Subelliptic

Subelliptic is a term used in mathematics, specifically in the field of partial differential equations (PDEs), to describe a class of operators that exhibit properties similar to those of elliptic operators but are not elliptic themselves. Elliptic operators are characterized by their ability to smooth out solutions, a property known as elliptic regularity. Subelliptic operators, while not elliptic, also possess a form of regularity that is weaker than elliptic regularity.

One of the key features of subelliptic operators is their connection to the heat equation. Specifically, they are associated with a heat equation that has a slower rate of diffusion compared to the standard heat equation. This slower diffusion rate is reflected in the behavior of the fundamental solution of the associated heat equation, which decays at a slower rate than the fundamental solution of the standard heat equation.

Subelliptic operators arise naturally in the study of hypoelliptic operators, which are operators that are not elliptic but still possess a form of smoothness in their solutions. The study of subelliptic operators is important in various areas of mathematics, including the theory of PDEs, harmonic analysis, and geometric analysis. They also have applications in physics, particularly in the study of quantum mechanics and the behavior of particles in magnetic fields.

In summary, subelliptic operators are a class of operators that exhibit a form of regularity similar to elliptic operators but are not elliptic themselves. They are characterized by their association with a heat equation that has a slower rate of diffusion and their connection to hypoelliptic operators. The study of subelliptic operators is an active area of research in mathematics with applications in various fields.