Resolution of operator singularities via the mixed-variable method

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.