scholarly journals Backward and forward filtering under the weak Hörmander condition

2022 ◽  
Author(s):  
Andrea Pascucci ◽  
Antonello Pesce
Keyword(s):  
Hörmander Condition

scholarly journals The Calderón–Zygmund Theorem with an $$L^1$$ Mean Hörmander Condition

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Soichiro Suzuki
Keyword(s):  
Singular Integral ◽  
Integral Operators ◽  
Acta Math ◽  
Limited Range ◽  
Singular Integral Operators ◽  
Convolution Type ◽  
Worst Case ◽  
Hörmander Condition

AbstractIn 2019, Grafakos and Stockdale introduced an $$L^q$$ L q mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the $$L^p$$ L p boundedness for the “limited-range” instead of $$1< p < \infty $$ 1 < p < ∞ . However, in this paper, we show that the $$L^q$$ L q mean Hörmander condition is actually enough to obtain the $$L^p$$ L p boundedness for all $$1< p < \infty $$ 1 < p < ∞ even in the worst case $$q=1$$ q = 1 . We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the $$L^2$$ L 2 boundedness for convolution type singular integral operators under the $$L^1$$ L 1 mean Hörmander condition.

scholarly journals Tube estimates for diffusions under a local strong Hörmander condition

2019 ◽  
Vol 55 (4) ◽  
pp. 2320-2369
Author(s):  
Vlad Bally ◽  
Lucia Caramellino ◽  
Paolo Pigato
Keyword(s):  
Hörmander Condition

scholarly journals Higher order asymptotics for large deviations — Part II

2020 ◽  
pp. 2150025
Author(s):  
Kasun Fernando ◽  
Pratima Hebbar
Keyword(s):  
Large Deviations ◽  
Asymptotic Expansions ◽  
Continuous Time ◽  
Compact Manifold ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Hörmander Condition ◽  
Additive Functionals ◽  
Higher Order Asymptotics ◽  
Weakly Dependent

We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly-dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a [Formula: see text]-dimensional compact manifold admit these asymptotic expansions of all orders.

scholarly journals Pointwise gradient bounds for degenerate semigroups (of UFG type)

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160442 ◽  
Author(s):  
D. Crisan ◽  
M. Ottobre
Keyword(s):  
Differential Operators ◽  
Hörmander Condition ◽  
Asymptotic Bounds ◽  
Long Time Behaviour ◽  
Time Asymptotics ◽  
Diffusion Semigroup ◽  
Long Time ◽  
Gradient Bounds ◽  
Degenerate Type ◽  
Derivatives Of

In this paper, we consider diffusion semigroups generated by second-order differential operators of degenerate type. The operators that we consider do not , in general, satisfy the Hörmander condition and are not hypoelliptic. In particular, instead of working under the Hörmander paradigm, we consider the so-called UFG (uniformly finitely generated) condition, introduced by Kusuoka and Strook in the 1980s. The UFG condition is weaker than the uniform Hörmander condition, the smoothing effect taking place only in certain directions (rather than in every direction, as it is the case when the Hörmander condition is assumed). Under the UFG condition, Kusuoka and Strook deduced sharp small time asymptotic bounds for the derivatives of the semigroup in the directions where smoothing occurs. In this paper, we study the large time asymptotics for the gradients of the diffusion semigroup in the same set of directions and under the same UFG condition. In particular, we identify conditions under which the derivatives of the diffusion semigroup in the smoothing directions decay exponentially in time. This paper constitutes, therefore, a stepping stone in the analysis of the long-time behaviour of diffusions which do not satisfy the Hörmander condition.

On Hörmander condition

1997 ◽  
Vol 42 (16) ◽  
pp. 1341-1345 ◽  
Author(s):  
Qixiang Yang ◽  
Lixin Yan ◽  
Donggao Deng
Keyword(s):  
Hörmander Condition

Gaussian estimates for heat kernels on Lie groups

2000 ◽  
Vol 128 (1) ◽  
pp. 45-64 ◽  
Author(s):  
SAMI MUSTAPHA
Keyword(s):  
Hilbert Space ◽  
Lie Algebra ◽  
Spectral Gap ◽  
Adjoint Action ◽  
Modular Function ◽  
Heat Kernels ◽  
Iwasawa Decomposition ◽  
Triangular Form ◽  
Hörmander Condition ◽  
Levi Decomposition

Let G be a connected real Lie group and let [gfr ] be its Lie algebra. We shall denote by [qfr ] ⊂ [gfr ] the radical of [gfr ]. Let [gfr ] = [qfr ] [ltimes ] [sfr ] (where [sfr ] is semisimple or 0) be a Levi decomposition of [gfr ] (cf. [11]). When [sfr ] ≠ 0 we can apply the Iwasawa decomposition on [sfr ] (cf. [8]) [sfr ] = [nfr ] [oplus ] [afr ] [oplus ] [kfr ], where [nfr ] is nilpotent and [afr ] is abelian and normalizes [nfr ] so that [nfr ] [oplus ] [afr ] is a soluble algebra. Since [nfr ] [oplus ] [afr ] normalizes [qfr ] it is clear that [rfr ] = [qfr ] [oplus ] [nfr ] [oplus ] [afr ] is a soluble Lie algebra of [gfr ]. By Lie's theorem (cf. [11]) we can find a basis on [rfr ]c = [rfr ] [otimes ] C for which the adjoint action of [rfr ] on [rfr ]c takes a triangular form. Let us denote by λ1(x); λ2(x), …, λn(x), x ∈ [rfr ] the corresponding eigenvalues. The λj's can be identified with elements of Homℝ([rfr ], C) and are called the roots of the adjoint action of [rfr ]. Let us denote by [Lscr ] = {L1, …, Lk} the set of the non zero real parts of the λj's. We shall say that the group G is a B-group if [Lscr ] ≠ &0slash; and if there exist α1, …, αk [ges ] 0, [sum ]kj=1 αj = 1, such that [sum ]kj=1 αjLj = 0. Otherwise we say that G is an NB-group. It can be shown that the above definition is independent of the particular choice of the Levi and Iwasawa decompositions that are used (cf. [13]).We shall denote by dlg = dg (resp. drg) the left (resp. right) Haar measure on G and by m(g) = drg/dlg the modular function.Let [Xscr ] = {X1, X2, …, Xn} be left invariant fields on G that verify the Hörmander condition (cf. [15]) and let Δ = −[sum ]X2j be the corresponding sub-Laplacian. Δ is formally self adjoint on the Hilbert space L2(G, drg) and the spectral gap of Δ is defined byformula here

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