Remarks on potential theory

Abstract

Although some of the results of the present note hold for potential theories constructed with a variety of kernels, we are interested only in the applications to Bessel potentials [2], Riesz potentials and the logarithmic potential. Accordingly we state everything in the terminology of Bessel potentials on the space ~, since the results are then easily extended to other classes. The kernel G~,(x,y) is defined on R for all real a by G~(x,y) = G~,(xy), where G~(x) = [2 (=+~-z'j~ ~t n12 F(a/2)]l ]xl (~-n)~2 g(=_~)/2 ( Ix]) , the function K,(z) being the modified Bessel function of the third kind. We remark tha t the kernel vanishes identically when a is a non-positive even integer. I n a neighborhood of the origin G~(x) is equivalent to Ix ]~= if a 0 form the space P~ of all functions which coincide except for a set of 2a-capaci ty zero with convolutions of the form u = G~,-)e/, where / is in L*; the integral exists, except, perhaps, for a set of the corresponding capaci ty zero. The norm of u in P~ equals the L 2 norm of the corresponding / and P is a Hilbert space which also appears as the perfect functional completion of the space of all (Bessel) potentials of order 2:r of measures of finite 2a energy. In contradistinct ion to Riesz potentials, the Bessel potentials are always L 2 functions, and we have the following convenient formula for the norm in terms of the Fourier transform: