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If $f$ and $g$ are functions on the real line, and $f$ is oscillatory, then an important technique for bounding the integral $\int fg$ is applying an integration by parts, writing

$$ \int_{-\infty}^\infty f(x) g(x) = - \int_{-\infty}^\infty F(x) g'(x) $$

where $F'(x) = f(x)$. If $f$ has rapid oscillation at a point, then this oscillation will cancel out in the integral formula for $F$ in terms of $f$, and thus have neglible impact on the values of $F$, leading to a useful bound. Thus it seems integration by parts is only useful if one wants to cancel out local cancellation in a particular integral / sum. For instance, one can use integration by parts to obtain strong bounds for oscillatory integrals of the form

$$ \left| \int \psi(x) e^{\lambda i \phi(x)} \right| $$

where $\lambda$ is a large parameter, by applying in integration by parts and then applying the triangle inequality to take in the absolute value sign into the integral. On the other hand, if we consider a bump function $\eta$ supported on $[N-1,N]$ and define $\psi(x) = \eta(|x|)$, then we have

$$ \int_{-\infty}^\infty x \psi(x)\; dx = 0. $$

Now integration by parts on $[-N,N]$ expresses this integral as

$$ 2N \int \eta(x)\; dx - \int_{-N}^N \Psi(x)\; dx $$

where $\Psi'(x) = \psi(x)$. The integral here is thus $\approx N$, and so it is not clear that integration by parts has expressed the cancellation in the integral.

My question is whether integration by parts is useful only for obtaining bounds via 'local cancellation', or whether it also captures 'global' cancellation that can occur on $f$ on different parts of the domain of $f$. If integration by parts cannot capture this bound, what techniques are available to study 'global cancellation' when estimating oscillatory quantities. Moreover, are there operators $T: L^p(X) \to L^q(Y)$ where global cancellation is useful when obtaining boundedness of these operators, or is local cancellation the only obstruction to boundedness? If local cancellation is only useful to $L^p$ estimates, does global cancellation become more relevant when proving the boundedness of operators in other norm spaces?

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    $\begingroup$ The short answer (in such generality as you ask about) is "it all depends on the particulars of the problem and your inventiveness in non-trivial cases". I guess some people will post some examples but there seems to be no general rule or method that always works or always does not. At least, that is what my personal experience suggests :-) $\endgroup$ – fedja Feb 12 at 23:44

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