Fundamental solutions and local solvability for nonsmooth Hörmander’s operators

Abstract

We consider operators of the form L = n i=1 X 2 i + X0 in a bounded domain of R p where X0, X1, . . . , Xn are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method we construct a local fundamental solution γ for L and provide growth estimates for γ and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that γ also possesses second derivatives, and we deduce the local solvability of L, constructing, by means of γ, a solution to Lu = f with Hölder continuous f . We also prove C 2,α X,loc estimates on this solution.

The Role of Fundamental Solution in Potential and Regularity Theory for Subelliptic PDE

“…The additional difficulty here is due to the fact that a function differentiable in the direction of the vectors X i will not necessarly be differentiable in the direction X i,w . We explicitly note that not even the more recent results of Bramanti Brandolini Manfredini Pedroni [29] could allow to obtain the result. On the contrary a completely new approach to singular integrals has been introduced in order to deal with these non linear vector fields.…”

Chapter 12 Hardy and Rellich Inequalities for Sums of Squares of Vector Fields

“…Concerning (E3), the validity of (A + y ) was shown by Bramanti, Brandolini, Manfredini and Pedroni [4,Theorem 4.8 and Theorem 5.9]. The validity of (1.3) and the strong admissibility for any domain with piecewise smooth simple boundary will be shown in Section 5.…”

“…For both cases (E2) and (E3) we present the following explicit example (see [4,Section 6]): In R 3 let N = 2 and let…”

The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method

“…Global fundamental solutions, without an explicit representation, are given e.g., in Folland [39]; Nagel, Ricci, Stein [56]; the second-named author and Lanconelli [21]. Existence results without an exact representation are also available, based on the so-called Levi's parametrix method [53]: the interested reader is referred to e.g., [29,42,48]; concerning Levi's parametrix method, we also highlight the recent paper by Bramanti, Brandolini, Manfredini, Pedroni [30], where a local Lifting technique and a local saturation argument are also applied. See also the paper [35] by Citti, Manfredini, where it is exploited a local Lifting technique involving hypoelliptic Hörmander operators and their local fundamental solutions.…”