By Brian Street
This e-book develops a brand new conception of multi-parameter singular integrals linked to Carnot-Carathéodory balls. Brian highway first info the classical concept of Calderón-Zygmund singular integrals and functions to linear partial differential equations. He then outlines the idea of multi-parameter Carnot-Carathéodory geometry, the place the most software is a quantitative model of the classical theorem of Frobenius. highway then offers numerous examples of multi-parameter singular integrals bobbing up evidently in a number of difficulties. the ultimate bankruptcy of the publication develops a common thought of singular integrals that generalizes and unifies those examples. this can be one of many first basic theories of multi-parameter singular integrals that is going past the product idea of singular integrals and their analogs. Multi-parameter Singular Integrals will curiosity graduate scholars and researchers operating in singular integrals and similar fields.
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This booklet develops a brand new idea of multi-parameter singular integrals linked to Carnot-Carathéodory balls. Brian highway first info the classical idea of Calderón-Zygmund singular integrals and purposes to linear partial differential equations. He then outlines the idea of multi-parameter Carnot-Carathéodory geometry, the place the most instrument is a quantitative model of the classical theorem of Frobenius.
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Additional info for Multi-parameter singular integrals
As pointed out in the previous remark, though, this only uniquely specifies the kernel as an element of S0 (Rn ) , and not as a distribution. 26. For instance, consider (ii). If Op (K) : S0 (Rn ) → S0 (Rn ),11 then it follows by continuity that for any bounded set B ⊂ S0 (Rn ), Op (K) B is also bounded. (ii) takes this automatic fact, and instead assumes a scale invariant version of it. This leads us directly to the first main property of Calder´on-Zygmund kernels. 31. Suppose K1 , K2 ∈ S0 (Rn ) are Calder´on-Zygmund kernels of order s, t ∈ R, respectively.
Take M = M (α, β, m) and E (x, z), so that E ranges m = m (m) large to be chosen later. Set E (x, z) = −M x over a bounded subset of P0 as E ranges over B. We have, 2−j ∂x α j F (2 ) (x, z) = 2−2M j−tj = 2−2M j−tj 2−j ∂x α + 2−2M j−tj 2−j ∂x α T M T M 2−j ∂x α T M j E (2 ) (·, z) (x) j φ 2j (· − x) E (2 ) (·, z) (x) 1 − φ 2j (· − x) j E (2 ) (·, z) (x) . We bound these two terms separately. Using the cancellation condition applied with φR,z replaced by 2−nj−2M j M j φ 2j (· − x) E (2 ) (·, z) =: ψ we see 2−2M j−tj 2−j ∂x 2nj−tj α T M 2−j ∂x j φ 2j (· − x) E (2 ) (·, z) (x) α Tψ 2nj 1 + 2j |x − z| −m ; ´ THE CALDERON-ZYGMUND THEORY I: ELLIPTICITY 25 −m here, we have used the rapid decrease of E to obtain the factor 1 + 2j |x − z| Using, now, the growth condition 2−2M j−tj 2−j ∂x α M T 1 − φ 2j (· − x) 2−2M j−tj M y T |y−x|>2−j j E (2 ) (·, z) (x) (x, y) 1 − φ 2j (y − x) −n−t−2M 2−2M j−tj |x − y| .
15, we have Gj j ≥ k, φ ∈ B ⊂ P is a bounded set. For j < k (2j ) (2j ) (2k ) 1. We have, define G =E F . A simple estimate shows |∂ α G (x, z)| j j x j for N sufficiently large, (2k ) 2jt 2−N |j−k| ∂xα Gj (x, z) j≥k 2jt 2−N |j−k| 2k(n+|α|) 2k(t+n+|α|) , j≥k (2j ) 2jt ∂xα Gj (x, z) j