Abstract
In this paper we consider the uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4 in two variables. The obtained estimate is sharp and the result is an analogue of the more general theorem of Karpushkin (Proc I.G.Petrovsky Seminar 9:3–39, 1983) for sufficiently smooth functions, thus, in particular, removing the analyticity assumption.
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Acknowledgements
This paper was supported in parts by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). The first author was also supported by EPSRC grant EP/R003025/2. The second author was supported by “El-yurt umidi”Foundation of Uzbekistan and partially supported in parts by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). The authors would like to thank the referees for numerous suggestions which greatly helped to improve the exposition.
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Michael Ruzhansky, Akbar Safarov and Gafurjan Khasanov contributed equally to the writing of this paper. Michael Ruzhansky, Akbar Safarov and Gafurjan Khasanov read and approved the final manuscript.
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Ruzhansky, M., Safarov, A.R. & Khasanov, G.A. Uniform estimates for oscillatory integrals with homogeneous polynomial phases of degree 4. Anal.Math.Phys. 12, 130 (2022). https://doi.org/10.1007/s13324-022-00747-w
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DOI: https://doi.org/10.1007/s13324-022-00747-w