Calderón–Zygmund lemma

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

The lemma shows a way, an integrable function in their split "small" and "large" shares and to control the "big" shares. This decomposition is as essential for the proof of the atomic decomposition of real Hardy functions.

Lemma of Calderón - Zygmund

Let be a non-negative, integrable function, and is a positive constant. Then there is a separation of the following properties:

Calderón - Zygmund decomposition

Be an integrable function and a positive constant with

Then there exists a decomposition with and a sequence of cubes (or balls ) with the following properties:

  • For almost all
  • Each function has its support in the cube ( ball ), and it is
Hardy space
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