Marco Bramanti

Marco Bramanti
Politecnico di Milano | Polimi · Department of Mathematics "Francesco Brioschi"

PhD

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55
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Publications

Publications (55)
Preprint
We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such that the corresponding model operator having constant $a_{ij}$ is hypoelliptic, translation invariant w.r.t. a...
Article
Let L=∑j=1mXj2 be a Hörmander sum of squares of vector fields in Rn, where any Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in Rn. Then, L is known to admit a global fundamental solution Γ(x;y) that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space Rn×Rp,...
Article
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Let \({\mathscr{H}}={\sum }_{j=1}^{m}{X_{j}^{2}}-\partial _{t}\) be a heat-type operator in \(\mathbb {R}^{n+1}\), where X = {X1,…,Xm} is a system of smooth Hörmander’s vector fields in \(\mathbb {R}^{n}\), and every Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in \(\mathbb {R}^{n}\), while no underlying group s...
Article
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In this note we point out and correct a mistake in our paper “Global Lp estimates for degenerate Ornstein–Uhlenbeck operators with variable coefficients”, published in Math. Nachr. 286 (2013), no. 11–12, 1087–1101.
Article
Let L=∑j=1mXj2 be a Hörmander sum of squares of vector fields in space Rn, where any Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for L in the X-Sobolev spaces WXk,p(Rn), where X={X1,…,Xm}. In our approach, we combine local results for g...
Preprint
Full-text available
Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying H\"{o}rmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie...
Preprint
Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth H\"{o}rmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$, while no underlying gr...
Preprint
Let $\mathcal{L}=\sum_{j=1}^{m}X_{j}^{2}$ be a H\"{o}rmander sum of squares of vector fields in $\mathbb{R}^{n}$, where any $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$. Then $\mathcal{L}$ is known to admit a global fundamental solution $\Gamma (x;y)$, that can be represented as the in...
Preprint
Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a H\"ormander sum of squares of vector fields in space $\mathbb{R}^n$, where any $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for $\mathcal{L}$ in the $X$-Sobolev spaces $W^{k,p}_X(\mathbb{R}^...
Preprint
We consider a heat-type operator L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, multiplied by a uniformly positive matrix of bounded measurable coefficients depending only on time. We prove that if Lu is smooth with respect to the space variables, the same is true for u, with quantitative regu...
Article
We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We g...
Article
Full-text available
We prove a local version of Fefferman-Stein inequality for the local sharp maximal function, and a local version of John-Nirenberg inequality for locally BMO functions, in the framework of locally homogeneous spaces, in the sense of Bramanti-Zhu [Manuscripta Math. 138 (2012), no. 3-4, 477-528].
Article
We give a self-contained analytical proof of Hörmander’s hypoellipticity theorem in the case of left invariant sub-laplacians of Carnot groups. The proof does not rely on pseudodifferential calculus and provides intermediate regularity estimates expressed in terms of Sobolev spaces induced by right invariant vector fields.
Chapter
This note describes the results of a joint research with L. Brandolini, M. Manfredini and M. Pedroni, contained in Bramanti et al. [Fundamental solutions and local solvability of nonsmooth Hörmander’s operators. Mem. Am. Math. Soc., in press. http:// arxiv. org/ abs/ 1305. 3398], with some background. We consider operators of the form \(L =\sum _{...
Chapter
The chapter describes some examples of PDEs written in the form of Hörmander’s operators which arise both from physical applications and from other fields of mathematics, to give some more motivations for the study of these equations. We deal with two different areas, which represent the main “historical” motivations for this field of research, nam...
Chapter
The chapter focusses of some basic ideas in the geometry of Hörmander’s vector fields. We start with a discussion of the basic concept of connectivity, and some of its physical meanings which have been known for a long time. We then pass to survey some ideas and fundamental results about the metric induced by a system of vector fields, which are co...
Chapter
With the creation of distribution theory, around 1950, a systematic study of properties of linear PDEs with smooth coefficients started. In this context, two basic concepts are those of solvability and hypoellipticity of an operator. Looking for a characterization of linear second order hypoelliptic operators with real coefficients, Hörmander’s pro...
Chapter
In this last chapter I want to discuss some developments which have taken place in the study of Hörmander’s operators and related topics since the 1990’s. As we will see, most of these developments have extended the class of operators under study, passing from classical Hörmander’s operators to operators “structured on Hörmander’s vector fields”, i...
Chapter
Here we deal with the theme of a-priori estimates, in suitable Sobolev spaces, for Hörmander’s operators. This involves the concept of homogeneous group, the construction of fundamental solutions, the use of abstract singular integral theories, and the development of suitable algebraic and differential geometric tools. The chapter surveys three fun...
Article
We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, 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 $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bound...
Article
We consider operators of the form $L=\sum_{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of R^p where X_0, X_1,...,X_n are nonsmooth H\"ormander's vector fields of step r such that the highest order commutators are only H\"older continuous. Applying Levi's parametrix method we construct a local fundamental solution \gamma\ for L and provide growth e...
Article
For a linear nonvariational operator structured on smooth H\"ormander's vector fields, with H\"older continuous coefficients, we prove a regularity result in the spaces of H\"older functions. We deduce an analogous regularity result for nonvariational degenerate quasilinear equations.
