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Restriction estimates via the derivatives of the heat semigroup and connexion with dispersive estimates
Classical Analysis and ODEs · Submitted 12 Apr 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We derive a characterization for the restriction estimate $\| dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} - \frac{1}{p'}) -1}$ in terms of higher order derivatives of the semigroup $e^{-tH}$.
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We consider an abstract non-negative self-adjoint operator $H$ on an $L^2$-space. We derive a characterization for the restriction estimate $\| dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} - \frac{1}{p'}) -1}$ in terms of higher order derivatives of the semigroup $e^{-tH}$. We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. We also prove $L^p-L^{p'}$ estimates for the derivatives of the spectral resolution of $H$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider an abstract non-negative self-adjoint operator $H$ on an $L^2$-space. | background 0.92 | 0.28 | 0.02 | asserted 0.85 |
| s1 | We derive a characterization for the restriction estimate $\| dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} - \frac{1}{p'}) -1}$ in terms of higher order derivatives of the semigroup $e^{-tH}$. | result 0.85 | 0.96 | 0.75 | asserted 1.00 |
| s2 | We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. | contribution 0.80 | 0.83 | 0.13 | asserted 1.00 |
| s3 | We also prove $L^p-L^{p'}$ estimates for the derivatives of the spectral resolution of $H$. | result 0.88 | 0.95 | 0.14 | asserted 1.00 |
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Original abstract
We consider an abstract non-negative self-adjoint operator $H$ on an $L^2$-space. We derive a characterization for the restriction estimate $\| dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} - \frac{1}{p'}) -1}$ in terms of higher order derivatives of the semigroup $e^{-tH}$. We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. We also prove $L^p-L^{p'}$ estimates for the derivatives of the spectral resolution of $H$.
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Compares with 1 paper
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The equivalence of ([ref]) and ([ref]) is in the spirit of Proposition 2.4, Section 2.2 in [ref] .
Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means · Introduction and main results · 0.85
Cites as background 2 papers
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We refer to the introductions of the papers [ref] and [ref] for more details about this.
Restriction and spectral multiplier theorems on asymptotically conic manifolds · Introduction and main results · 1.00
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We refer to the introductions of the papers [ref] and [ref] for more details about this.
Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means · Introduction and main results · 1.00
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