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N. Garofalo

Publications and source records attributed to N. Garofalo.

7 recordsLinked to original sources
arXiv2011

A note on the boundedness of Riesz transform for some subelliptic operators

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter $ρ_1$, the generalized curvature inequality in \eqref{CD} below, then the Riesz transform is bounded in $L^p (\bM)$ for every $p>1$, that is \[\| \sqrt{Γ((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM), \] where $Γ$ is the \textit{carré du champ} associated to $L$. Our results apply in particular to all Sasakian manifolds whose horizontal Tanaka-Webster Ricci curvature is nonnegative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

math.FA↗
arXiv2010

Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions

In a cylinder $Ω_T=Ω\times (0,T)\subset \R^{n+1}_+$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form \[ Hu =\sum_{i,j=1}^ma_{ij}(x,t) X_iX_ju - \p_tu = 0, \ (x,t)\in\R^{n+1}_+, \] where $X=\{X_1,...,X_m\}$ is a system of $C^\infty$ vector fields in $\Rn$ satisfying Hörmander's finite rank condition \eqref{frc}, and $Ω$ is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance $d$ induced by $X$. Concerning the matrix-valued function $A=\{a_{ij}\}$, we assume that it be real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries $a_{ij}$ be Hölder continuous with respect to the parabolic distance associated with $d$. Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem \ref{T:back}); 2) the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem \ref{T:quotients}); 3) the doubling property for the parabolic measure associated with the operator $H$ (Theorem \ref{T:doubling}). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [FSY] and [SY]. With one proviso: in those papers the authors assume that the coefficients $a_{ij}$ be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.

math.AP↗
arXiv2006

Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group

Let S be a C^2 H-minimal noncharacteristic hypersurface in the first Heisenberg group. We show that if S contains a graphical strip, then it is not a stable minimal surface. Moreover, we show that if S is a C^2 H-minimal noncharacteristic entire graph which is not itself a vertical plane, then S contains a graphical strip. Thus, as a corollary, we obtain an analogue of the Bernstein theorem: the only stable C^2 H-minimal noncharacteristic entire graphs are the vertical planes.

math.DG↗