A formula for the non-integer powers of the Laplacian (Q1284786)

An elementary formula for the non-integer powers of the Laplace operator \(\Delta^\alpha\), \(0<\alpha<1\) acting on the Sobolev space \(H^\alpha({\mathbb R}^n;X)\) with \(X = {\mathbb R}^m\) or \(X = {\mathbb C}^m\) was earlier known. In the paper the author gives a similar construction in a more general case, namely, for the operator \(\Delta^{\alpha}\), \(k<\alpha<k+1\), \(k=0,1,2,\dots\) on the Sobolev space \(H^\alpha({\mathbb R}^n;X)\), \(k<\alpha<k+1\), \(k=0,1,2,\dots\). The applications of these formulas are expected in the establishing of the elliptic nature of some pseudo-differential operators arising from the study of energies of knotted loops in space.