Balls and metrics defined by vector fields I: Basic properties

Nagel, Alexander; Stein, Elias M.; Wainger, Stephen

doi:10.1007/bf02392539

Cited by 854 publications

(940 citation statements)

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“…By the way, we note that the results of [37], [33] about subelliptic metrics, balls, etc., play a minor role in this paper: these are used only in §1.4, to formulate the VMO assumption on the coefficients in terms of the "natural" metric (instead of the metric which is "natural" for the proof).…”

Section: Introduction and Main Resultsmentioning

confidence: 91%

“…It turns out that the metric balls are well shaped to describe geometrical properties related to the operator. For instance, the fundamental solution of L can be estimated pointwise through the size of these metric balls (see Sanchez-Calle, '84, [37], and Nagel-Stein-Weinger, '85, [33]). Moreover, these metric balls define a structure of homogeneous space, in the sense of Coifman-Weiss, '71, [15], so that many tools from singular integrals and real-variable theory can be naturally employed in the study of hypoelliptic operators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Since our final goal is to study functions and differential operators defined on R n , it is desirable to express also the VMO assumption on the coefficients in terms of the structure induced in R n by the original fields X i . This is possible in view of the results of [37], [33], who have studied the relationship between the geometry of balls defined by the X i 's in R n and theX i 's in R N . We now recall some known results on this subject.…”

Section: Proof Of (C)mentioning

confidence: 99%

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

Bramanti

Brandolini

1999

Trans. Amer. Math. Soc.

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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

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Section: Introduction and Main Resultsmentioning

confidence: 91%