This paper applies a modern method of singularity resolution in algebraic geometry
to resolving singularities of integral operators in Fourier analysis. This is
achieved by introducing a method of mixed variables that is equivalent to changing
coordinates for integral operators. We decompose the integral operator into dyadic
pieces via monomial transforms and the mixed-variable method so as to obtain its
sharp estimates on different domains. These sharp estimates can be written in an
elegant form in terms of continued fractions.
Primary Subjects: 42B20, 47G10, 35S30, 35S05, 45P05, 47B38
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