An invariant Harnack inequality for a class of subelliptic operators under global doubling and Poincaré assumptions, and applications

Battaglia, Erika; Bonfiglioli, Andrea

doi:10.1016/j.jmaa.2017.11.044

Cited by 6 publications

(12 citation statements)

References 18 publications

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“…Once the results in Theorem 1.3 are established, we consider some possible applications. Firstly, in Section 7 we deal with potential-theoretic properties of L. Indeed, the estimates of Γ and the presence of a blowing-up pole (see (V) in Theorem 1.3) allow us to verify, for our operators L, all the axioms of potential theory required for the analysis contained in the series of papers [1,2,4,12]. Secondly, in Section 8 we shall show that the kernel k (x, y) = X x i X x j Γ (x; y) satisfies, globally in R n , the so-called standard estimates of singular integrals, together with a suitable cancelation property, with respect to both variables.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global estimates for the fundamental solution of homogeneous Hörmander operators

Biagi¹,

Bonfiglioli²,

Bramanti³

2019

Preprint

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Let L = m j=1 X 2 j be a Hörmander sum of squares of vector fields in R n , where any X j is homogeneous of degree 1 with respect to a family of non-isotropic dilations in R n . Then L is known to admit a global fundamental solution Γ(x; y), that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space R n × R p , equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of Γ, in terms of the Carnot-Carathéodory distance induced by X = {X 1 , . . . , Xm} on R n , as well as global pointwise (upper) estimates for the X-derivatives of any order of Γ, together with suitable integral representations of these derivatives. The least dimensional case n = 2 presents several peculiarities which are also investigated. Applications to the potential theory for L and to singular-integral estimates for the kernel X i X j Γ are also provided.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Global estimates for the fundamental solution of homogeneous Hörmander operators

Biagi¹,

Bonfiglioli²,

Bramanti³

2019

Preprint

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“…If the second inequality in (3) holds, the proof is completed. Thus we suppose S Ω (B θR , f ) < εαcγ p M0 which implies (4) S Ω (B r (ỹ), f ) < εαcγ p M 0 for every B r (ỹ) ⊆ B θR .…”

Section: Abstract Harnack Inequalitymentioning

confidence: 98%

“…We now define suitable sublevel sets of a specific functions g and h in which we are able to construct barriers to prove the critical density and the double ball property respectively. We construct these functions modifying the fundamental solution Γ(x, 0) = x 4 1 + 4x 2 2…”

Section: Application To Grushin Type Operatorsmentioning

confidence: 99%

“…We highlight that the d−diameter of the set Ω is required to be sufficiently small and the doubling and the Poincaré inequality are local conditions, so the Harnack inequality is obtained for balls with small enough radius and the constants appearing depend on the compact set fixed. Very recently, in [4] Battaglia and Bonfiglioli obtained an invariant non homogeneus Harnack inequality for solutions of a class of sub-elliptic operators in divergence form under global doubling and Poincaré assumptions but no restrictions on the diameter of the set Ω.…”

Section: Application To X-elliptic Operatorsmentioning

confidence: 99%

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Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces

Guidi¹,

Montanari²

2017

Preprint

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We develop an abstract theory to obtain Harnack inequality for non homogeneous PDEs in the setting of quasi metric spaces. The main idea is to adapt the notion of double ball and critical density property given by Di Fazio, Gutiérrez, Lanconelli, taking into account the right hand side of the equation. Then we apply the abstract procedure to the case of subelliptic equations in non divergence form involving Grushin vector fields and to the case of X-elliptic operators in divergence form.

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“…For instance, we drop our previous hypoellipticity assumptions (HY) and (HY) ε , in favor of more geometrical hypotheses. We shall skip any detail here, referring the interested reader to [8].…”

Section: An Invariant Harnack Inequalitymentioning

confidence: 99%

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

Bonfiglioli

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We outline several results of Potential Theory for a class of linear partial differential operators L of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for L ; under different geometrical assumptions on L (mainly, under global doubling/Poincaré assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When L is equipped with a global fundamental solution Γ , further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on L ensuring that such a Γ exists.

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An invariant Harnack inequality for a class of subelliptic operators under global doubling and Poincaré assumptions, and applications

Cited by 6 publications

References 18 publications

Global estimates for the fundamental solution of homogeneous Hörmander operators

Global estimates for the fundamental solution of homogeneous Hörmander operators

Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces

Potential Theory Results for a Class of PDOs Admitting a Global Fundamental Solution

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