Semi-elliptic operators generated by vector fields

E. Shargorodsky. Semi-elliptic operators generated by vector fields. 2002. <http://eudml.org/doc/285992>.

@book{E2002,
abstract = {Let M be a $C^\{∞\}$-smooth n-dimensional manifold and ν₁, ..., νₙ be $C^\{∞\}$-smooth vector fields on M which span the tangent space $T_\{x\}M$ at each point x ∈ M. The vector fields ν₁, ..., νₙ may have nonzero commutators. We construct a calculus of pseudodifferential operators (ψDOs) which act on sections of vector bundles over M and have symbols belonging to anisotropic analogues of the Hörmander classes $S^\{r\}_\{ϱ,δ\}$, and apply it to semi-elliptic operators generated by ν₁,...,νₙ. The results obtained include the formula expressing the symbol of a ψDO in terms of its amplitude, the formula for the symbol of the adjoint ψDO, the theorem on composition of ψDOs, the L₂-boundedness of ψDOs with symbols from $S⁰_\{ϱ,δ\}$, 0 ≤ δ < ϱ ≤ 1, and the $L_\{p\}$-boundedness, 1 < p < ∞, of ψDOs with symbols from $S⁰_\{1,δ\}$. We prove that a semi-elliptic ψDO A is Fredholm if M is compact and obtain analogues of the well known “elliptic” results concerning the resolvent and complex powers of A and the exponential $e^\{-tA\}$. We also prove an asymptotic formula for the spectral function of A with a remainder estimate and more precise, in particular two-term, asymptotic formulae for the Riesz means of the spectral function.},
author = {E. Shargorodsky},
keywords = {analysis on compact manifolds; pseudodifferential operators; semi-elliptic operators},
language = {eng},
title = {Semi-elliptic operators generated by vector fields},
url = {http://eudml.org/doc/285992},
year = {2002},
}

TY - BOOK
AU - E. Shargorodsky
TI - Semi-elliptic operators generated by vector fields
PY - 2002
AB - Let M be a $C^{∞}$-smooth n-dimensional manifold and ν₁, ..., νₙ be $C^{∞}$-smooth vector fields on M which span the tangent space $T_{x}M$ at each point x ∈ M. The vector fields ν₁, ..., νₙ may have nonzero commutators. We construct a calculus of pseudodifferential operators (ψDOs) which act on sections of vector bundles over M and have symbols belonging to anisotropic analogues of the Hörmander classes $S^{r}_{ϱ,δ}$, and apply it to semi-elliptic operators generated by ν₁,...,νₙ. The results obtained include the formula expressing the symbol of a ψDO in terms of its amplitude, the formula for the symbol of the adjoint ψDO, the theorem on composition of ψDOs, the L₂-boundedness of ψDOs with symbols from $S⁰_{ϱ,δ}$, 0 ≤ δ < ϱ ≤ 1, and the $L_{p}$-boundedness, 1 < p < ∞, of ψDOs with symbols from $S⁰_{1,δ}$. We prove that a semi-elliptic ψDO A is Fredholm if M is compact and obtain analogues of the well known “elliptic” results concerning the resolvent and complex powers of A and the exponential $e^{-tA}$. We also prove an asymptotic formula for the spectral function of A with a remainder estimate and more precise, in particular two-term, asymptotic formulae for the Riesz means of the spectral function.
LA - eng
KW - analysis on compact manifolds; pseudodifferential operators; semi-elliptic operators
UR - http://eudml.org/doc/285992
ER -