Abstract
In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays some challenging features. In the opening of his 1967 hypoellipticity paper Hörmander discussed a general class of degenerate Ornstein–Uhlenbeck operators that includes Kolmogorov’s as a special case. In this note we combine semigroup theory with a nonlocal calculus for these hypoelliptic operators to establish new inequalities of Hardy–Littlewood–Sobolev type in the situation when the drift matrix has nonnegative trace. Our work has been influenced by ideas of E. Stein and Varopoulos in the framework of symmetric semigroups. One of our objectives is to show that such ideas can be pushed to successfully handle the present degenerate non-symmetric setting.
Highlights
Sobolev inequalities occupy a central position in analysis, geometry and physics
Hardy–Littlewood–Sobolev condition, and the if and only if character is connected with the interplay between the differential operator ∇ and the homogeneous structure (Euclidean dilations) of the ambient space
In this paper we are concerned with a scale of global inequalities such as the one above for the following class of second-order partial differential equations in RN+1, K u = A u − ∂t u d=ef tr(Q∇2u) + B X, ∇u − ∂t u = 0, (1.1)
Summary
Sobolev inequalities occupy a central position in analysis, geometry and physics. Typically, in such a priori estimates one is able to control a certain Lq norm of a derivative of a function in terms of a L p norm of derivatives of higher order. For the class of Hörmander operators i Xi Xi and i Xi Xi −∂t , where Xi are smooth vector fields in RN satisfying Hörmander’s finite rank condition These operators, which are seemingly close relatives of (1.1), have a well-understood underlying geometry which stands in stark contrast with the one discussed in the present paper. For kinetic Fokker-Planck equations (where in particular we have X = (v, x), with v indicating velocity and x position), we mention the recent papers [3,32] In the former the authors prove a local gain of integrability for nonnegative sub-solutions via a nontrivial adaptation of the so-called velocity averaging method.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
