𝐿^{𝑝} estimates for nonvariational hypoelliptic operators with 𝑉𝑀𝑂 coefficients

Abstract

Abstract. Let X 1 , X 2 , . . . , Xq be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain Ω ⊂ R n (n > q). We consider the differential operatorwhere the coefficients a ij (x) are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:for a.e. x ∈ Ω, every ξ ∈ R q , some constant µ. Moreover, we assume that the coefficients a ij belong to the space VMO ("Vanishing Mean Oscillation"), defined with respect to the subelliptic metric induced by the vector fields X 1 , X 2 , . . . , Xq. We prove the following local L p -estimate:for every Ω ⊂⊂ Ω, 1 < p < ∞. We also prove the local Hölder continuity for solutions to Lf = g for any g ∈ L p with p large enough. Finally, we prove L p -estimates for higher order derivatives of f , whenever g and the coefficients a ij are more regular. Introduction and main resultsLet X 0 , X 1 ,. . . , X q be a system of C ∞ real vector fields defined in R n (n ≥ q+1), that is, . . . , q; j = 1, 2, . . . , n), and letRecall that a linear differential operator P with C ∞ coefficients is said to be hypoelliptic in an open set Ω if, whenever the equation P u = f is satisfied, in the distributional sense, in the neighborhood of a point in Ω and f is C ∞ in that neighborhood, then u is also C ∞ in that neighborhood. A famous theorem proved by

Subelliptic Cordes Estimates in the Grušin Plane

“…We point out that the statement in our previous theorem is contained in Theorem 3.2 in [1]. However we provided our proof to show that in our (simpler) case there is no dependence on the constant c on the radius of the support of u. Theorem 3.2.…”

“…Now, using Theorem 8 and Corollary of Theorem 9, §14 in [20] (see also Lemma 2.9 and Theorem 2.10 in [1]) it follows…”

“…In [20] (see also [1]) it has been proved that for every test function a (for the existence of test functions the reader can consult e.g. [11]) there exist an operator of type two P * and q operators of type one S * j such that for every u ∈ C ∞ 0 (R n )…”

“…Subelliptic p-harmonic functions are weak solutions of the corresponding p-Laplace equation (1) − ∆ p X u = −X 1 |Xu| p−2 X 1 u − X 2 |Xu| p−2 X 2 u = 0 , in Ω . Our goal is to obtain new Talenti-Pucci type inequalities (see Remark 3.2 and Theorem 3.4) and use them to obtain regularity for the weak solution of the quasilinear equation (1). For example, we show that the following sharp inequality holds:…”

Estimates in generalized Morrey spaces for nondivergence degenerate elliptic operators with discontinuous coefficients

“…Bramanti and Brandolini in [5] proved L p estimates for nonvariational operators L = q i, j=1 a i j (x)X i X j with discontinuous coefficients and vector fields satisfying Hörmander's condition by following the original idea of [10] and applying Rothschild-Stein's technique of "lifting and approximating" [30] as well as Folland's results for the homogeneous situation. The line of the proof in [5] is: they apply the lifting theorem to the Hörmander's vector fields on a domain ⊂ R n (n > q) to obtain free vector fields on a larger domain˜ ⊂ R N (N > n) and lift the operator L to an operatorL on˜ ; then coefficients ofL are frozen at some point, getting an operator L 0 ; by the approximating theorem in [30],L 0 can be approximated by an operator L * 0 on a homogeneous group, for which a homogeneous fundamental solution exists; an explicit representation formula for X i X j f can be written by utilizing singular integrals and their commutators; by establishing L p estimates of singular integrals and commutators on spaces of homogeneous type, the authors obtained the L p estimates of the operator L. Bramanti and Brandolini in [2] studied also the L p estimates for operators with discontinuous coefficients on homogeneous groups. Estimates of BMO type for singular integrals on spaces of homogeneous type and Schauder estimates for parabolic nondivergence operators of Hörmander type have been disscussed by Bramanti and Brandolini (see [3,4]).…”