Lemma of Calderón-Zygmund

The Calderón-Zygmund lemma is a mathematical result from the field of Fourier analysis or harmonic analysis . It was named after the mathematicians Alberto Calderón and Antoni Zygmund .

The lemma shows a possibility to split an integrable function into its "small" and "large" parts and to control the "large" parts. This decomposition is, for example, essential for the proof of the atomic decomposition of real Hardy functions .

Lemma of Calderón-Zygmund

Let be a non-negative, integrable function, and be a positive constant. Then there is a decomposition of with the following properties: ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ ${\displaystyle \alpha }$ ${\displaystyle \mathbb {R} ^{n}}$

1. ${\displaystyle \mathbb {R} ^{n}=F\cup \Omega }$ With ${\displaystyle F\cap \Omega =\emptyset ;}$
2. ${\displaystyle f(x)\leq \alpha }$ almost everywhere in ${\displaystyle F;}$
3. ${\displaystyle \Omega }$ is the union of cubes

${\displaystyle \Omega =\bigcup _{k}Q_{k},}$

where the interior of each cube is disjoint to the interior of every other cube. In addition, the inequality holds for every cube ${\displaystyle Q_{k}}$

${\displaystyle \alpha <{\frac {1}{m(Q_{k})}}\int _{Q_{k}}f(x)\mathrm {d} \,m(x)\leq 2^{n}\alpha .}$

Here is a measure of . ${\displaystyle m(Q_{k})}$ ${\displaystyle Q_{k}}$

Calderón-Zygmund decomposition

Let be an integrable function and a positive constant with ${\displaystyle f\in L^{1}}$ ${\displaystyle \beta }$

${\displaystyle \beta >{\frac {1}{m(\mathbb {R} ^{n})}}\int _{\mathbb {R} ^{n}}|f(x)|\mathrm {d} \,m(x).}$

Then there is a decomposition with and a sequence of cubes (or balls) with the following properties: ${\displaystyle f=g+b}$ ${\displaystyle \textstyle b=\sum _{k=1}^{\infty }b_{k}}$ ${\displaystyle (Q_{k})_{k\in \mathbb {N} }}$

• ${\displaystyle |g(x)|\leq c\beta }$ for almost everyone ${\displaystyle x\in \mathbb {R} ^{n};}$
• Every function has its bearer in the cube (ball) , and it applies ${\displaystyle b_{k}}$ ${\displaystyle Q_{k}}$

${\displaystyle \int _{Q_{k}}|b_{k}(x)|\mathrm {d} \,m(x)\leq c\beta m(B_{k})\ }$ and ${\displaystyle \int _{Q_{k}}b_{k}(x)\mathrm {d} \,m(x)=0.}$

• ${\displaystyle \sum _{k}m(Q_{k})\leq {\frac {c}{\beta }}\int _{\mathbb {R} ^{n}}|f(x)|\mathrm {d} \,m(x)}$

literature

• Elias M. Stein: Harmonic Analysis: Real-Variable Mathods, Orthogonality, and Oscillatory Integrals . Princeton University Press 1993, ISBN 0-691-03216-5 .
• Elias M. Stein: Singular Integrals And Differentiability Properties Of Functions . Princeton University Press 1970, ISBN 0-691-08079-8 .