# Help in Analysis

I'm studying the article "On W1;p Estimates for Elliptic Equations in Divergence Form" of L. A. CAFFARELLI and I. PERAL. There, you can find

We use the classical Hardy-Littlewood maximal operator, namely, $$Mf(x)= \sup_{x\ \in Q, Q \mbox{cube}} \dfrac{1}{|Q|} \int_{Q} |f(y)|dy$$ which satisfies the (1,1) weak-type inequality and obviously, by interpolation, the $L^{p}$-estimate (see, for intance [8]="Guzman, M. de, Differentiation of Integrals in Rn, Lecture Notes in Mathematics No. 481,Springer, New York, 1975.")

I did not find [8] and I would like to know the (1,1) weak-type inequality and the $L^{p}$-estimate above. Moreover, there is the statement

If $M(|\nabla u|^{p}) \in L^{q/p}$ a fortiori $\nabla u \in L^{q}$.

I would like to understand this too. I thank if you can help me.

-
The second is easy: $M(f)\ge |f|$ for every function $f$, by definition. So, $|\nabla u|^p\in L^{q/p}$, which is the same as $|\nabla u|\in L^q$. –  user31373 Jul 14 '12 at 0:22
Also, have you checked Wikipedia? –  user31373 Jul 14 '12 at 0:30