Ricerc@Sapienza

A general class of probability density functions (1 (‖x‖) )γ β u(x, t) = Ct−αd − ctα, x ∈ Rd,t >0, + is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions u(x, t) are proposed. In the one-dimensional case, by means of this approach, u(x, t) can be connected with the wave propagation.

In \cite{LM} we considered a nontrivial example of eigenfunction in the sense of distribution for the planar Fourier transform. Here a method to obtain other eigenfunctions is proposed.
Moreover we consider positive homogeneous distributions in \(\R^n\) of order \(-n/2\). It is shown that \({F(\om)}{|\bx|^{-n/2}}, |\om|=1\) is an eigenfunction in the sense of distribution of the Fourier transform if and only if \(F(\om)\) is an eigenfunction of a certain singular integral operator on the unit sphere of \(\R^n\).