Hörmander’s Hypoelliptic Theorem for Nonlocal Operators

Journal of Theoretical Probability - In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking,...     

Hörmander index in finite-dimensional case

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.     

Hölder Estimates for the ∂¯-Operator on Polycylinders

Hölder estimates for the ?¯-operator are obtained on polycylinders and applied to a number of problems-cohomology with bounds, the corona problem and approximation of Hölder functions by holomorphic functions.     

Hörmander Representations and Hill Trajectories for Certain Partial Differential Operators

We consider some partial differential operators associated with Markov projections and describe their Hörmander representations. The corresponding Hill diffusion and drift trajectories are also studied.     

Hörmander Multipliers on the Heisenberg Group

 Let be the Heisenberg group of dimension . Let be the partial sub-Laplacians on and T the central element of the Lie algebra of . For any we prove that the operator is bounded on the Hardy spaces , if the function m satisfies a Hrmander-type condition on which depends on . We also obtain analogous results for the operators and , where the function m satisfies analogous H?rmander-type conditions on and on , respectively. Here is the Kohn-Laplacian on . (Received 28 July 1999; in final form 6 March 2000)     

On elliptic systems in Hörmander spaces

We study a linear system of pseudodifferential equations uniformly elliptic in Petrovskii’s sense in the Hilbert scale of H?rmander functional spaces defined in ℝ _n . An a priori estimate is proved for the solution of the system and its interior smoothness in this scale of spaces is investigated. As an application, we establish a sufficient condition for the existence of continuous bounded derivatives of the solution.     

A Remark on the Hörmander Inequality

A sufficient condition is given in order to ensure the Hörmander inequality for a (formally) self-adjoint transversally elliptic pseudodifferential operator P in the case of the symplectic form changing rank on the characteristic manifold.     

Smoothing of Commutators for a Hörmander Class of Bilinear Pseudodifferential Operators

Commutators of bilinear pseudodifferential operators with symbols in the Hörmander class $BS_{1, 0}^{1}$ and multiplication by Lipschitz functions are shown to be bilinear Calderón-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.     

Regularity results for the minimum time function with Hörmander vector fields

In a bounded domain of ${\mathbb{R}}^n$ with boundary given by a smooth $(n?1)$-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\dots ,X_N\}$ subject to Hörmander's bracket generating condition. We investigate the regularity of the viscosity solution T of such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we prove that the local Lipschitz continuity of T, the local semiconcavity of T, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied whenever the characteristic set of $\{X_1,\dots ,X_N\}$ is a symplectic manifold. We apply our results to several examples.     

The Hörmander index of symmetric periodic orbits

A symmetric periodic orbit is a special kind of periodic orbit that can also be regarded as a Lagrangian intersection point. Therefore it has two Maslov indices whose difference is the Hörmander index. In this paper we provide a formula for the Hörmander index of a symmetric periodic orbit and its iterates in terms of Chebyshev polynomials.     

Estimates for a Group of Deviations in Generalized Hölder Metric

We present order relations for a group of deviations of a function f(·) H in terms of partial Fourier sums of this function in a generalized Hölder metric defined in a generalized Hölder space H _* H .     

Roughness of Exponential Estimates for Linear Functional Differential Equations

<正> Assume that the fundamental solution matrix U (t, s ) of x'(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ   

A Note on Variable Exponent Hörmander Spaces

In this paper we introduce the variable exponent Hörmander spaces and we study some of their properties. In particular, it is shown that ${{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}$ is isomorphic to ${{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}$ (Ω open set in ${{\mathbb{R}^n, p? > 1}}$ and the Hardy–Littlewood maximal operator M is bounded in ${L_p(\cdot))}$ extending a Hörmander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hörmander spaces are given.     

A Nash–Hörmander iteration and boundary elements for the Molodensky problem

We investigate the numerical approximation of the nonlinear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. The method, based on a smoothed Nash–Hörmander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of derivatives in the surface update. In particular, we obtain a quantitative a priori estimate for the error after $m$ steps, justify the use of smoothing operators based on the heat equation, and comment on the accurate evaluation of the Hessian of the gravitational potential on the surface, using a representation in terms of a hypersingular integral. A boundary element method is used to solve the exterior problem. Numerical results compare the error between the approximation and the exact solution in a model problem.     

Improved Hörmander’s theorem and new methods for oscillatory integral operators

This paper proves a theorem on the decay rate of the oscillatory integral operator with a degenerate C^∞ phase function, thus improving a classical theorem of HSrmander. The proof invokes two new methods to resolve the singularity of such kind of operators： a delicate method to decompose the operator and balance the L^2 norm estimates; and a method for resolution of singularity of the convolution type. The operator is decomposed into four major pieces instead of infinite dyadic pieces, which reveals that Cotlar＇s Lemma is not essential for the L^2 estimate of the operator. In the end the conclusion is further improved from the degenerate C^∞ phase function to the degenerate C^4 phase function.     

On the Fedosov-Hörmander Formula for Differential Operators

The Fedosov-H?rmander formula gives the Fredholm index of some pseudodifferential operators of order 0 on L ². It is well known that it can be used to calculate the index of elliptic systems under the assumption that, among other things, the coefficients are smooth and their partial derivatives of all orders satisfy specific asymptotic conditions at infinity. We prove that the formula remains valid when the coefficients are only and bounded and have vanishing oscillation at infinity. In turn, this generalization is used to obtain a nonstandard invariance property of the index as well as various sufficient conditions for the index to be 0, when the coefficients are merely continuous and bounded with vanishing oscillation.