Fractional thoughts

Abstract

This work was supported in part by a grant "Progetti d'Ateneo, 2013", University of Padova Contents 1. Introduction 2. The fractional Laplacean 3. Maximum principle, Harnack inequality and Liouville theorem 4. A brief interlude about very classical stuff 5. Fourier transform, Bessel functions and (−∆) s 6. The fractional Laplacean and Riesz transforms 7. The fractional Laplacean of a radial function 8. The fundamental solution of (−∆) s 9. The nonlocal Yamabe equation 10. Traces of Bessel processes: the extension problem 11. Fractional Laplacean and subelliptic equations 12. Hypoellipticity of (−∆) s 13. Regularity at the boundary 14. Monotonicity formulas and unique continuation for (−∆) s 15. Nonlocal Poisson kernel and mean-value formulas 16. The heat semigroup P (s) t = e t(−∆) s 17. Bochner's subordination: from P t to (−∆) s 18. More subordination: from P t to P (s) t 19. A chain rule for (−∆) s 20. The Gamma calculus for (−∆) s 21. Are there nonlocal Li-Yau inequalities? 22. A Li-Yau inequality for Bessel operators 23. The fractional p-Laplacean Bibliography iii

Harmonic extension technique for non-symmetric operators with completely monotone kernels

“…This representation of fractional powers of the Laplace operator was studied already in 1960s (see [32,33]), and was definitely stated in the above form by Caffarelli and Silvestre in [7]. We refer to Section 10 in the survey article [15] for further discussion.…”

Li--Yau Inequalities for Dunkl Heat Equations

“…Recently, in [27], sharp Li-Yau inequalities for the Laplace-Beltrami operator on hyperbolic spaces were obtained by employing the explicit formula for the corresponding heat kernel. Very recently, in [26], similar to the idea of [27], the Li-Yau inequality in the sense of (1.1) for the fractional Laplacian has been proved; however, the Li-Yau inequality of gradient type in the sense of (1.2) has not been mentioned, where the "gradient" should be understood as the the carré du champ operator induced by the fractional Laplacian (see also the conjectures at the end of [13,Section 21]). We should mention that there are works on the Li-Yau inequality in the setting of graphs via various curvature-dimension conditions in the sense of Barky-Emery [3]; see e.g.…”

Half-Harmonic Gradient Flow: Aspects of a Non-Local Geometric PDE

“…The presentation of this subsection follows [12] and we refer to it and the references mentioned therein for details.…”

“…The fractional heat semigroup may be used for various things, such as a formula for the fractional Laplacians by subordination, see [12]. We are more interested in the immediate regularity properties.…”