Marcinkiewicz interpolation theorem: Difference between revisions

Revision as of 23:10, 14 October 2011

Revision as of 18:06, 23 October 2011

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==Preliminaries==

Let ''f'' be a [[measurable function]] with real or complex values, defined on a [[measure space]] (''X'', ''F'', ωω). The [[cumulative distribution function|distribution function]] of ''f'' is defined by

:<math>\lambda_f(t) = \omega\left\{x\in X\mid |f(x)| > t\right\}.</math>

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:<math>\lambda_f(t)\leq \frac{C}{t}.</math>

The smallest constant ''C'' in the inequality above is called the '''weak <math>L^1</math> norm''' and is usually denoted by ||''f''||_1,''w'' or ||''f''||_1,&infin;∞. Similarly the space is usually denoted by ''L''^1,''w'' or ''L''^1,&infin;∞.

(Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on <math> (0,1) </math> given by <math> 1/x </math> and <math> 1/(1-x) </math>, which has norm 4 not 2.)

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:<math>\|Tf\|_{p,w} \le N_p\|f\|_p,</math>

:<math>\|Tf\|_{q,w} \le N_q\|f\|_q,</math>

so that the [[operator norm]] of ''T'' from ''L''^''p'' to ''L''^''p'',''w'' is at most ''N''<sub>''p''</sup>, and the operator norm of ''T'' from ''L''^''q'' to ''L''^''q'',''w'' is at most ''N''<sub>''q''</sup>. Then the following '''interpolation inequality''' holds for all ''r'' between ''p'' and ''q'' and all ''f''&nbsp;&isin;∈&nbsp;''L''^''r'':

:<math>\|Tf\|_r\le \gamma N_p^\delta N_q^{1-\delta}\|f\|_r</math>

where

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and

:<math>\gamma=2\left(\frac{r(q-p)}{(r-p)(q-r)}\right)^{1/r}.</math>

The constants &delta;δ and &gamma;γ can also be given for ''q''&nbsp;=&nbsp;∞ by passing to the limit.

A version of the theorem also holds more generally if ''T'' is only assumed to be a [[quasilinear]] operator. That is, there exists a constant ''C''&nbsp;>&nbsp;0 such that ''T'' satisfies

:<math>|T(f+g)(x)| \le C(|Tf(x)|+|Tg(x)|)</math>

for [[almost everywhere|almost every]] ''x''. The theorem holds precisely as stated, except with &gamma;γ replaced by

:<math>\gamma=2C\left(\frac{r(q-p)}{(r-p)(q-r)}\right)^{1/r}.</math>

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:<math>\|Tf\|_q\le C\|f\|_p.</math>

A more general formulation of the interpolation theorem is as follows:

* If ''T'' is a quasilinear operator of weak type (''p''₀, ''q''₀) and of weak type (''p''<sub>1</sup>, ''q''₁) where ''q''₀&nbsp;≠&nbsp;''q''₁, then for each &theta;θ&nbsp;&isin;∈&nbsp;(0,1), ''T'' is of type (''p'',''q''), for ''p'' and ''q'' with ''p'' ≤ ''q'' of the form

:<math>\frac{1}{p} = \frac{1-\theta}{p_0}+\frac{\theta}{p_1},\quad \frac{1}{q} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}.</math>

The latter formulation follows from the former through an application of [[Hölder's inequality]] and a duality argument.

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A famous application example is the [[Hilbert transform]]. Viewed as a [[multiplier (Fourier analysis)|multiplier]], the Hilbert transform of a function ''f'' can be computed by first taking the [[Fourier transform]] of ''f'', then multiplying by the [[sign function]], and finally applying the [[inverse Fourier transform]].

Hence [[Parseval's theorem]] easily shows that the Hilbert transform is bounded from <math>L^2</math> to <math>L^2</math>. A much less obvious fact is that it is bounded from <math>L^1</math> to <math>L^{1,w}</math>. Hence Marcinkiewicz's theorem shows that it is bounded from <math>L^p</math> to <math>L^p</math> for any 1 < ''p'' < 2. [[dual space|Duality]] arguments show that it is also bounded for 2 < ''p'' < &infin;∞. In fact, the Hilbert transform is really unbounded for ''p'' equal to 1 or &infin;∞.

Another famous example is the [[Hardy–Littlewood maximal function]], which is only quasilinear rather than linear. While <math>L^p</math> to <math>L^p</math> bounds can be derived immediately from the <math>L^1</math> to weak <math>L^1</math> estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the Hardy–Littlewood Maximal Function is trivially bounded from <math>L^\infty</math> to <math>L^\infty</math>, strong boundedness for all <math>p>1</math> follows immediately from the weak (1,1) estimate and interpolation.