The question is $1<p<\infty$, prove that $\|Mf\|_{L^{p,\infty}}\lesssim\|f\|_{L^{p,\infty}}$
I want to prove that for $p=1$, $Mf$ is of restricted weak type (1,1),
i.e.$\exists C, \;s.t.\;\forall \;\text{measurable set}\;A $ satisfying $\mu(A)<\infty, \|M\chi_A\|_{L^{1,\infty}} \le C {\mu(A)}$
It suffices to consider $A $ cube. (1)
By the rearrangement invariance of $L^{1,\infty}$ norm, WLOG A is a cube centered at $O$ of length 2$a$. $$M\chi_A(x)\sim_n M\chi_{B(0,a)}(x)\sim_n (\frac{a}{\lvert x \rvert +a})^n$$ $$\Rightarrow\mu\{x: M\chi_A(x) >\lambda\}\sim_n \frac{a^n}{\lambda}\;\Rightarrow\|M\chi_A\|_{L^{1,\infty}} \lesssim_n {\mu(A)}$$ I am wondering whether my proof is correct, especially (1). If (1) is wrong, then I want to know how to prove the origin problem. Thank you all!