Applications of the Funk–Hecke theorem to smoothing and trace estimates

Bez, Neal; Saito, Hiroki; Sugimoto, M.

doi:10.1016/j.aim.2015.08.025

Cited by 9 publications

(22 citation statements)

References 16 publications

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“…(this latter paper also addresses the Hölder continuity of the trace and its behaviour for small values of r ). Our Theorem 3.1 partly overlaps with [, Theorem 1.4] and [, pp. 1779, 1792].…”

Section: Trace Theorems With Explicit Uniform Boundsmentioning

confidence: 63%

“…In particular it does not rely on assumptions on the Fourier transform of

1 / μ

. Some overlap with previous results, notably in and , is addressed in Remark . Our wish to find a uniform resolvent estimate for the free Dirac operators with mass

m \geq 0

in any dimensions

n \geq 2

motivated us to replace the weight primarily employed in such estimates,

\frac{1}{μ (τ)} = {(() 1 + τ^{2})}^{- s} false(τ \geq 0; s > 1 / 2 false),

(note that μ is the weight in the inhomogeneous Sobolev space

H^{s} false({boldR}^{n} false)

and that the Fourier transform of

1 / μ

is the Bessel potential of order s ), by the function

\frac{1}{μ (τ)} = τ^{- 2 t} {(1 + τ^{2})}^{- s} false(τ > 0; 0 \leq t < 1 / 2, t + s > 1 / 2 false) .

It will be essential to have an explicit bound on the trace that does not depend on the radius r of the sphere; moreover one needs to know that the trace tends to zero as r tends to zero [Remark 3.3].…”

Section: Introductionmentioning

confidence: 87%

“…Our Theorem says that

A (n, w)

is indeed finite when w is non‐increasing with

0true {false∥ w false∥}_{1} = \int_{0}^{\infty} w < \infty .

In fact, in this case

A false(n, w false) \leq {∥ w ∥}_{1} max [a_{\frac{n - 2}{2}}, a_{\frac{n}{2}}],

where

a_{ν} : = {trueprefixsup}_{r > 0} r J_{ν}^{2} false(r false)

. Using different ideas, cases where

A (n, w)

is finite are established in (we refer in particular to their Theorem and its role in the proof of their Theorem 1.7) and in [, Theorem 1.1–4] (here the positivity of the Fourier transform of w is important). These papers also provide an extensive list of references.…”

Section: Sharp Uniform Bounds On Integrals Of Bessel Functionsmentioning

confidence: 99%

“…];

ν > 0

is assumed there, but the case

ν = 0

can easily be settled as well), which proves our claim. For a different treatment of this important case see [, p. 1778].…”

Section: Sharp Uniform Bounds On Integrals Of Bessel Functionsmentioning

confidence: 99%

“…In recent years there has been an extensive interest in the interplay between smoothness and decay estimates for solutions of certain (pseudo)differential equations and restriction theorems for the Fourier transform onto the sphere in

{boldR}^{n}

or more general smooth manifolds with certain curvature properties (see and the literature cited there). The results we present here depend on a bound, given in Theorem , of an integral over Bessel functions which is uniform with respect to the order of the Bessel functions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Kalf

Okaji

Yamada

2018

Mathematische Nachrichten

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Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for some smoothing estimates as well as for the L2 restrictions of Fourier transforms onto spheres in boldRn which are independent of the radius of the sphere. For more special weights these restrictions are shown to be Hölder continuous with a Hölder constant having this independence as well. To illustrate the use of these results a uniform resolvent estimate of the free Dirac operator with mass m≥0 in dimensions n≥2 is derived.

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Section: Trace Theorems With Explicit Uniform Boundsmentioning

confidence: 63%

“…In particular it does not rely on assumptions on the Fourier transform of

1 / μ

. Some overlap with previous results, notably in and , is addressed in Remark . Our wish to find a uniform resolvent estimate for the free Dirac operators with mass

m \geq 0

in any dimensions

n \geq 2

motivated us to replace the weight primarily employed in such estimates,

\frac{1}{μ (τ)} = {(() 1 + τ^{2})}^{- s} false(τ \geq 0; s > 1 / 2 false),

(note that μ is the weight in the inhomogeneous Sobolev space

H^{s} false({boldR}^{n} false)

and that the Fourier transform of

1 / μ

is the Bessel potential of order s ), by the function

\frac{1}{μ (τ)} = τ^{- 2 t} {(1 + τ^{2})}^{- s} false(τ > 0; 0 \leq t < 1 / 2, t + s > 1 / 2 false) .

Section: Introductionmentioning

confidence: 87%

“…Our Theorem says that

A (n, w)

is indeed finite when w is non‐increasing with

0true {false∥ w false∥}_{1} = \int_{0}^{\infty} w < \infty .

In fact, in this case

A false(n, w false) \leq {∥ w ∥}_{1} max [a_{\frac{n - 2}{2}}, a_{\frac{n}{2}}],

where

a_{ν} : = {trueprefixsup}_{r > 0} r J_{ν}^{2} false(r false)

. Using different ideas, cases where

A (n, w)

Section: Sharp Uniform Bounds On Integrals Of Bessel Functionsmentioning

confidence: 99%

“…];

ν > 0

is assumed there, but the case

ν = 0

can easily be settled as well), which proves our claim. For a different treatment of this important case see [, p. 1778].…”

Section: Sharp Uniform Bounds On Integrals Of Bessel Functionsmentioning

confidence: 99%

{boldR}^{n}

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Kalf

Okaji

Yamada

2018

Mathematische Nachrichten

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show abstract

Optimal Constants of Smoothing Estimates for the 2D Dirac Equation

Ikoma

2022

J Fourier Anal Appl

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Recently, Ikoma (2022) considered optimal constants and extremisers for the 2-dimensional Dirac equation using the spherical harmonics decomposition. Though its argument is valid in any dimensions d ≥ 2, the case d ≥ 3 remains open since it leads us to too complicated calculation: determining all eigenvalues and eigenvectors of infinite dimensional matrices. In this paper, we give optimal constants and extremisers of smoothing estimates for the 3-dimensional Dirac equation. In order to prove this, we construct a certain orthonormal basis of spherical harmonics. With respect to this basis, infinite dimensional matrices actually become block diagonal and so that eigenvalues and eigenvectors can be easily found.

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Stability of Trace Theorems on the Sphere

et al. 2017

Self Cite

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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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Applications of the Funk–Hecke theorem to smoothing and trace estimates

Cited by 9 publications

References 16 publications

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Optimal Constants of Smoothing Estimates for the 2D Dirac Equation

Stability of Trace Theorems on the Sphere

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