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If your problem can be expressed in terms of convolutions and inner products then take the Fourier transform
Additive combinatorics front page
Fourier transforms front page
Use Fourier identities
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Littlewood-Paley heuristic for derivative
Geometric view of Hölder's inequality
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Try to prove a stronger result
Use self-similarity to get a limit from an inferior or superior limit.
Prove a consequence first
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