Lp boundedness of Calderón–Zygmund operators

Statement

Every Calderón–Zygmund operator $T$ extends uniquely from $L^2\cap L^p$ to a bounded operator on $L^p(\mathbb R^n)$ for $1<p<\infty$ , with

\|Tf\|_p\le C_p\|f\|_p.

The constant depends on dimension, $p$ , the kernel size and regularity bounds, its Hölder exponent, and the $L^2$ operator norm.

Proof mechanism and endpoints

The Calderón–Zygmund decomposition splits an integrable input into a bounded part and localized pieces of mean zero. The $L^2$ bound handles the bounded part; cancellation and kernel regularity control the others off enlarged supporting cubes. This gives the distribution-function estimate $|\{|Tf|>\lambda\}|\le C\|f\|_1/\lambda$ . Interpolation gives $1<p<2$ , and applying the same argument to the adjoint and using duality gives $2<p<\infty$ . The cited lecture notes supply the full decomposition and interpolation proof.

Neither strong $L^1$ nor strong $L^\infty$ boundedness is part of this conclusion.

References

Tao, Fourier analysis lecture notes 4, §2.