Article
Full-text available
We consider a class of nonvariational linear operators formed by homogeneous left invariant Hormander's vector fields with respect to a structure of Carnot group. The bounded coefficients of the operators belong to "vanishing logarithmic mean oscillation" class with respect to the distance induced by the vector fields (in particular they can be dis...
Article
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...
Article
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...
Article
Full-text available
In this article, we give some a priori Lp(\mathbbRn){L^{p}(\mathbb{R}^{n})} estimates for elliptic operators in nondivergence form with VMO coefficients and a potential V satisfying an appropriate reverse Hölder condition, generalizing previous results due to Chiarenza–Frasca–Longo to the scope of Schrödinger-type operators. In particular, our cl...
Article
Full-text available
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_{i...
Article
Full-text available
We introduce the concept of locally homogeneous space, and prove in this context L^p and Holder estimates for singular and fractional integrals, as well as L^p estimates on the commutator of a singular or fractional integral with a BMO or VMO function. These results are motivated by local a-priori estimates for subelliptic equations.
Article
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...
Article
We consider a class of degenerate Ornstein-Uhlenbeck operators in, ℝ, of the kind where (aij), (bij) are constant matrices, (aij) is symmetric positive definite on (p0 ≤ N), and (bij) is such that is hypoelliptic. For this class of operators we prove global Lp estimates (1 <p <∞) of the kind: and corresponding weak type (1,1) estimates. This result...
Article
Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat genera...
Article
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.
Article
We present a result of $L^p$ continuity of singular integrals of Calderón-Zygmund type in the context of bounded nonhomogeneous spaces, well suited to be applied to problems of a priori estimates for partial differential equations. First, an easy and selfcontained proof of $L^2$ continuity is got by means of $C^{\alpha}$ continuity, thanks to an ab...
Article
After a brief explanation of what the DNA test consists in, we discuss some probabilistic problems related to this test and we try to establish some formulas for computing the probabilities of relevant events in this context. This allows to point out some interesting remarks, both of qualitative and of quantitative kind.
Article
In this work we deal with linear second order partial differential operators of the following type: (Equation) where X 1, X 2,..., X q is a system of real Hörmander's vector fields in some bounded domain Ω ⊆ ℝ n, A = {a ij (t, x)} qi,j=1 is a real symmetric uniformly positive definite matrix such that: (Equation) for a suitable constant λ > 0 a for...
Article
Full-text available
We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties...
Article
Let X1,X2,…,Xq be a system of real smooth vector fields satisfying Hörmander's rank condition in a bounded domain Ω of Rn. Let A={aij(t,x)}i,j=1q be a symmetric, uniformly positive definite matrix of real functions defined in a domain U⊂R×Ω. For operators of kindH=∂t−∑i,j=1qaij(t,x)XiXj−∑i=1qbi(t,x)Xi−c(t,x) we prove local a-priori estimates of Sch...
Article
We prove the existence of a fundamental solution for a class of Hörmander heat-type operators. For this fundamental solution and its derivatives we obtain sharp Gaussian bounds that allow to prove an invariant Harnack inequality. To cite this article: M. Bramanti et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).
Article
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 of H\"ormander vector fields in $\mathbb{R}^{n}$ ($n>q$), $\{a_{ij}\}$ is a $q\times q$ uniformly elliptic matrix, and the functions $a_{ij} (x)$ are continuous, with a suit...
Article
Let X-1,X-2,...,X-q be a system of real smooth vector fields, satisfying Hormander's condition in some bounded domain Omega subset of R-n (n > q). We consider the differential operator [GRAPHICS] where the coefficients a(ij)(x) are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: [GRAPHICS] for a.e. x is an e...
Article
Let G be a homogeneous group and let X0,X1,..., Xq be left invariant real vector fields on G, satisfying Hormander's condition. Assume that X1,..., Xq be homogeneous of degree one and X0 be homogeneous of degree two. We study operators of the kind: (Equation) where aij(x) and a0(x) are real valued, bounded measurable functions belonging to the spac...
Article
Let X 1 , X 2 , … , X q X_1,X_2,\ldots ,X_q be a system of real smooth vector fields, satisfying Hörmander’s condition in some bounded domain Ω ⊂ R n \Omega \subset \mathbb {R}^n ( n > q n>q ). We consider the differential operator L = ∑ i = 1 q a i j ( x ) X i X j , \begin{equation*} \mathcal {L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*} where...
Article
We consider a class of ultraparabolic operators of the kind[formula](z=(x,t)∈RN+1), where the principal part is uniformly elliptic on Rq,q≤N, and the constant matrixBis upper triangular and such that the operator obtained by freezing the coefficientsaijat any pointz0∈RN+1is hypoelliptic. We prove local Lp-estimates for the derivatives ∂xixju(i,j=1,...
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In this paper we deal with a uniformly elliptic operator of the kind: Lu  Au + Vu, where the principal part A is in divergence form, and V is a function assumed in a “Kato class”. This operator has been studied in different contexts, especially using probabilistic techniques. The aim of the present work is to give a unified and simplified presenta...
Article
We consider the Cauchy-Neumann problem for parabolic operators of the kind: on a smooth cylinder [0,T]×Ω. By symmetrization techniques we establish for the solution u of this problem an estimate of the kind: where U is the solution of a symmetrized problem and u(t)*(·) is the decreasing rearrangement of u(t,.). We also obtain an “energy inequality”...
Article
Let X 1 ,X 2 ,⋯,X q be a system of real smooth vector fields satisfying Hörmander’s rank condition in a bounded domain Ω of ℝ n . Let A={a ij (t,x)} i,j=1 q be a symmetric, uniformly positive definite matrix of real functions defined in a domain U⊂ℝ×Ω. For operators of the kind H=∂ t -∑ i,j=1 q a ij (t,x)X i X j -∑ i=1 q b i (t,x)X i -c(t,x) we pro...

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