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Keep parameters unspecified until it is clear how to optimize them
Complete the square
Do an extreme case first
How to use the Cauchy-Schwarz inequality
If a parameter is generating undesirable boundary terms, try averaging over many choices of that parameter
If you don't know how to make a decision, then don't make it
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Littlewood-Paley heuristic for derivative
Littlewood-Paley heuristic
Geometric view of Hölder's inequality
Diagonal arguments
Finding an interval for rational numbers with a high denominator
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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