𝐿² boundedness of highly oscillatory integrals on product domains

Abstract

ABSTRACT. We prove L2 boundedness of the oscillatory singular integral r/(x, y) = H ««rty ^ f(x -x', y -y') dx' dy'where N(y) is an arbitrary integer-valued L°° function and where nothing is assumed on the dependency upon y of the domain of integration Dy. We also prove L2 boundedness of the corresponding maximal opertaor.Operators of this kind appear in a problem of a.e. convergence of double Fourier series.