Stability of Trace Theorems on the Sphere

Bez, Neal; Jeavons, Chris; Ozawa, Tohru; Sugimoto, M.

doi:10.1007/s12220-017-9870-8

Cited by 6 publications

(5 citation statements)

References 43 publications

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“…By Lemma 3.8 of [27], we obtain the existence of ðg k Þ k ! 1 & C 1 0 ðR þ Þ such that the quantity in (36) is equal to 1 for all k and the quantity in (37) converges to 4 n 2 as k ! 1.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Hardy type inequalities with spherical derivatives

Bez

Machihara

Ozawa

2020

SN Partial Differ. Equ. Appl.

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A Hardy type inequality is presented with spherical derivatives in R n with n ! 2 in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments. Mathematics Subject Classification Primary 26D10 Á Secondary 35A23 Á 46E35 This article is part of the section ''Theory of PDEs'' edited by Eduardo Teixeira.

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Section: Proof Of Theoremmentioning

confidence: 99%

Hardy type inequalities with spherical derivatives

Bez

Machihara

Ozawa

2020

SN Partial Differ. Equ. Appl.

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“…Sometimes the improved versions of different inequalities, or remainder estimates, are called stability of the inequality if the estimates depend on certain distances: see, e.g., [BJOS16] for stability of trace theorems, [CFW13] for stability of Sobolev inequalities, etc.…”

Section: Moreovermentioning

confidence: 99%

Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for 𝐿^{𝑝}-weighted Hardy inequalities

Ruzhansky

Сураган

Yessirkegenov

2018

Trans. Amer. Math. Soc. Ser. B

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Abstract. In this paper we give an extension of the classical Caffarelli-KohnNirenberg inequalities: we show that for 1 < p, q < ∞, 0 < r < ∞ with , where the constantMoreover, we also obtain anisotropic versions of these inequalities which can be conveniently formulated in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for L p -weighted Hardy inequalities on homogeneous groups, which are also new in the Euclidean setting of R n . The critical Hardy inequalities of logarithmic type and uncertainty type principles on homogeneous groups are obtained. Moreover, we investigate another improved version of L p -weighted Hardy inequalities involving a distance and stability estimate. The relation between the critical and the subcritical Hardy inequalities on homogeneous groups is also investigated. We also establish sharp Hardy type inequalities in L p , 1 < p < ∞, with superweights, i.e., with the weights of the formallowing for different choices of α and β. There are two reasons why we call the appearing weights the superweights: the arbitrariness of the choice of any homogeneous quasi-norm and a wide range of parameters.

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“…When (d, β + , β − ) = (2, 1 4 , 1 4 ) we have C = 4π and the function M is identically equal to 1 [0,1] ; hence extremisers exist. To give an identification of the class of extremisers, note that (7.5) holds if and only if T = F −1 m ρf is a bounded operator L 2 → L 2 , with m(ξ, τ ) = (|ξ| 2 − |τ | 2 ) 1/4 , and one can show that the class of extremisers for T coincides with the image under T * of the class of extremisers for the dual inequality T * : L 2 → L 2 (see, for example, [5]). By (4.1) it follows that g is an extremiser for T * if and only if g is an extremiser for the multiplier estimate F −1 (m −1 g) 2 2 ≤ g 2 2 .…”

Section: Symmetric Datamentioning

confidence: 99%

Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

Bennett

Bez

Gutiérrez

et al. 2018

Journal de Mathématiques Pures et Appliquées

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We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator ρ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents q and r, we find a characterisation of the exponents β + and β − , except possibly for an endpoint case, for which DHere, D + and D − are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on L 2 . We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp ℓ p decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.

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Stability of Trace Theorems on the Sphere

Cited by 6 publications

References 43 publications

Hardy type inequalities with spherical derivatives

Hardy type inequalities with spherical derivatives

Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for 𝐿^{𝑝}-weighted Hardy inequalities

Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

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