RECORD
Fourier transform of surface--carried measures of two-dimensional generic surfaces and applications
Analysis of PDEs · Submitted 28 Jul 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schr\"odinger operators on the cubic lattice.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. | contribution 0.74 | 0.96 | 0.76 | asserted 1.00 |
| s1 | The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. | contribution 0.55 | 0.91 | 0.15 | asserted 1.00 |
| s2 | We also obtain new results on the spectral and scattering theory of discrete Schr\"odinger operators on the cubic lattice. | result 0.57 | 0.95 | 0.13 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose characteristic surfaces satisfy the generic assumptions. We also obtain new results on the spectral and scattering theory of discrete Schr\"odinger operators on the cubic lattice.
Human review
References
How the literature uses it
1 arXiv paper cite this in 2 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 1 paper
-
(For a more detailed discussion and references to generic results, see, e.g., [ref] by Schippa and the first author.)
On the number and sums of eigenvalues of Schr\"odinger-type operators with degenerate kinetic energy · Fourier restriction and extension · 0.91
What it cites · 7 sentences without a confident label
Uses the method or tool of 2 papers
-
The expression is estimated as in [ref] by suitable decompositions in Fourier space and crucially depending on the Fourier restriction estimates for Proposition [ref]; see [ref] for $p(ξ) = |ξ|^α$.
Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces · Resolvent estimates · 0.97
-
We estimate the contribution of $β_22$ by properties of the Bessel kernel (cf. [ref] ) [equation] for $1 ≤p,q ≤∞$ and $0 ≤1/p - 1/q ≤α/3$ with the endpoints excluded for $α≤3$.
A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations · Resolvent estimates · 0.89
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.