(1993). Wp 1,2solvability for the cauchy-dirichlet problem parabolic equations with vmo coefficients. Communications in Partial Differential Equations: Vol. 18, No. 9-10, pp. 1735-1763.

In this work the authors deal with linear second order partial differential operators of following type $H=\partial_{t}-L=\partial_{t}-\sum_{i,j=1}^{q}a_{ij}(t,x) X_{i}X_{j}-\sum_{k=1}^{q}a_{k}(t,x)X_{k}-a_{0}(t,x)$ where $X_{1},X_{2},\ldots,X_{q}$ is a system real Hormander's vector fields in some bounded domain $\Omega\subseteq\mathbb{R}^{n}$, $A=\left\{ a_{ij}\left( t,x\right) \right\} _{i,j=1}^{q}$ symmetric uniformly positive definite matrix such that...

Abstract We consider degenerate Kolmogorov–Fokker–Planck operators $$\begin{aligned} \mathcal {L}u&amp;=\sum _{i,j=1}^{m_{0}}a_{ij}(x,t)\partial _{x_{i}x_{j}} ^{2}u+\sum _{k,j=1}^{N}b_{jk}x_{k}\partial _{x_{j}}u-\partial _{t}u\\&amp;\equiv \sum _{x_{i}x_{j}}^{2}u+Yu \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> </mml:mtd> <mml:mo>=</mml:mo> <mml:munderover>...

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 1 comma upper 2 ellipsis Subscript q Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>…</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X_1,X_2,\ldots ,X_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a system of real...

Abstract We consider a class of degenerate Ornstein‐Uhlenbeck operators in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^{N}\!$\end{document} , the kind where ( ij ) is symmetric uniformly positive definite on {R}^{p_{0}}$\end{document} p 0 ≤ N ), with continuous and bounded entries, b constant matrix such that frozen operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {A}_{x_{0}}$\end{document}...

() defined in some bounded domain and assume that the Xi satisfy Hörmander's rank condition of step r , . We extend to this nonsmooth context results which are well known for smooth vector fields, namely: basic properties distance induced by doubling condition, Chow's connectivity theorem, and, under stronger assumption Poincaré's inequality. By results, these facts also imply a Sobolev embedding. All tools allow us draw consequences about second order differential operators modeled on fields:

We consider degenerate Kolmogorov-Fokker-Planck operatorsLu=∑i,j=1qaij(x,t)∂xixj2u+∑k,j=1Nbjkxk∂xju−∂tu,(x,t)∈RN+1,N≥q≥1 such that the corresponding model operator having constant aij is hypoelliptic, translation invariant w.r.t. a Lie group operation in RN+1 and 2-homogeneous family of nonisotropic dilations. The coefficients are bounded Hölder continuous space (w.r.t. some distance induced by L RN) only measurable time; matrix {aij}i,j=1q symmetric uniformly positive on Rq. prove "partial...

Let us consider the class of “nonvariational uniformly hypoelliptic operators”: Lu\equiv\sum_{i,j=1}^{q}a_{ij} (x) X_{i} X_{j} u where: X_1,X_2,\ldots,X_q is a system Hörmander vector fields in \mathbb{R}^{n} ( n&gt;q ), \{a_{ij}\} q\times q elliptic matrix, and functions a_{ij} are continuous, with suitable control on modulus continuity. We prove that: \| \|_{BMO(\Omega^{\prime})} \leq c \left\{\left\|Lu\right\|_{BMO(\Omega)} + \left\| u\right\|_{BMO(\Omega)} \right\} for domains...

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain R^{n}, where X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) valued, measurable functions, such that matrix {a_{ij}} is symmetric uniformly positive. prove if coefficients H\"older continuous with respect to distance induced by fields, then local Schauder estimates X_{i}X_{j}u, X_{0}u...

We present a result of L^p continuity singular integrals Calderón-Zygmund type in the context bounded nonhomogeneous spaces, well suited to be applied problems priori estimates for partial differential equations. First, an easy and selfcontained proof L^2 is got by means C^{\alpha} continuity, thanks abstract theorem Krein. Then derived adapting known results Nazarov-Treil-Volberg about spaces.

In this paper we provide two different characterizations of sets with finite perimeter and functions bounded variation in Carnot groups, analogous to those that hold Euclidean spaces, terms the short-time behavior heat semigroup. The second one holds under hypothesis reduced boundary a set is rectifiable, result presently known Step 2 groups.

We prove a version of Rothschild-Stein's theorem lifting and approximation some related results in the context nonsmooth Hörmander's vector fields for which highest order commutators are only Hölder continuous.The theory explicitly covers case one field having weight two while others have one.

We consider a Kolmogorov-Fokker-Planck operator of the kind: $ \mathcal{L}u = \sum\limits_{i,j 1}^{q}a_{ij}\left( t\right) \partial_{x_{i}x_{j}} ^{2}u+\sum\limits_{k,j 1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1} where $\left\{ a_{ij}\left(t\right) \right\} _{i, j 1}^{q}$ is symmetric uniformly positive matrix on $\mathbb{R}^{q}$, $q\leq N$, bounded measurable coefficients defined for $t\in\mathbb{R}$ and $B \left\{ b_{ij}\right\} 1}^{N}$ satisfies...