𝐿² boundedness of highly oscillatory integrals on product domains

Prestini, Elena

doi:10.1090/S0002-9939-1988-0962818-0

$L^ 2$ boundedness of highly oscillatory integrals on product domains
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Proc. Amer. Math. Soc. 104 (1988), 493-497 Request permission

Abstract:

We prove ${L^2}$ boundedness of the oscillatory singular integral \[ Tf(x,y) = \iint \limits _{{D_y}} {\frac {{\operatorname {exp} (2\pi iN(y)x’)}}{{x’y’}}}f(x - x’,y - y’)dx’dy’\] where $N(y)$ is an arbitrary integer-valued ${L^\infty }$ function and where nothing is assumed on the dependency upon $y$ of the domain of integration ${D_y}$. We also prove ${L^2}$ boundedness of the corresponding maximal opertaor. Operators of this kind appear in a problem of a.e. convergence of double Fourier series.

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