Related papers: Fractional Laplacians and Nilpotent Lie Groups
Through duality it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
This paper is a survey of recent results on comparison of various fractional Laplacians prepared for Proceedings of ICM2022.
In this note we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems and the corresponding Euler-Lagrange and Hamilton equations are analyzed.…
The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the…
We establish partial semigroup property of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order…
We discuss the recent developments of semi-classical and micro-local analysis in the context of nilpotent Lie groups and for sub-elliptic operators. In particular, we give an overview of pseudo-differential calculi recently defined on…
We present a method derived from Laplace transform theory that enables the evaluation of fractional integrals. This method is adapted and extended in a variety of ways to demonstrate its utility in deriving alternative representations for…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…
We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential…
This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to…
We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant…
We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular…
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation…
This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on…
We review some recent results on eigenvalues of fractional Laplacians and fractional Schr\"odinger operators. We discuss, in particular, Lieb-Thirring inequalities and their generalizations, as well as semi-classical asymptotics.
In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in…
We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena. Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations,…
Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…