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

T(1) theorem

Builds on Calderón–Zygmund kernel Calderón–Zygmund operator Bounded mean oscillation Lp space

Statement

Let $n \ge 1$ , let $\mathcal{D}$ be the space of smooth functions with compact support on $\R^n$ , and let $\mathcal{D}'$ be its dual, the space of distributions. Let $T \colon \mathcal{D} \to \mathcal{D}'$ be a continuous linear operator with transpose $T^t$ , defined by $\langle T^t f, g \rangle = \langle Tg, f \rangle$ for $f, g \in \mathcal{D}$ . Assume that $T$ is associated with a Calderón–Zygmund kernel $K$ : for $f, g \in \mathcal{D}$ with disjoint supports, $\langle Tf, g \rangle = \int\!\!\int K(x, y) f(y) g(x) \, dy \, dx.$ Then $T$ extends to a bounded operator on $L^2(\R^n)$ if and only if the following three conditions hold:

$T(1)$ lies in the space $\mathrm{BMO}$ of functions of bounded mean oscillation;
$T^t(1)$ lies in $\mathrm{BMO}$ ;
$T$ has the weak boundedness property.

Here $T(1)$ and $T^t(1)$ are defined as elements of $\mathrm{BMO}$ modulo constants, using the kernel bounds. The weak boundedness property means that there are constants $C$ and $N$ with the following property. Let $B$ be a ball of radius $R$ , and let $\phi, \psi \in \mathcal{D}$ be supported in $B$ with $\norm{\partial^\alpha\phi}_\infty, \norm{\partial^\alpha\psi}_\infty \le R^{-\abs{\alpha}}$ for $\abs{\alpha} \le N$ . Then $\abs{\langle T\phi, \psi \rangle} \le C R^n$ .

Remarks

Guy David and Jean-Lin Journé proved the theorem in 1984. The three conditions are necessary. An $L^2$ -bounded operator with a Calderón–Zygmund kernel maps $L^\infty$ into $\mathrm{BMO}$ , and so does its transpose. The weak boundedness property follows from the $L^2$ bound and the Cauchy–Schwarz inequality. For sufficiency, one subtracts paraproducts with symbols $T(1)$ and $T^t(1)$ , which are $L^2$ bounded, and reduces to the case $T(1) = T^t(1) = 0$ . That case is handled by an almost orthogonality argument, such as the Cotlar–Stein lemma, applied to a Littlewood–Paley decomposition of $T$ .

Example: for an antisymmetric kernel, $K(x, y) = -K(y, x)$ , with $T$ defined as a principal value, the weak boundedness property holds automatically. Then the theorem reduces to the single condition $T(1) \in \mathrm{BMO}$ . A model case is the Hilbert transform, for which $T(1) = 0$ .

Without conditions 1 and 2 the conclusion fails: the kernel bounds alone do not give $L^2$ boundedness, since they carry no information on the cancellation of $T$ .

Going deeper

For a student

The theorem turns $L^2$ bounds for non-convolution operators, which cannot be checked with the Fourier transform, into simple tests. It gives a short proof of the $L^2$ boundedness of the Cauchy integral on Lipschitz curves and of the Calderón commutators. Its output, an $L^2$ -bounded operator with a Calderón–Zygmund kernel, is a Calderón–Zygmund operator, hence bounded on $L^p$ for $1 < p < \infty$ . There are versions with $T(b)$ in place of $T(1)$ , for suitable accretive functions $b$ .

Reading path

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Tag	`3FBF628XEF`
URI	`m2://3FBF628XEF`
Kind	statement, theorem
Writing	W0 stub
MSC	42B20, 47G10
Status	This copy of the site has no build data. Run `m2 status 3FBF628XEF`.
About	Calderón–Zygmund operator
Uses	Calderón–Zygmund kernel, Calderón–Zygmund operator, Bounded mean oscillation, Lp space
Same as	Wikidata Q7667741 Wikipedia T(1) theorem