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

Question

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.

Thank you in advance.

Answer 1

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.

Answer 2

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.