M ‐elliptic pseudo‐differential operators on L^p(ℝⁿ)

Abstract

Key words M -ellipticity, minimal and maximal pseudo-differential operators, spectrum, essential spectrum MSC (2000) 47G30, 35S05We construct the minimal and maximal extensions in L p (R n ), 1 < p < ∞, for M -elliptic pseudo-differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M -elliptic pseudo-differential operators. For M -elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given.

M-Hypoelliptic pseudo-differential operators onLp(ℝn)

“…We give in this article similar results as Wong's ones in [5], but in the context of the symbol classes HM m, '…”

“…Using the same arguments as in the paper [5] by Wong we can prove that the range R(T ,0 À I) of T ,0 À I is dense in L p (R n ). So, by Theorem 4.1 the proof is complete.…”

“…In this section we shall recall some definitions and results in the paper [5] by Wong. For 2 M m , Ã , m > 0, we can consider the linear continuous mapping T : S ! S as a linear operator,…”

“…These classes of symbols are both subclasses of M m , Ã and have been introduced by Garello and Morando in [3], [4]. Some definitions and results from Wong's paper [5] concerning minimal and maximal operators are recalled in Section 2.…”

“…Wong's results in the paper [5] have been stated in the framework of the symbol classes EM m , Ã of M-elliptic symbols. We give in this article similar results as Wong's ones in [5], but in the context of the symbol classes HM m, '…”