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By S Ponnusamy; Herb Silverman

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1. 3. Most of our continuity results will be consequences of the following result of G. Bourdaud [BG], itself following pioneering work of Stein [S2]. A. 1) p(x, D) : H s+m,p −→ H s,p provided 0 < s < r. 2) p(x, D) : C∗s+m −→ C∗s . We will present Bourdaud’s proof below. First we record some implications. m satisfies the hypotheses for all r > 0. B. 2) for all s ∈ R. 0 It is known that elements of OP S1,1 need not be bounded on Lp , even for p = 2, but by duality and interpolation we have the following.

8) supp fˆk ⊂ {ξ : |ξ| ≤ A · 2k+1 }, k ≥ 0. 9) ∞ fk k=0 4ks |fk |2 ≤C H s,p 1/2 Lp k=0 . 16). This is necm m than for X s Scl . 10) ∞ p(x, ξ) = p(x, ξ)ψj (ξ) = j=0 pj (x, ξ). 11) pjkl (x)βkl (2−j ξ)ψj# (ξ), pj (x, ξ) = k,l where ψ1# (ξ) has support on 1/2 < |ξ| < 2 and is 1 on supp ψ1 , ψj# (ξ) = ψ1# (2−j+1 ξ), with an analogous decomposition for p0 (ξ). 10) and summing over j, we obtain p(x, ξ) as a sum of a rapidly 47 decreasing sequence of elementary symbols. 13) |Qk (x)| ≤ C, Qk Cr ≤ C · 2krδ .

22) Dξα P (ξ) L(H1 ,H2 ) ≤ Cα ξ −|α| for all α ≥ 0. 23) P (D) : Lp (Rn , H1 ) −→ Lp (Rn , H2 ) for 1 < p < ∞. This leads to an important circle of results known as Littlewood-Paley Theory. 24) ϕj (ξ)2 1= j=0 where ϕj ∈ C ∞ , ϕ0 (ξ) is supported on |ξ| ≤ 1, ϕ1 (ξ) is supported on 1/2 ≤ |ξ| ≤ 2, and ϕj (ξ) = ϕ1 (21−j ξ) for j ≥ 2. 26) Φ(f ) = ϕ0 (D)f, ϕ1 (D)f, ϕ2 (D)f, . . This is clearly an isometry, though of course it is not surjective. 27) Φ∗ (g0 , g1 , g2 , . . 28) on L2 (Rn ). 29) Φ(ξ) = (ϕ0 (ξ), ϕ1 (ξ), ϕ2 (ξ), .

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