Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther. "Estimates with global range for oscillatory integrals with concave phase." Colloquium Mathematicae 91.2 (2002): 157-165. <http://eudml.org/doc/283522>.

@article{BjornGabrielWalther2002,
abstract = {We consider the maximal function $||(S^\{a\}f)[x]||_\{L^\{∞\}[-1,1]\}$ where $(S^\{a\}f)(t)^\{∧\}(ξ) = e^\{it|ξ|^a\}f̂(ξ)$ and 0 < a < 1. We prove the global estimate $||\{S^\{a\}f\}||_\{L²(ℝ,L^\{∞\}[-1,1])\} ≤ C||f||_\{H^\{s\}(ℝ)\}$, s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.},
author = {Bjorn Gabriel Walther},
journal = {Colloquium Mathematicae},
keywords = {oscillatory integral; summability of Fourier integral; maximal function; Sobolev regularity},
language = {eng},
number = {2},
pages = {157-165},
title = {Estimates with global range for oscillatory integrals with concave phase},
url = {http://eudml.org/doc/283522},
volume = {91},
year = {2002},
}

TY - JOUR
AU - Bjorn Gabriel Walther
TI - Estimates with global range for oscillatory integrals with concave phase
JO - Colloquium Mathematicae
PY - 2002
VL - 91
IS - 2
SP - 157
EP - 165
AB - We consider the maximal function $||(S^{a}f)[x]||_{L^{∞}[-1,1]}$ where $(S^{a}f)(t)^{∧}(ξ) = e^{it|ξ|^a}f̂(ξ)$ and 0 < a < 1. We prove the global estimate $||{S^{a}f}||_{L²(ℝ,L^{∞}[-1,1])} ≤ C||f||_{H^{s}(ℝ)}$, s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.
LA - eng
KW - oscillatory integral; summability of Fourier integral; maximal function; Sobolev regularity
UR - http://eudml.org/doc/283522
ER -