Gagliardo–Nirenberg type inequality for variable exponent Lebesgue spaces

Kopaliani, Tengiz; Chelidze, G.

doi:10.1016/j.jmaa.2009.03.012

Cited by 19 publications

(13 citation statements)

References 16 publications

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“…We start the paper by reformulating (or reproving) the version of Gagliardo-Nirenberg inequality for rearrangement invariant Banach function spaces stated already in [9] (cf. [21]) in another form. We do so, since the statement from [9] allows only very limited analysis of optimality, as we explain in Section 4.…”

Section: Resultsmentioning

confidence: 99%

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Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

Leśnik¹,

Roskovec²,

Soudský³

2021

Preprint

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We prove optimality of the Gagliardo-Nirenberg inequalityZ , where Y, Z are rearrangement invariant Banach function spaces and X = Y 1/2 Z 1/2 is the Calderón-Lozanovskii space. By optimality we mean that for a certain pair of spaces on the right-hand side, one cannot reduce the space on the left-hand, remaining in the class of rearrangement invariant spaces. The optimality for the Lorentz and Orlicz spaces is given as a consequence, exceeding previous results. We also discuss pointwise inequalities, their importance and counterexample prohibiting an improvement.

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Section: Resultsmentioning

confidence: 99%

“…Relaxations from the Lebesgue spaces to the Lorentz spaces scale were given for j = 0 in [32] and, recently, in [7]. See also [21] for variable Lebesgue spaces setting and [3,6,5] for fractional Sobolev and Besov spaces setting.…”

Section: Introductionmentioning

confidence: 99%

Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

Leśnik¹,

Roskovec²,

Soudský³

2021

Preprint

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“…and 0 < k/m < 1 under the assumption that the exponents p(•) and r (•) are in B(Ω). B(Ω) denotes there the class of all exponents p(•) for which the Hardy-Littlewood operator is bounded in L p(•) (Ω) (see [15]). Without this assumption on the exponents, it is possible to prove the following Sobolev generalized inequality…”

Section: Preliminariesmentioning

confidence: 99%

“…See the proof in Ladyzhenskaya et al[16, p. 62]. The extension of Gagliardo-Nirenberg inequality(2.15) to generalized Lebesgue spaces was proved by Kopaliani and Chelidze[15] for…”

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The Oberbeck–Boussinesq problem modified by a thermo-absorption term

Antont︠s︡ev

Oliveira

2011

Journal of Mathematical Analysis and Applications

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We consider the Oberbeck-Boussinesq problem with an extra coupling, establishing a suitable relation between the velocity and the temperature. Our model involves a system of equations given by the transient Navier-Stokes equations modified by introducing the thermo-absorption term. The model involves also the transient temperature equation with nonlinear diffusion. For the obtained problem, we prove the existence of weak solutions for any N 2 and its uniqueness if N = 2. Then, considering a low range of temperature, but upper than the phase changing one, we study several properties related with vanishing in time of the velocity component of the weak solutions. First, assuming the buoyancy forces field extinct after a finite time, we prove the velocity component will extinct in a later finite time, provided the thermo-absorption term is sublinear. In this case, considering a suitable buoyancy forces field which vanishes at some instant of time, we prove the velocity component extinct at the same instant. We prove also that for non-zero buoyancy forces, but decaying at a power time rate, the velocity component decay at analogous power time rates, provided the thermo-absorption term is superlinear. At last, we prove that for a general non-zero bounded buoyancy force, the velocity component exponentially decay in time whether the thermo-absorption term is sub or superlinear.

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On the interpolation constants for variable Lebesgue spaces

Karlovych

Shargorodsky

2023

Mathematische Nachrichten

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For θ∈false(0,1false)$\theta \in (0,1)$ and variable exponents p0(·),q0(·)$p_0(\cdot ),q_0(\cdot )$ and p1(·),q1(·)$p_1(\cdot ),q_1(\cdot )$ with values in [1, ∞], let the variable exponents pθ(·),qθ(·)$p_\theta (\cdot ),q_\theta (\cdot )$ be defined by 1/pθ(·):=(1−θ)/p0(·)goodbreak+θ/p1(·),1em1/qθ(·):=(1−θ)/q0(·)goodbreak+θ/q1(·).$$\begin{equation*} 1/p_\theta (\cdot ):=(1-\theta )/p_0(\cdot )+\theta /p_1(\cdot ), \quad 1/q_\theta (\cdot ):=(1-\theta )/q_0(\cdot )+\theta /q_1(\cdot ). \end{equation*}$$The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space Lpjfalse(·false)$L^{p_j(\cdot )}$ to the variable Lebesgue space Lqjfalse(·false)$L^{q_j(\cdot )}$ for j=0,1$j=0,1$, then false∥Tfalse∥Lpθ(·)→Lqθ(·)badbreak≤Cfalse∥Tfalse∥Lp0(·)→Lq0(·)1−θfalse∥Tfalse∥Lp1(·)→Lq1(·)θ,$$\begin{equation*} \Vert T\Vert _{L^{p_\theta (\cdot )}\rightarrow L^{q_\theta (\cdot )}} \le C \Vert T\Vert _{L^{p_0(\cdot )}\rightarrow L^{q_0(\cdot )}}^{1-\theta } \Vert T\Vert _{L^{p_1(\cdot )}\rightarrow L^{q_1(\cdot )}}^{\theta }, \end{equation*}$$where C is an interpolation constant independent of T. We consider two different modulars ϱmax(·)$\varrho ^{\max }(\cdot )$ and ϱsum(·)$\varrho ^{\rm sum}(\cdot )$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that Cmax≤2$C_{\rm max}\le 2$ and Csum≤4$C_{\rm sum}\le 4$, as well as, lead to sufficient conditions for Cmax=1$C_{\rm max}=1$ and Csum=1$C_{\rm sum}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that pj(·)=qj(·)$p_j(\cdot )=q_j(\cdot )$, j=0,1$j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ϱmax(·)=ϱsum(·)$\varrho ^{\rm max}(\cdot )=\varrho ^{\rm sum}(\cdot )$).

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Gagliardo–Nirenberg type inequality for variable exponent Lebesgue spaces

Abstract: We prove analogies of the classical Gagliardo-Nirenberg inequalitieswhen usual L p norms are replaced by variable L p(·) norms.

Cited by 19 publications

References 16 publications

Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

Optimal Gagliardo-Nirenberg interpolation inequality for rearrangement invariant spaces

The Oberbeck–Boussinesq problem modified by a thermo-absorption term

On the interpolation constants for variable Lebesgue spaces

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