Problem: If $a$ is a smooth homogeneous function of degree $m$, show that $$ex |dot lap_ju(x)|leq C2^{jm}(Mu)(x), eex$$ where $$ex (Mf)(x)=sup_{r>0}f{1}{|B(x,r)|} int_{B(x,r)}|u(y)|
d y eex$$ is the Hardy-Littlewood maximal function.
Proof: $$eex ea dot lap_ja(D)u &=calF^{-1}sex{phi(2^{-j}xi)a(xi)hat u(xi)}\ &=c_dcalF^{-1} sex{phi(2^{-j}xi)a(xi)}*u\ &=c_dcalF^{-1}sex{phi(2^{-j}xi)a(2^{-j}cdot 2^jxi)}*u\ &=c_d 2^{jd} calF^{-1}sex{phi(xi)a(2^jxi)}(2^jcdot)*u\ &=c_d 2^{jd} calF^{-1}(phi(xi)2^{jm}a(xi))(2^jcdot)*u\ &=c_d2^{jm}2^{jd} h_a(2^jcdot)*uqx{ h_a=calF^{-1}(phi a)in calS }. eea eeex$$ And consequently by 1.17 in the book, $$eex ea &quad|dotlap_ja(D)u(x)| leq 2^{jm} int_{bR^d} sez{ sup_{|y'|geq |y|} 2^{jd} |h_a(2^jy')|} d ycdot Mu(x)\ &leq 2^{jm} int_{bR^d} sup_{|z'|geq |z|} |h_a(z')| d zcdot Mu(x)\ &=2^{jm} int_{bR^d} sup_{|z'|geq |z|} f{1}{(1+|z'|)^{d+1}} cdot (1+|z'|)^{d+1}|h_a(z')| d zcdot Mu(x)\ &leq C2^{jm} int_{bR^d} f{1}{(1+|z|)^{d+1}} d zcdot Mu(x) leq C2^{jm}Mu(x). eea eeex$$