바벤코-베크너 부등식

수학에서, 바벤코-베크너 부등식 (K.Ivan Babenko 및 William E.Beckner의 이름을 따서 지어짐)은 하우스도르프-영 부등식의 더 정확한 형태 중 하나이다.

개념

n차원 푸리에 변환의 (q, p norm)은 다음과 같이 정의된다.

$\|{\mathcal {F}}\|_{q,p}=\sup _{f\in L^{p}(\mathbb {R} ^{n})}{\frac {\|{\mathcal {F}}f\|_{q}}{\|f\|_{p}}},{\text{ where }}1<p\leq 2,{\text{ and }}{\frac {1}{p}}+{\frac {1}{q}}=1.$

1961년, Babenko는 이 norm의 값을 q가 짝수일 때에 한정하여 구하였고. 1975년에, Beckner는 에르미트 함수가 푸리에 변환의 고유벡터라는 것을 이용하여 이 norm의 값을 2보다 작지 않은 모든 실수 q에 대하여 정확하게 구하였다. 해당 값은 다음과 같다:

$\|{\mathcal {F}}\|_{q,p}=\left(p^{1/p}/q^{1/q}\right)^{n/2}.$

엄밀히 말하자면, 이 부등식은, 사용되는 푸리에 변환의 정의가, 1차원에서는 아래와 같이 정의되고 n차원에서는 seperable kernel n개를 곱한 것을 kernel로 하는 방식으로 정의되어야만 성립한다.

$g(y)\approx \int _{\mathbb {R} }e^{-2\pi ixy}f(x)\,dx{\text{ and }}f(x)\approx \int _{\mathbb {R} }e^{2\pi ixy}g(y)\,dy,$

then we have

$\left(\int _{\mathbb {R} }|g(y)|^{q}\,dy\right)^{1/q}\leq \left(p^{1/p}/q^{1/q}\right)^{1/2}\left(\int _{\mathbb {R} }|f(x)|^{p}\,dx\right)^{1/p}$

아래의 증명은 르베그 측도와 Bernoulli trial 및 중심 극한 정리를 이용한다:

Main ideas of proof

Throughout this sketch of a proof, let

$1<p\leq 2,\quad {\frac {1}{p}}+{\frac {1}{q}}=1,\quad {\text{and}}\quad \omega ={\sqrt {1-p}}=i{\sqrt {p-1}}.$

(Except for q, we will more or less follow the notation of Beckner.)

The two-point lemma

Let $d\nu (x)$ be the discrete measure with weight $1/2$ at the points $x=\pm 1.$ Then the operator

$C:a+bx\rightarrow a+\omega bx$

maps $L^{p}(d\nu )$ to $L^{q}(d\nu )$ with norm 1; that is,

$\left[\int |a+\omega bx|^{q}d\nu (x)\right]^{1/q}\leq \left[\int |a+bx|^{p}d\nu (x)\right]^{1/p},$

or more explicitly,

$\left[{\frac {|a+\omega b|^{q}+|a-\omega b|^{q}}{2}}\right]^{1/q}\leq \left[{\frac {|a+b|^{p}+|a-b|^{p}}{2}}\right]^{1/p}$

for any complex a, b. (See Beckner's paper for the proof of his "two-point lemma".)

A sequence of Bernoulli trials

The measure $d\nu$ that was introduced above is actually a fair Bernoulli trial with mean 0 and variance 1. Consider the sum of a sequence of n such Bernoulli trials, independent and normalized so that the standard deviation remains 1. We obtain the measure $d\nu _{n}(x)$ which is the n-fold convolution of $d\nu ({\sqrt {n}}x)$ with itself. The next step is to extend the operator C defined on the two-point space above to an operator defined on the (n + 1)-point space of $d\nu _{n}(x)$ with respect to the elementary symmetric polynomials.

Convergence to standard normal distribution

The sequence $d\nu _{n}(x)$ converges weakly to the standard normal probability distribution $d\mu (x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}\,dx$ with respect to functions of polynomial growth. In the limit, the extension of the operator C above in terms of the elementary symmetric polynomials with respect to the measure $d\nu _{n}(x)$ is expressed as an operator T in terms of the Hermite polynomials with respect to the standard normal distribution. These Hermite functions are the eigenfunctions of the Fourier transform, and the (q, p)-norm of the Fourier transform is obtained as a result after some renormalization.