Marcinkiewicz interpolation theorem: Difference between revisions

Revision as of 16:37, 6 December 2018

Revision as of 12:32, 1 January 2019

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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 <math>\||''f''\||<sub>_{1,''w''}</submath> or <math>\||''f''\||<sub>_{1,∞\infty}.</submath>. Similarly the space is usually denoted by ''L''^1,''w'' or ''L''^1,∞.

(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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Similarly, one may define the [[Lp space#Weak Lp|'''weak <math>L^p</math> space''']] as the space of all functions ''f'' such that <math>|f|^p</math> belong to ''L''^1,''w'', and the '''weak <math>L^p</math> norm''' using

:<math>\|f\|_{p,w}= \|left \,||f|^p \right \|_{1,w}^{\frac{1/}{p}}.</math>

More directly, the ''L''^''p'',''w'' norm is defined as the best constant ''C'' in the inequality

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Informally, Marcinkiewicz's theorem is

:'''Theorem.''': ''Let ''T'' be a [[bounded linear operator]] from <math>L^p</math> to <math>L^{p,w}</math> and at the same time from <math>L^q</math> to <math>L^{q,w}</math>. Then ''T'' is also a bounded operator from <math>L^r</math> to <math>L^r</math> for any ''r'' between ''p'' and ''q.''.

In other words, even if you only require weak boundedness on the extremes ''p'' and ''q'', you still get regular boundedness inside. To make this more formal, one has to explain that ''T'' is bounded only on a [[Dense set|dense]] subset and can be completed. See [[Riesz-Thorin theorem]] for these details.

Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the <math>L^r</math> norm of ''T'' but this bound increases to infinity as ''r'' converges to either ''p'' or ''q''. Specifically {{harv|DiBenedetto|2002|loc=Theorem VIII.9.2}}, suppose that

:<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''_''p'', and the operator norm of ''T'' from ''L''^''q'' to ''L''^''q'',''w'' is at most ''N''_''q''. Then the following '''interpolation inequality''' holds for all ''r'' between ''p'' and ''q'' and all ''f''&nbsp;∈&nbsp;''L''^''r'':

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

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A version of the theorem also holds more generally if ''T'' is only assumed to be a quasilinear operator in the following sense: 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 γ replaced by

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

An operator ''T'' (possibly quasilinear) satisfying an estimate of the form

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

is said to be of '''weak type (''p'',''q'')'''. An operator is simply of type (''p'',''q'') if ''T'' is a bounded transformation from ''L^p'' to ''L^q'':

:<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''₁, ''q''₁) where ''q''₀&nbsp;≠&nbsp;''q''₁, then for each θ&nbsp;∈&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.{{Citation needed|reason=How to use Hölder's inequality and the special case?|date=June 2016}}

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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'' < ∞. In fact, the Hilbert transform is really unbounded for ''p'' equal to 1 or ∞.

Another famous example is the [[Hardy–Littlewood maximal function]], which is only [[sublinear operator]] 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. The weak (1,1) estimate can be obtained from the [[Vitali covering lemma]].

==History==