“Van der Corput lemma (harmonic analysis)”的意思、由来-开放百科全书

词条

Van der Corput lemma (harmonic analysis)

释义

Sublevel set estimates
References

In mathematics, in the field of harmonic analysis,

the van der Corput lemma is an estimate for oscillatory integrals

named after the Dutch mathematician J. G. van der Corput.

The following result

is stated by E. Stein:^[1]

Suppose that a real-valued function is smooth in an open interval ,

and that for all .

Assume that either , or that

and is monotone for .

There is a constant , which does not depend on ,

such that

for any .

Sublevel set estimates

The van der Corput lemma is closely related to the sublevel set estimates

(see for example

^[2]),

which give the upper bound on the measure of the set

where a function takes values not larger than .

Suppose that a real-valued function is smooth

on a finite or infinite interval ,

and that for all .

There is a constant , which does not depend on ,

such that

for any

the measure of the sublevel set

is bounded by .

References

1. ^Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993. {{ISBN|0-691-03216-5}}
2. ^M. Christ, Hilbert transforms along curves, Ann. of Math. 122 (1985), 575--596

3 : Inequalities|Harmonic analysis|Fourier analysis

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。