本文对变系数自伴形式部分亚椭圆微分算子作了基本的讨论。一个推广的结果伪微分算子，在下面的文章中给出ArXiv。接近N。Nilsson在Arkiv för matematik，10，1972中的“Some estimates for spectral functions connected with形式亚椭圆微分算子“中，给出了该算子在适当拓扑中的基本解的构造和估计。我们进一步给出相应的谱核的估计。
英文摘要：
This article gives a fundamental discussion on variable coefficients, self-adjoint, formally partially hypoelliptic differential operators. A generalization of the results to pseudo differential operators, is given in a following article in ArXiv. Close to N. Nilsson "Some estimates for spectral functions connected with formally hypoelliptic differential operators" in Arkiv för matematik ,10, 1972, we give a construction and estimates for a fundamental solution to the operator in a suitable topology. We further give estimates of the corresponding spectral kernel.
We discuss the polar in symbol space to hypoelliptic and partially hypoelliptic operators, assuming a transmission property related to a rectifiable boundary and using a representation based on two scalar products.
This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity. Conversely, the sufficiency is approached ana\-lyzing three types of hypotheses: a Hörmander condition, logarithmic growth and super-logarithmic growth.
Boundary conditions for Bismut's hypoelliptic Laplacian which naturally correspond to Dirichlet and Neumann boundary conditions for Hodge Laplacians are considered. Those are related with specific boundary conditions for the differential and its various adjoints. Once the closed realizations of those operators are well understood, the commutation of the differential with the resolvent of the hypoelliptic Laplacian is checked with other properties like the PT-symmetry, which are important for the spectral analysis.
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^ε\}_{ε> 0}$ generated by a class of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter $ε$. By scaling, this regime implies corresponding optimal results both for fixed dissipation and lar…
This article presents an investigation on the global hypoellipticity problem for systems belonging to the class $P = D_t + Q(t,D_x)$, where $Q(t,D_x)$ is a $m\times m$ matrix with entries $c_{j,k}(t)Q_{j,k}(D_x)$. The coefficients $c_{j,k}(t)$ are smooth, complex-valued functions on the torus $\mathbb{T} \simeq \mathbb{R}/2π\mathbb{Z}$ and $Q_{j,k}(D_x)$ are pseudo-differential operators on $ \mathbb{T}^n$. The approach consists in establishing conditions on the matrix symbol $Q(t,ξ)$ such that it can be transformed into a suitable triangular form $Λ(t,ξ) + \mathcal{N}(t,ξ)$, where $Λ(t,ξ)$ is…
We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator $P(D) = P(D_x, D_t)$ on $\mathbb{R}^{d+1}$. Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of several results of Langenbruch [23] about distributional boundary values of zero solutions of $P(D)$.
We give an index formula for the class of all *-maximally hypoelliptic differential operators on any closed manifold with vector bundle coefficients, generalising previous index formulas by Atiyah-Singer and van Erp. Using this formula, we give new explicit index computations for Hormander's sum of squares operators of arbitrary rank.
We introduce a class of time-periodic Gelfand-Shilov spaces of functions on $\mathbb{T} \times \mathbb{R}^n$, where $\mathbb{T} \sim \mathbb{R} /2π\mathbb{Z}$ is the one-dimensional torus. We develop a Fourier analysis inspired by the characterization of the Gelfand-Shilov spaces in terms of the eigenfunction expansions given by a fixed normal, globally elliptic differential operator on $\mathbb{R}^n$. In this setting, as an application, we characterize the global hypoellipticity for a class of linear differential evolution operators on $\mathbb{T} \times \mathbb{R}^n$.
We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and …
We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main …
In the symbol space for differential operators, we discuss a scalar type for change of local coordinates, using vector valued distributions. In particular, we discuss P-convexity for hypoelliptic operators.
Some recent works have shown that the heat equation posed on the whole Euclidean space is null-controllable in any positive time if and only if the control subset is a thick set. This necessary and sufficient condition for null-controllability is linked to some uncertainty principles as the Logvinenko-Sereda theorem which give limitations on the simultaneous concentration of a function and its Fourier transform. In the present work, we prove new uncertainty principles for finite combinations of Hermite functions and establish an analogue of the Logvinenko-Sereda theorem with an explicit contro…
