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Semi-elliptic operators generated by vector fields

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Shargorodsky E.

Instytut Matematyczny Polskiej Akademii Nauk

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.

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1 Shargorodsky E.

1 2002

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Semi-elliptic operators generated by vector fields