Finite-sheeted Cauchy operator at rational corners

Abstract

We study Cauchy singular integral operators on planar wedges whose opening angle is a rational multiple of $π$. For $θ=pπ/q$, the covering $w=ζ^q$ yields an exact finite-sheeted factorization of the wedge Cauchy transform into $2q$ interval Cauchy transforms with explicit algebraic recombination coefficients. The factorization is formulated on weighted conormal Hölder spaces. We prove that the lifting operator preserves conormal order, lowers the Hölder exponent from $β$ to $β/q$, and has sharp $\ell^1$ sheet norm $q$. Combining this operator factorization with a Mellin model for interval Cauchy transforms, we derive a mode-by-mode propagation rule for polyhomogeneous endpoint expansions. Nonresonant powers preserve their logarithmic order, while integer exponents raise it by one. The results also give a local singular decomposition for Cauchy operators on piecewise analytic curves with rational corner angles.