruska-Pałuba, Katarzyna

doi:10.4064/cm106-1-8

Cited by 7 publications

(3 citation statements)

References 26 publications

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“…Original results by Gagliardo [13] and Nirenberg [35] are focused on X, Y, Z being the Lebesgue spaces. It is natural to extend these results to finer scales, considering, for instance, the Orlicz spaces (see [20,18,17,19] by Kałamajska and Pietruska-Pałuba), the Lorentz spaces (see [30] by Martín and Milman, [24] by Kolyada and Pérez Lázaro, [9] by Dao, Díaz and Nguyen, even if such papers deal with a slightly different setting) or the fractional Sobolev cases (see [6] by Brezis and Mironescu). There are also results where BMO or BV spaces are used [34,16,40,25,9].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces

Fiorenza

Formica

Roskovec

et al. 2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces

Fiorenza

Formica

Roskovec

et al. 2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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“…By Gagliardo-Nirenberg's inequality (see [16,23,33]) and the Young inequality, we see for the second term of (9) that…”

Section: Lemma 23 (Hardy-littlewood-sobolev's Inequalitymentioning

confidence: 96%

Spatial-decay of solutions to the quasi-geostrophic equation with the critical and supercritical dissipation

Yamamoto

Sugiyama

2019

Nonlinearity

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The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.

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“…for any ϕ ∈ L p (R 2 ). Lemma 2.3 (Gagliardo-Nirenberg inequality [11,15,25]). Let 0 < σ < s < 2, 1 < p 1 , p 2 < ∞ and…”

Section: Preliminariesmentioning

confidence: 99%

Optimal estimates for far field asymptotics of solutions to the quasi-geostrophic equation

Yamamoto

Sugiyama

2019

Preprint

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Gagliardo–Nirenberg inequalities in logarithmic spaces

Abstract: M q,α (|∇f (x)|) dx ≤ C M p,β (Φ 1 (x, |f |, |∇ (2) f |)) dx + M r,γ (Φ 2 (x, |f |, |∇ (2) f |)) dx , Ò Ø Ö ÓÙÒØ ÖÔ ÖØ× ÜÔÖ ×× Ò ÇÖÐ Þ ÒÓÖÑ× ∇f 2 (q,α) ≤ C Φ 1 (x, |f |, |∇ (2) f |) (p,β) Φ 2

Cited by 7 publications

References 26 publications

Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces

Gagliardo–Nirenberg Inequality for rearrangement-invariant Banach function spaces

Spatial-decay of solutions to the quasi-geostrophic equation with the critical and supercritical dissipation

Optimal estimates for far field asymptotics of solutions to the quasi-geostrophic equation

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