Master of Science / Mathematics / Nathan Albin / Pietro Poggi-Corradini / This report explores singular integration, both real and complex, focusing on the the Cauchy type integral, culminating in the proof of generalized Sokhotski-Plemelj formulae and the applications of such to a Riemann-Hilbert problem.

We show that the singular numbers of single layer potentials on smooth curves asymptotically behave like O(1/n). For the curves with singularities, as long as they contain a smooth sub-arc, the resulting single layer potentials are never trace-class. We provide upper bounds for the operator and the Hilbert-Schmidt norms of single layer potentials on smooth and chord-arc curves. Regarding the injectivity of single layer potentials on planar curves, we prove that among single layer potentials on dilations of a given curve, only one yields a non-injective single layer potential. A criterion for injectivity of single layer potentials on ellipses is given. We establish an isoperimetric inequality for Schatten p−norms of logarithmic potentials over quadrilaterals and its analogue for Newtonian potentials on parallelepipeds.