van der Corput lemma, Bessel function, asymptotic estimate, wave equation, oscillatory integrals


In this article, we study analogues of the van der Corput lemmas [19] involving Bessel functions. In harmonic analysis, one of the most important estimates is the van der Corput lemma, which is an estimate of the oscillatory integrals. This estimate was first obtained by the Dutch mathematician Johannes Gaultherus van der Corput. Van der Corput interested in the behavior for large positive λ of the oscillatory integral R b a e iλφ(x)ψ(x)dx, where φ is a real-valued smooth function (the phase) and ψ is complex valued smooth function (amplitude). In case a = −∞, b = +∞, it is assumed that ψ has a compact support in R. In our case we replace the exponential function with the Bessel functions, to study oscillatory integrals appearing in the analysis of wave equation with singular damping. More specifically, we study integral of the form I(λ) = R b a Jn(λφ(x))ψ(x)dx for the range n = 0, where ψ ∈ C and smooth, and λ is a positive real number that can vary.The generalisations of the van der Corput lemma is proved. As an application of the above results, the generalised Riemann-Lebesgue lemma is considered.


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