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Who in history introduced the notation $X\lesssim Y$ for meaning $X\leq CY$ for some constant $C$? I've seen this notation in modern literature in PDE a lot. (See for instance the notation section of Nonlinear dispersive equations: local and global analysis by Terence Tao.) But it seldom appears in undergraduate textbooks.

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    $\begingroup$ I never saw this notation with this meaning (but with a different one). But then I do not do PDE. Still can you give a specific example where it is used with the meaning you claim (and this is apparent). $\endgroup$ – quid May 20 '16 at 19:55
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    $\begingroup$ Never actually saw it, but I could rather imagine it has similar meaning to $X \approx Y$; just transferred to relative symbols. $\endgroup$ – qwertz May 20 '16 at 20:02
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Terry Tao says the following in his lecture notes on real analysis:

... we use ${X \lesssim_p Y}$ as shorthand for the inequality ${X \leq C_p Y}$ for some constant ${C_p}$ depending only on ${p}$ (it can be a different constant at each use of the ${\lesssim_p}$ notation). [Note: in analytic number theory, it is more customary to use ${\ll_p}$ instead of ${\lesssim_p}$, following Vinogradov. However, in analysis ${\ll}$ is sometimes used instead to denote “much smaller than”, e.g. ${X \ll Y}$ denotes the assertion ${X \leq cY}$ for some sufficiently small constant ${c}$.]

See also this set of notes on harmonic analysis regarding the Vinogradov notation.

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