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What is the best intuition for Calderon-Zygmund operators? Why are they so important in singular integrals, and complementary, which singular integrals don't they cover?

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Whether there is such a thing as a "best" intuition for anything is debatable. People's brains are different, so what works well as intuition for one person may not work well for another. –  Qiaochu Yuan Jan 31 '11 at 23:13
Have you read Stein's book? Differentiability properties and so on? –  Jonas Teuwen Jan 31 '11 at 23:18
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