Lower Bounds for Pseudodifferential Operators

Abstract

We start off by fixing some notation (see Sjöstrand [6]). Let X be an open subset of R ⁿ (more generally, X can be a C ^∞ n-dimensional manifold without boundary) and let ∑ ⊂ T * (X\0 ≃. X × (R ⁿ \{0}) be a C^∞ conic sub-manifold. With µ∈ R and h ∈ Z ₊ = {0, 1, 2,…}, we denote by N ^µ,h (X, ∑) the set of all classical symbols of order µ, p(x,ξ) ∼ ∑ _{j ≥0} p _µ-j (x, ξ), such that for any j ≥ 0 one has

$$\left| {{{p}_{{\mu - j}}}\left( {x,\xi } \right)} \right|{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ < }}{{\left| \xi \right|}^{{\mu - j}}}dis{{t}_{\Sigma }}{{\left( {x,\xi } \right)}^{{{{{\left( {h - 2j} \right)}}_{ + }}}}} $$

where t ₊ =max{t, 0} and dist_∑(x,ξ) denotes the distance of x,ξ/∣ξ∣) to{ ( y,η) ∈ ∑;∣η∣ =1}. OPN^μ,h (X, ∑) will then denote the corresponding

class of (properly-supported) pseudodifferential operators.

Recall that the notation f ≲ g, stands for: for any conic subset U of T*(X)\ 0 with compact base, there exists a constant Cu > 0, for which f (x,ξ) ≤ Cug(x,ξ), ∀(x, ξ) ∈ U.