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93
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"Bramanti, Marco"
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Non-divergence equations structured on Hörmander vector fields: heat kernels and Harnack inequalities
In this work we deal with linear second order partial differential operators of the following type:
eBook
Fundamental solutions and local solvability for nonsmooth Hörmander’s operators
The authors consider operators of the form L=\\sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of \\mathbb{R}^{p} where X_{0},X_{1},\\ldots,X_{n} are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \\gamma for L and provide growth estimates for \\gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \\gamma also possesses second derivatives, and they deduce the local solvability of L, constructing, by means of \\gamma, a solution to Lu=f with Hölder continuous f. The authors also prove C_{X,loc}^{2,\\alpha} estimates on this solution.
eBook
Schauder estimates on bounded domains for KFP operators with coefficients measurable in time and Hölder continuous in space
We consider degenerate Kolmogorov-Fokker-Planck operators
such that the corresponding model operator having constant
is hypoelliptic, translation invariant w.r.t. a Lie group operation in
and 2-homogeneous w.r.t. a family of nonisotropic dilations. We assume that the
’s are globally bounded and Hölder continuous in
(w.r.t. some intrinsic distance induced by
); the matrix
is symmetric and uniformly positive on
. We prove “partial Schauder
estimates” on a bounded open set
, of the kind
for suitable functions
, where
Here
is a homogeneous norm in
, while
and
. We also prove that the derivatives
are locally Hölder continuous in space and time.
Journal Article
Fundamental solutions for Kolmogorov-Fokker-Planck operators with time-depending measurable coefficients
We consider a Kolmogorov-Fokker-Planck operator of the kind: Lu=q∑i,j=1aij(t)∂2xixju+N∑k,j=1bjkxk∂xju−∂tu,(x,t)∈RN+1 where {aij(t)}qi,j=1 is a symmetric uniformly positive matrix on Rq , q≤N , of bounded measurable coefficients defined for t∈R and the matrix B={bij}Ni,j=1 satisfies a structural assumption which makes the corresponding operator with constant aij hypoelliptic. We construct an explicit fundamental solution Γ for L , study its properties, show a comparison result between Γ and the fundamental solution of some model operators with constant aij , and show the unique solvability of the Cauchy problem for Lunder various assumptions on the initial datum.
Journal Article
Fundamental solutions and local solvability for nonsmooth Hèormander's operators
The authors consider operators of the form $L=\\sum_{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of $\\mathbb{R}^{p}$ where $X_{0},X_{1},\\ldots,X_{n}$ are nonsmooth Hormander's vector fields of step $r$ such that the highest order commutators are only Holder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution $\\gamma$ for $L$ and provide growth estimates for $\\gamma$ and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that $\\gamma$ also possesses second derivatives, and they deduce the local solvability of $L$, constructing, by means of $\\gamma$, a solution to $Lu=f$ with Holder continuous $f$. The authors also prove $C_{X,loc}^{2,\\alpha}$ estimates on this solution.
eBook
Non-divergence equations structured on Hèormander vector fields: heat kernels and Harnack inequalities
In this work the authors deal with linear second order partial differential operators of the 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 of real Hormander's vector fields in some bounded domain $\\Omega\\subseteq\\mathbb{R}^{n}$, $A=\\left\\{ a_{ij}\\left( t,x\\right) \\right\\} _{i,j=1}^{q}$ is a real symmetric uniformly positive definite matrix such that $\\lambda^{-1}\\vert\\xi\\vert^{2}\\leq\\sum_{i,j=1}^{q}a_{ij}(t,x) \\xi_{i}\\xi_{j}\\leq\\lambda\\vert\\xi\\vert^{2}\\text{}\\forall\\xi\\in\\mathbb{R}^{q}, x \\in\\Omega,t\\in(T_{1},T_{2})$ for a suitable constant $\\lambda>0$ a for some real numbers $T_{1}
eBook
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 Hörmander's vector fields for which the highest order commutators are only Hölder continuous. The theory explicitly covers the case of one vector field having weight two while the others have weight one.
Journal Article
Fundamental solution for higher order homogeneous hypoelliptic operators structured on Hörmander vector fields
We introduce and study a new class of higher order differential operators defined on \\(\\mathbb{R}^{n}\\), which are built with H\"{o}rmander vector fields, homogeneous w.r.t. a family of dilations (but not left invariant w.r.t. any structure of Lie group) and have a structure such that a suitably lifted version of the operator is hypoelliptic. We call these operators ''generalized Rockland operators''. We prove that these operators are themselves hypoelliptic and, under a natural condition on the homogeneity degree, possess a global fundamental solution \\(\\Gamma\\left( x,y\\right) \\) which is jointly homogeneous in \\(\\left( x,y\\right) \\) and satisfies sharp pointwise estimates. Our theory can be applied also to some higher order heat-type operators and their fundamental solutions.
Paper
Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and \\(VMO\\) in space
We consider Kolmogorov-Fokker-Planck operators of the form $$ \\mathcal{L}u=\\sum_{i,j=1}^{q}a_{ij}(x,t)u_{x_{i}x_{j}}+\\sum_{k,j=1}^{N} b_{jk}x_{k}u_{x_{j}}-\\partial_{t}u, $$ with \\(\\left( x,t\\right) \\in\\mathbb{R}^{N+1},N\\geq q\\geq1\\). We assume that \\(a_{ij}\\in L^{\\infty}\\left( \\mathbb{R}^{N+1}\\right) \\), the matrix \\(\\left\\{ a_{ij}\\right\\} \\) is symmetric and uniformly positive on \\(\\mathbb{R}^{q}\\), and the drift \\[ Y=\\sum_{k,j=1}^{N}b_{jk}x_{k}\\partial_{x_{j}}-\\partial_{t} \\] has a structure which makes the model operator with constant \\(a_{ij}\\) hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and \\(2\\)-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients \\(a_{ij}\\) are \\(VMO\\) w.r.t. the space variable, and only bounded measurable in \\(t\\). We prove, for every \\(p\\in\\left( 1,\\infty\\right) \\), global Sobolev estimates of the kind: \\begin{align*} \\Vert u\\Vert _{W_{X}^{2,p}(S_{T})} \\equiv & \\sum_{i,j=1}^{q}\\Vert u_{x_{i}x_{j}}\\Vert_{L^{p}(S_{T})} +\\Vert Yu\\Vert _{L^{p}(S_{T})} +\\sum_{i=1}^{q}\\Vert u_{x_{i}}\\Vert _{L^{p}(S_{T})} +\\Vert u\\Vert _{L^{p}(S_{T})} \\\ & \\leq c\\big\\{ \\Vert \\mathcal{L}u\\Vert _{L^{p}(S_{T})}+\\Vert u\\Vert_{L^{p}(S_{T})}\\big\\} \\end{align*} with \\(S_{T}=\\mathbb{R}^{N}\\times\\left( -\\infty,T\\right) \\) for any \\(T\\in(-\\infty,+\\infty]\\). Also, the well-posedness in \\(W_{X}^{2,p}(\\Omega_{T})\\), with \\(\\Omega_{T}=\\mathbb{R}^{N}\\times(0,T) \\) and \\(T\\in\\mathbb{R}\\), of the Cauchy problem% $$ \\begin{cases} \\mathcal{L}u=f & \\text{in \\(\\Omega_{T}\\)} \\\ u(\\cdot,0) =g & \\text{in \\(\\mathbb{R}^{N}\\)} \\end{cases} $$ is proved, for \\(f\\in L^{p}(\\Omega_{T}), g\\in W_{X}^{2,p}(\\mathbb{R}^{N})\\).
Paper