Inversion of parabolic potentials with \(L_ p\)-densities
The authors show a general construction of inverse operators (called parabolic hypersingular integrals) for the parabolic potentials \((H^{\alpha}\phi)(x,t)\), (\({\mathcal H}^{\alpha}\phi)(x,t)\), \(\alpha >0\). The latter are defined by \[ F[H^{\alpha}\phi](\xi,\tau)=(| \xi |^ 2+i\tau)^{-\alpha /2}F[\phi](\xi,\tau), \] \[ F[{\mathcal H}^{\alpha}\phi](\xi,\tau)=(1+| \xi |^ 2+i\tau)^{-\alpha /2}F[\phi](\xi,\tau), \] where F[\(\phi\) ] is the Fourier transform of \(\phi\) (x,t). (The authors give references for applications of such potentials.) The construction of the inverse of \(H^{\alpha}\phi\) and \({\mathcal H}^{\alpha}\phi\) with \(L_ p\)-densities gives explicit formulae for positive powers of the differential operators \(-\Delta_ x+\partial /\partial t\) and \(I-\Delta_ x+\partial /\partial t\) in \(L_ p(R^{n+1})\).
positive powers of the differential operators
scientific article; zbMATH DE number 4023803