Flow Monotonicity and Strichartz Inequalities

Bennett, Jonathan; Bez, Neal; Iliopoulou, Marina

doi:10.1093/imrn/rnu230

Cited by 14 publications

(28 citation statements)

References 33 publications

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“…In each case (1.4) and (1.5), the constant is optimal since there is equality when u 0 (x) = exp(−|x| 2 ). These sharp estimates were proved by Foschi [8] and also Hundertmark and Zharnitsky [10]; we also note that (1.5) follows from (1.2) in the case d = 2 by choosing u 0 = v 0 , which means we have a number of proofs of this sharp estimate (see also the proofs in [2] and [3], where the emphasis is on underlying heat-flow monotonicity phenomena).…”

Section: Introductionsupporting

confidence: 56%

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On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

Bennett

Bez

Jeavons

et al. 2017

J. Math. Soc. Japan

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Abstract. We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schrödinger equation, which give sharp control on |u| 2 in classical Sobolev spaces. In particular, we generalise their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon-Vega, via entirely different methods, by seeing them all as special cases of a one-parameter family of sharp estimates. The extremal functions are solutions of the MaxwellBoltzmann functional equation and hence Gaussian. For u 2 we argue that the natural analogous results involve certain dispersive Sobolev norms; in particular, despite the validity of the classical Ozawa-Tsutsumi estimates for both |u| 2 and u 2 in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.

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Section: Introductionsupporting

confidence: 56%

“…A new proof of the Ozawa-Tsutsumi estimate (1.2) was given in [3]. An advantage of this new proof was that it exposed an underlying heat-flow monotonicity phenomenon.…”

Section: Introductionmentioning

confidence: 99%

On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

Bennett

Bez

Jeavons

et al. 2017

J. Math. Soc. Japan

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“…holds for any f ∈ H s (R n ). The constant C is as in Theorem 1, is positive and optimal, and there exists an extremiser for (9). If in addition s = 1 then we have…”

Section: Remarksmentioning

confidence: 94%

“…(1) − I n 2 (1)K n 2 (1) where I µ and K µ are modified Bessel functions of the first kind of order µ. In this case, equality holds in (9) if and only if there exists c ∈ C and Y 1 ∈ H 1 such that |ξ| n−2…”

Section: Remarksmentioning

confidence: 99%

“…has an alternative interpretation as an estimate for the classical X-ray transform on R n+1 ; see [9] for discussion and further references regarding these estimates (and natural generalisations such as the Radon transform) from this perspective. In particular, it follows from the results of [22], [27], and [29] that the optimal constant in (17) is attained when…”

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confidence: 99%

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Stability of Trace Theorems on the Sphere

et al. 2017

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We prove stable versions of trace theorems on the sphere in L 2 with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the trace theorem into L q for q > 2, by combining a refined Hardy-Littlewood-Sobolev inequality on the sphere with a duality-stability result proved very recently by Carlen. Finally, we extend a local version of Carlen's duality theorem to establish local stability of certain Strichartz estimates for the kinetic transport equation.1 2 , R denotes the operation of restriction to S n−1 , and the function w : (0, ∞) → (0, ∞) is such that the Fourier transform w(| · |) makes sense

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Stability of Sharp Fourier Restriction to Spheres

Carneiro,

Negro,

e Silva

2024

J Fourier Anal Appl

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In dimensions $$d \in \{3,4,5,6,7\}$$ d ∈ { 3 , 4 , 5 , 6 , 7 } , we prove that the constant functions on the unit sphere $$\mathbb {S}^{d-1}\subset \mathbb {R}^d$$ S d - 1 ⊂ R d maximize the weighted adjoint Fourier restriction inequality $$\begin{aligned} \left| \int _{\mathbb {R}^d} |\widehat{f\sigma }(x)|^4\,\big (1 + g(x)\big )\,\textrm{d}x\right| ^{1/4} \leqslant \textbf{C} \, \Vert f\Vert _{L^2(\mathbb {S}^{d-1})}, \end{aligned}$$ ∫ R d | f σ ^ ( x ) | 4 ( 1 + g ( x ) ) d x 1 / 4 ⩽ C ‖ f ‖ L 2 ( S d - 1 ) , where $$\sigma $$ σ is the surface measure on $$\mathbb {S}^{d-1}$$ S d - 1 , for a suitable class of bounded perturbations $$g:\mathbb {R}^d \rightarrow \mathbb {C}$$ g : R d → C . In such cases we also fully classify the complex-valued maximizers of the inequality. In the unperturbed setting ($$g = \textbf{0}$$ g = 0 ), this was established by Foschi ($$d=3$$ d = 3 ) and by the first and third authors ($$d \in \{4,5,6,7\}$$ d ∈ { 4 , 5 , 6 , 7 } ) in 2015. Our methods also yield a new sharp adjoint restriction inequality on $$\mathbb S^7\subset \mathbb {R}^8$$ S 7 ⊂ R 8 .

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Flow Monotonicity and Strichartz Inequalities

Abstract: Abstract. We identify complete monotonicity properties underlying a variety of well-known sharp Strichartz inequalities in euclidean space.

Cited by 14 publications

References 33 publications

On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

Stability of Trace Theorems on the Sphere

Stability of Sharp Fourier Restriction to Spheres

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