Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Lu, Guozhen; Tang, Hanli

doi:10.1007/s12220-015-9573-y

Cited by 52 publications

(20 citation statements)

References 23 publications

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“…This inequality is proved in [Mancini-Sandeep-Tintarev] in the case of the gradient in the ball model (really a consequence of Hardy's inequality) and for even α in [Tat]. In this setup sharp versions of the Moser-Trudinger inequality for W α, n α (H n ) are only known in the case α = 1 for the gradient ( [MS], [MST], [LT1], [LT2]), and with the same sharp constant as in the Euclidean case. In the following theorem we give the general version of this result for arbitrary α :…”

Section: Moser-trudinger Inequalities In Hyperbolic Spacementioning

confidence: 99%

Adams inequalities for Riesz subcritical potentials

Fontana

Morpurgo

2020

Nonlinear Analysis

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We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results in [FM1]. The integral operators involved, which we call "Riesz subcritical", have kernels whose decreasing rearrangements are not worse than that of the Riesz kernel on R n , where the kernel is large, but they behave better where the kernel is small. The new element is a "critical integrability" condition on the kernel at infinity. Typical examples of such kernels are fundamental solutions of nonhomogeneous differential, or pseudo-differential, operators. Another example is the Riesz kernel itself restricted to suitable measurable sets, which we name "Riesz subcritical domains". Such domains are characterized in terms of their growth at infinity. As a consequence of the general results we obtain several new sharp Adams and Moser-Trudinger inequalities on R n , on the hyperbolic space, on Riesz subcritical domains, and on domains where the Poincaré inequality holds.

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Section: Moser-trudinger Inequalities In Hyperbolic Spacementioning

confidence: 99%

Adams inequalities for Riesz subcritical potentials

Fontana

Morpurgo

2020

Nonlinear Analysis

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“…This improves the Moser-Trudinger inequality with exact growth in H n established by Lu and Tang [24]. These inequalities are achieved from the comparison of the symmetric nonincreasing rearrangement of a function both in the hyperbolic and the Euclidean spaces, and the same inequalities in the Euclidean space.…”

mentioning

confidence: 54%

“…It was also shown in [32] that the inequality (1.9) is sharp in the sense that if we replace α n by any constant α > α n or the power n n−1 in the denominator by any p < n n−1 then the supremum will be infinity. This kind of inequality was extended to the hyperbolic spaces by Lu and Tang [24] in the form sup…”

Section: Introductionmentioning

confidence: 99%

Improved Moser–Trudinger type inequalities in the hyperbolic spaceHn

Nguyen

2018

Nonlinear Analysis

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“…Later, (1.4) was extended to the general case n ≥ 3 by Masmoudi and Sani [39] (see [23]) for more general form) and to the framework of hyperbolic space by Lu and Tang in [32]. The second order Adams' inequality with the exact growth condition was obtained by Masmoudi and Sani [38] in dimension 4:…”

Section: Introductionmentioning

confidence: 99%

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
Chen
¹
,
Lu
²
,
Zhu
³

2020
Advances in Mathematics
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Though much work has been done with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W 1,n (R n ) and higher order Adams inequalities on finite domain Ω ⊂ R n , whether there exists an extremal function for the critical higher order Adams inequalities on the entire space R n still remains open. The current paper represents the first attempt in this direction. The classical blowup procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [24]), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R 4 of the formwhere α ∈ (−∞, 32π 2 ). We establish the existence of the threshold α * , where α * ≥ (32π 2 ) 2 B2 2 and B 2 ≥ 1 24π 2 , such that S (α) is attained if 32π 2 − α < α * , and is not attained if 32π 2 − α > α * . This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R 2 . Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

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Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Cited by 52 publications

References 23 publications

Adams inequalities for Riesz subcritical potentials

Adams inequalities for Riesz subcritical potentials

Improved Moser–Trudinger type inequalities in the hyperbolic spaceHn

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
Chen
¹
,
Lu
²
,
Zhu
³

2020
Advances in Mathematics
Self Cite
22
0
4
0
View full text Add to dashboard Cite

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Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Cited by 52 publications

References 23 publications

Adams inequalities for Riesz subcritical potentials

Adams inequalities for Riesz subcritical potentials

Improved Moser–Trudinger type inequalities in the hyperbolic spaceHn

Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2Chen1, Lu2, Zhu3 2020Advances in MathematicsSelf Cite22040View full textAdd to dashboardCite

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Existence and nonexistence of extremals for critical Adams inequalities in R4 and Trudinger-Moser inequalities in R2
Chen
¹
,
Lu
²
,
Zhu
³

2020
Advances in Mathematics
Self Cite
22
0
4
0
View full text Add to dashboard Cite