Hypoelliptic differential operators and nilpotent groups

Semilinear Hypoelliptic Differential Operators with Multiple Characteristics

In this paper we consider the regularity of solutions of semilinear differential equations principal parts of which consist of linear polynomial operators constructed from real vector fields. Based on the study of fine properties of the principal linear parts we then obtain the regularity of solutions of the nonlinear equations. Some results obtained in this article are also new in the frame of...

Differential Operators on Some Nilpotent Orbits

In recent work, Astashkevich and Brylinski construct some differential operators of Euler degree −1 (thus, they lower the degree of polynomials by one) on the coordinate ring O(Omin(g)) of the minimal nilpotent orbit Omin(g) for any classical, complex simple Lie algebra g. They term these operators “exotic” since there is “(apparently) no geometric or algebraic theory that explains them”. In th...

Spectral Properties of Hypoelliptic Operators

We study hypoelliptic operators with polynomially bounded coefficients that are of the form K = ∑m i=1 X i Xi + X0 + f , where the Xj denote first order differential operators, f is a function with at most polynomial growth, and X i denotes the formal adjoint of Xi in L. For any ε > 0 we show that an inequality of the form ‖u‖δ,δ ≤ C(‖u‖0,ε + ‖(K + iy)u‖0,0) holds for suitable δ and C which are...

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their λ-extensions. The univariate analytic functions defined by these integrals are holonomic and even Gfunctions: they ...

Degenerate Hypoelliptic Operators and the Gaussian Kernel