# On Oscillatory Integral Operators in Higher Dimensions

# On Oscillatory Integral Operators in Higher Dimensions

This chapter discusses the progress made towards problems originating from Stein's seminal paper, “Some problems in harmonic analysis.” It is by now well-known that the mapping properties of Fourier restriction operators to hypersurfaces in **R**^{n} and their variable coefficient generalizations are intimately related to questions of a combinatorial nature. Over recent years there has been quite a bit of research around these underlying issues. In some way, it became interdisciplinary with connections towards geometric measure theory, the theory of finite fields, incidence geometry, and mathematical computer science. While the central original problems remain unsolved, this line of research has produced many new results of independent interest, though the chapter focuses primarily on developments around the theory of oscillatory integrals.

*Keywords:*
harmonic analysis, oscillatory integrals, Fourier restriction operators, higher dimensions, oscillatory integral operators

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