CHAPTER 2 Main result 2.1. Definitions and main results. In this paper we need several types of pseudo-differential operators, depending on the parameter α 0. For the symbol a = a(x, ξ), amplitude p = p(x, y, ξ), and any function u from the Schwartz class on Rd we define (2.1) (Opa α p)u(x) = α 2π d eiα(x−y)ξp(x, y ξ)u(y)dξdy, (2.2) (Opl α a)u(x) = α 2π d eiα(x−y)ξa(x, ξ)u(y)dξdy, (2.3) (Opαa)u(x) r = α 2π d eiα(x−y)ξa(y, ξ)u(y)dξdy. If the function a depends only on ξ, then the operators Opα(a), l Opα(a) r and Opα(a)a coincide with each other, and we simply write Opα(a). Later we formulate condi- tions on a and p which ensure boundedness of the above operators uniformly in the parameter α ≥ 1. Let Λ, Ω be two domains in Rd, and let χΛ(x), χΩ(ξ) be their characteristic functions. We always use the notation PΩ,α = Opα(χΩ). We study the operator (2.4) Tα(a) = Tα(a Λ, Ω) = χΛPΩ,αOpα(a)PΩ,αχΛ,l and its symmetrized version: Sα(a) = Sα(a Λ, Ω) = χΛPΩ,α Re Opl α (a) PΩ,αχΛ. Note that Tα(a) differs from the operator (1.1) by the presence of an extra projection PΩ,α on the left of Opα(a). l As we shall see later in Chapter 4, this difference does not affect the first two terms of the asymptotics (1.2). Let us now specify the class of symbols and amplitudes used throughout the paper. We denote by S(n1,n2,m) the set of all (complex-valued) functions p = p(x, y, ξ), which are bounded together with their partial derivatives up to order n1 w.r.t. x, n2 w.r.t. y and m w.r.t. ξ. It is convenient to define the norm in this class in the following way. For arbitrary numbers 0 and ρ 0 define (2.5) N(n1,n2,m)(p , ρ) = max 0≤n≤n1 0≤k≤n2 0≤r≤m sup x,y,ξ n+k ρr|∇n∇k∇rp(x, x y ξ y, ξ)|. 5

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