Narrowelliptic

Narrowelliptic is a term used in differential geometry and analysis to describe a variant of ellipticity defined with respect to a restricted set of directions in the cotangent bundle. Informally, an operator L is called narrowelliptic on a manifold M with boundary if its principal symbol sigma_L(x, xi) is invertible for all nonzero covectors xi lying outside a prescribed narrow cone C_x in T*_x M, while ellipticity may fail for xi in C_x. The restriction to a narrow region distinguishes narrowelliptic operators from standard elliptic operators.

The concept is typically introduced in contexts where full ellipticity is either too strong or not preserved under natural geometric restrictions. Narrowelliptic operators often arise by projecting a classical elliptic operator to a subbundle, or by studying the operator after a dimensional reduction that concentrates dynamics in transverse directions while allowing degeneracy along a controlled set of directions. Concrete constructions provide model examples where the operator remains elliptic in transverse directions but loses ellipticity along the cone, yielding a spectrum and regularity theory that interpolate between elliptic and degenerate cases.

Properties and implications vary with the precise definition of the narrow region. In many treatments, one can develop partial regularity results, microlocal analysis techniques, and a modified index theory that accounts for the restricted ellipticity. Stability under perturbations and boundary value problems often require adapted function spaces and boundary conditions tailored to the narrow geometry.

See also elliptic operator, microlocal analysis, spectral theory, and partial differential equations.