Van Der Corput Lemma (harmonic Analysis)
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harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...
, the van der Corput lemma is an estimate for
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mathematician J. G. van der Corput. The following result is stated by E. Stein: Suppose that a real-valued function \phi(x) is smooth in an open interval (a, b), and that , \phi^(x), \ge 1 for all x \in (a, b). Assume that either k \ge 2, or that k = 1 and \phi'(x) is monotone for x \in \R. Then there is a constant c_k, which does not depend on \phi, such that : \bigg, \int_a^b e^\bigg, \le c_k\lambda^ for any \lambda \in \R.


Sublevel set estimates

The van der Corput lemma is closely related to the sublevel set estimates,M. Christ, ''Hilbert transforms along curves'', Ann. of Math. 122 (1985), 575–596 which give the upper bound on the measure of the set where a function takes values not larger than \epsilon. Suppose that a real-valued function \phi(x) is smooth on a finite or infinite interval I \subset \R, and that , \phi^(x), \ge 1 for all x \in I. There is a constant c_k, which does not depend on \phi, such that for any \epsilon \ge 0 the measure of the sublevel set \ is bounded by c_k\epsilon^{1/k}.


References

Inequalities (mathematics) Harmonic analysis Fourier analysis