Linear and nonlinear Bessel potentials associated with the Poly-axially operator

In this work, we will study a Bessel potential spaces W-alpha(s,p) (R-+(n)), (s is an element of R, 1 <= p < +infinity, alpha = (alpha(1), ... , alpha(n)) is an element of R-n, alpha(1) > - 1/2, ... , alpha(n) > - 1/2) associated with the Poly-axially operator Delta(alpha) and their properties. We define the Sobolev type spaces E-alpha(m,p) (R-+(n)) (m is an element of N, 0 is an element of N, 1 <= p < +infinity) and we prove an analogue of Calderon's theorem for Fourier-Bessel transform. We define the nonlinear Bessel potentials associated with a measure. and the L-alpha(p)-capacity. Finally, we establish a classical problem of existence of an extremal function for the L-alpha(p)-capacity.