Representing functions as difference of Steiner potentials (Q1397536)

In connection to a problem of Blaschke the class of subharmonic functions of the type \[ H(x)= \int_{S^{m-1}}|\langle x,w\rangle|\,d\nu(w), \] was investigated, where \(\nu\) is positive measure. The author studies the more general class of subharmonic functions in the whole space \(\mathbb{R}^m\) having the representation \[ u(x)=\int_{[0,R]\times S^{m-1}}|t-\langle x,w\rangle| \,d\nu(t,w)+H_R(x), \] where \(R\) is an arbitrary positive number and \(H_R\) is a harmonic function. He calls such functions Steiner potentials. In 1996 he found a criterium for \(u\) to be a Steiner potential. Here he proves that any function of the class \(C^{\infty}\) is a difference of Steiner potentials of class \(C^{\infty}\).