Hypersingular operators as positive fractional powers of some operators of mathematical physics

We wish to show that in terms of hypersingular integrals (or their modification), it is possible to construct operators inverse to some classical potential operators other than the Riesz or generalized Riesz potential. We consider

i) Bessel potentials Ba<p denned in (7.28); ii) the parabolic Jones-Sampson fractional potentials (see their definition

(9.53)-(9.54) below); hi) hyperbolic Riesz potentials I^<p (see definition (9.76) below).