Khintchine inequality

In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Roman alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick $N$ complex numbers $x\_1,...,x\_N\; inmathbb\{C\}$, and add them together each multiplied by a random sign $pm\; 1$, then the expected value of its size, or the size it will be closest to on average, will be not too far off from $sqrt\{|x\_1|^\{2\}+cdots\; +\; |x\_N|^\{2$.

tatement of theorem

Let $\{epsilon\_\{n\}\}\_\{n=1\}^\{N\}$ be i.i.d. random variables with $P(epsilon\_n=pm1)=frac12$ for every $n=1ldots\; N$, i.e., a Rademacher sequence. Let

:$A\_p\; left(\; sum\_\{n=1\}^\{N\}|x\_\{n\}|^\{2\}\; ight)^\{frac\{1\}\{2\; leq\; left(mathbb\{E\}Big|sum\_\{n=1\}^\{N\}epsilon\_\{n\}x\_\{n\}Big|^\{p\}\; ight)^\{1/p\}\; leq\; B\_p\; left(sum\_\{n=1\}^\{N\}|x\_\{n\}|^\{2\}\; ight)^\{frac\{1\}\{2$

for some constants $A\_p,B\_p>0$ depending only on $p$ (see Expected value for notation).

Uses in analysis

The uses of this inequality are not limited to applications in probability theory. One example of its use in analysis is the following: if we let $T$ be a linear operator between two L^"p" spaces $L^p(X,mu)$ and $L^p(Y,\; u)$,

:$left|left(sum\_\{n=1\}^\{N\}|Tf\_n|^\{2\}\; ight)^\{frac\{1\}\{2\; ight|\_\{L^p(Y,\; u)\}leq\; C\_pleft|left(sum\_\{n=1\}^\{N\}|f\_\{n\}|^\{2\}\; ight)^\{frac\{1\}\{2\; ight|\_\{L^p(X,mu)\}$

for some constant $C\_p>0$ depending only on $p$ and $|T|$.

References

*Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5