Harmonic analysis on Euclidean spaces Singular integrals and Calderón–Zygmund theory Theorem

Kolmogorov's weak-type estimate for the conjugate function

Builds on Conjugate function Lebesgue measure Weak Lp space Calderón–Zygmund lemma

Statement

Let $\mathbb{T} = \R / 2\pi\Z$ , identified with $[-\pi, \pi)$ with Lebesgue measure $\abs{\cdot}$ . There is an absolute constant $C$ with the following property. For every integrable function $f$ on $\mathbb{T}$ , with conjugate function $\tilde f$ , and every $\lambda > 0$ , $\abs{\set{\theta \in \mathbb{T} : \abs{\tilde f(\theta)} > \lambda}} \le \frac{C}{\lambda} \int_{\mathbb{T}} \abs{f(\theta)} \, d\theta.$ That is, $f \mapsto \tilde f$ is of weak type $(1, 1)$ .

Remarks

A real-variable proof uses the Calderón–Zygmund lemma at height $\lambda$ to write $f = g + b$ . The good part $g$ is bounded by $2\lambda$ and is handled by the $L^2$ bound and Chebyshev's inequality. The bad part $b$ has mean zero on each interval $Q_k$ . So the smoothness of the kernel $\cot(t/2)$ away from $0$ makes $\tilde b$ small off the doubled intervals. These have total length at most $C\lambda^{-1}\norm{f}_1$ .

The strong bound $\norm{\tilde f}_{L^1} \le C\norm{f}_{L^1}$ fails. For example, if $f$ is the Poisson kernel $P_r$ and $r \to 1^-$ , then $\norm{P_r}_{L^1}$ stays bounded while $\tilde P_r$ , the conjugate Poisson kernel, has $L^1$ norm comparable to $\log\frac{1}{1 - r}$ .

So $\tilde f$ is finite almost everywhere for $f \in L^1$ , and it lies in weak $L^1$ , but it need not be integrable.

Going deeper

For a student

Together with the $L^2$ bound from Parseval's identity, the Marcinkiewicz interpolation theorem turns this estimate into the $L^p$ bounds of the M. Riesz theorem for $1 < p \le 2$ , and duality covers $2 \le p < \infty$ . It is the model for the weak type $(1, 1)$ bounds of all Calderón–Zygmund operators.

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Borel sigma-algebra concept
Conjugate function concept
Euclidean space object
Lebesgue integral concept
Outer measure concept
Weak Lp space concept
Lebesgue measure object
Calderón–Zygmund lemma statement
Kolmogorov's weak-type estimate for the conjugate function statement

Tag	`6VJDJZAYKA`
URI	`m2://6VJDJZAYKA`
Kind	statement, theorem
Writing	W0 stub
MSC	42A50
Status	This copy of the site has no build data. Run `m2 status 6VJDJZAYKA`.
About	Conjugate function
Uses	Conjugate function, Lebesgue measure, Weak Lp space, Calderón–Zygmund lemma