2,998 research outputs found
Global estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients
We consider a class of degenerate Ornstein-Uhlenbeck operators in
, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x)
\partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where
is symmetric uniformly positive definite on
(), with uniformly continuous and bounded entries, and
is a constant matrix such that the frozen operator
corresponding to is hypoelliptic. For this class of operators
we prove global estimates () of the kind:%
[|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leq
c{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} for
i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the
following one, which is of interest in its own:%
[|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leq
c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any where is the strip
, small, and is the
Kolmogorov-Fokker-Planck operator% [L\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x,t)
\partial_{x_{i}x_{j}}%
^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}-\partial_{t}%] with uniformly
continuous and bounded 's
Interior HW^{1,p} estimates for divergence degenerate elliptic systems in Carnot groups
Let X_1,...,X_q be the basis of the space of horizontal vector fields on a
homogeneous Carnot group in R^n (q<n). We consider a degenerate elliptic system
of N equations, in divergence form, structured on these vector fields, where
the coefficients a_{ab}^{ij} (i,j=1,2,...,q, a,b=1,2,...,N) are real valued
bounded measurable functions defined in a bounded domain A of R^n, satisfying
the strong Legendre condition and belonging to the space VMO_{loc}(A) (defined
by the Carnot-Caratheodory distance induced by the X_i's). We prove interior
HW^{1,p} estimates (2<p<\infty) for weak solutions to the system
On the lifting and approximation theorem for nonsmooth vector fields
We prove a version of Rothschild-Stein's theorem of lifting and approximation
and some related results in the context of nonsmooth Hormander's vector fields
for which the highest order commutators are only Holder continuous. The theory
explicitly covers the case of one vector field having weight two while the
others have weight one.Comment: 46 pages, LaTeX. Minor changes in Section
Two characterization of BV functions on Carnot groups via the heat semigroup
In this paper we provide two different characterizations of sets with finite
perimeter and functions of bounded variation in Carnot groups, analogous to
those which hold in Euclidean spaces, in terms of the short-time behaviour of
the heat semigroup. The second one holds under the hypothesis that the reduced
boundary of a set of finite perimeter is rectifiable, a result that presently
is known in Step 2 Carnot groups
L^p and Schauder estimates for nonvariational operators structured on H\"ormander vector fields with drift
We consider linear second order nonvariational partial differential operators
of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the
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) are real valued, bounded
measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly
positive. We prove that if the coefficients a_{ij} are H\"older continuous with
respect to the distance induced by the vector fields, then local Schauder
estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space
VMO with respect to the distance induced by the vector fields, then local L^{p}
estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the
presence of the drift term X_{0}, so that our class of operators covers, for
instance, Kolmogorov-Fokker-Planck operators
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