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Abstract

Let $M$ be a $C^{\infty}$-smooth $n$-dimensional manifold and $\nu_1, \dots, \nu_n$ be $C^{\infty}$-smooth vector fields on $M$ which span the tangent space $T_xM$ at each point $x \in M$. The vector fields $\nu_1, \dots, \nu_n$ may have nonzero commutators. We construct a calculus of pseudodifferential operators ($\psi$DOs) which act on sections of vector bundles over~$M$ and have symbols belonging to anisotropic analogues of the Hörmander classes~$S^r_{\varrho, \delta}$, and apply it to semi-elliptic operators generated by $\nu_1, \dots, \nu_n$. The results obtained include the formula expressing the symbol of a $\psi$DO in terms of its amplitude, the formula for the symbol of the adjoint $\psi$DO, the theorem on composition of $\psi$DOs, the $L_2$-boundedness of $\psi$DOs with symbols from $S^0_{\varrho, \delta}$, $0 \le \delta < \varrho \le 1$, and the $L_p$-boundedness, $1 < p < \infty$, of $\psi$DOs with symbols from~$S^0_{1, \delta}$. We prove that a semi-elliptic $\psi$DO $A$ 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.