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I added an example of the divide and conquer trick when estimating a (not completely trivial) series. Also, a dyadic decomposition is used to take advantage of the fact that a function remains more or less constant in a dyadic annulus. However the example turned out to be more messy than I originally thought and the result itself is quite elementary. Tell me what you think.
Also, the exponent in the denominator has to be "beta" instead of "alpha" (the whole typo looked like an uncorrected copy-paste)
Typo: In region II, the denominator in the integral has to be "x-y" instead of "y".
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Tricki