By Smirnov V.I.

By Smirnov V.I.

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Extra resources for A course of higher mathematics, vol. 2

Sample text

For E ∈ B, j ∈ R, define F (2 ) = 2−jt T E (2 ) . Then, F E ∈ B, j ∈ R ⊂ P j j is a bounded set. The same result holds for F (2 ) = 2−jt E (2 ) T . j j P ROOF. We prove the result for 2−jt T E (2 ) ; the result for 2−jt E (2 ) T follows by taking adjoints. Fix multi-indices α, β and fix m ∈ N. We wish to show 2−j ∂x α 2−j ∂z β j F (2 ) (x, z) 2nj 1 + 2j |x − z| −m . As E ranges over B, ∂zβ E ranges over a bounded subset of P0 . Thus we may, without loss of generality, assume that β = 0. Fix φ ∈ C0∞ (B n (2)), with φ ≡ 1 on B n (1).

Taking the supremum over f ∈ B and R > 0 shows that supgˆ∈T gˆ α,β < ∞, and it follows that T is a bounded subset of S (Rn ). We wish to show that T ⊂ S0 (Rn ). Indeed, let gˆ ∈ T , so that gˆ (ξ) = K (Rξ) fˆ (ξ). 12, we wish to show ∂ξα K (Rξ) fˆ (ξ) ξ=0 = 0, ∀α. 5. We turn to (ii)⇒(iii); let K be as in (ii). 22 to decompose j j δ0 = j∈Z ς (2 ) . 20 shows that I = j∈Z Op ς (2 ) , where the sum is taken in the topology of bounded convergence as operators S0 (Rn ) → S0 (Rn ), and I j (2j ) denotes the identity operator S (Rn ) → S (Rn ).

We will prove more general analogs of these results in later chapters. 24. We say K ∈ C0∞ (Rn ) is a Calder´on-Zygmund kernel of order t ∈ (−n, ∞) if: −n−t−|α| (i) (Growth Condition) For every multi-index α, |∂xα K (x)| ≤ Cα |x| particular, we assume that K (x) is a C ∞ function for x = 0. In ´ THE CALDERON-ZYGMUND THEORY I: ELLIPTICITY 17 (ii) (Cancellation Condition) For every bounded set B ⊂ C0∞ (Rn ), we assume sup R−t K (x) φ (Rx) dx ≤ CB . 25 When −n < t < 0, the Cancellation Condition follows from the Growth Condition and the weaker assumption that K has no part supported at 0.