# Is there a Calderon-Zygmund decomposition for $L^p$ function

The Calderon-Zygmund decomposition for a $L^1$ function is well known, which says for any $f\in L^1$, then we can decompose $f$ into a good term $f$ and a bad term $\sum b_k$, such that for any $\alpha>0$, we can make $f(x)\leq C\alpha$ and for each $b_k$, supported in non-overlapping cubes $Q_k$ and we have the cancellation property $\int{b_kdx=0}$, furthermore, we have $\sum|Q_k|\leq\frac{C}{\alpha}\|f\|_1$.

I wonder if there is similar result valid for $L^p$ ($1<p\leq \infty$) function with the above $L^1$ norm replaced by $L^p$ norm. I think the extreme case $p=\infty$ will fail, but for $1<p<\infty$, I hope there exist some related result.

Through google, I do find some C-Z decomposition in other spaces such as the sobolev type space, but I have not found result for $L^p$ case yet.

One can also find this in Loukas Grafakos's $\textit{Classical Fourier Analysis}$ page 303 exercise 4.3.8. The question is broken up into parts that should be easy to handle.
Several people have considered with this question. An excellent paper that comes to mind is Anthony Carbery's Variants of the Calderon--Zygmund theory for $L^p$-spaces which appeared in Revista Matematica Iberoamericana, Volume 2, Number 4 in 1986. There are also several useful references that appear in Carbery's paper.