It is well--known that certain properties of continuous functions on the circle T, related to the Fourier expansion, can be improved by a change of variable, i.e., by a homeomorphism of the circle onto itself. One of the results in this area is the Jurkat--Waterman theorem on conjugate functions, which improves the classical Bohr--Pal theorem.  In the present work we propose a short and technically very simple proof of the Jurkat--Waterman theorem. Our approach yields a stronger result.

We compute the Coulomb branch of a multiloop quiver gauge theory for the quiver with a single vertex, r loops, one-dimensional framing, and dim V = 2. We identify it with a Slodowy slice in the nilpotent cone of the symplectic Lie algebra of rank r. Hence it possesses a symplectic resolution with 2r fixed points with respect to a Hamiltonian torus action. We also idenfity its flavor deformation with a base change of the full Slodowy slice.

We consider bounded analytic functions in domains generated by sets that have Littlewood--Paley property. We show that each such function is an lp -multiplier.