The extension problem for fractional powers of higher order of some evolutive operators

TL;DR

This thesis extends the fractional powers of the heat operator to higher orders using an extension problem approach, generalizing the Caffarelli-Silvestre method and employing evolutive semigroups for analysis.

Contribution

It introduces a generalized extension problem for higher-order fractional heat operators, connecting them to Dirichlet-to-Neumann maps and expanding analytical tools beyond Fourier transform reliance.

Findings

Generalization of the Caffarelli-Silvestre extension to higher-order fractional operators.

Establishment of a link between the extension problem and evolutive semigroups.

Framework applicable even without Fourier transform availability.

Abstract

This thesis studies the extension problem for higher-order fractional powers of the heat operator $H=\Delta-\partial_t$ in $\mathbb{R}^{n+1}$. Specifically, given $s>0$ and indicating with $[s]$ its integral part, we study the following degenerate partial differential equation in the thick space $\mathbb{R}^{n+1}\times \mathbb{R}_y^+$, \begin{equation} \label{a:1} \mathscr{H}^{[s]+1}U= \left( \partial_{yy} +\frac{a}{y}\partial_y +H \right)^{[s]+1}U=0. \quad \quad (1) \end{equation} The connection between the Bessel parameter $a$ in (1) and the fractional parameter $s>0$ is given by the equation \begin{equation*} a= 1-2(s-[s]). \end{equation*} When $s\in(0,1)$ this equation reduces to the well-known relation $a=1-2s$, and in such case (1) becomes the famous Caffarelli-Silvestre extension problem. Generalising their result, in this thesis we show that the nonlocal operator $(-H)^{\,s}$ can be realised as the Dirichlet-to-Neumann map associated with the solution $U$ of the extension equation (1). In this thesis we systematically exploit the evolutive semigroup $\{P_{\tau}^H \}_{\tau>0}$, associated with the Cauchy problem \begin{equation*} \begin{cases} \partial_{\tau}u-Hu=0 u((x,t),0)=f(x,t). \end{cases} \end{equation*} This approach provides a powerful tool in analysis, and it has the twofold advantage of allowing an independent treatment of several complex calculations involving the Fourier transform, while at same time extending to frameworks where the Fourier transform is not available.