I have as an exercise to prove some properties of the Hardy-Littlewood maximal function, namely positivity, sublinearity, homogeneity, translation invariance and scaling invariance. The first three are easy but the last two is what I am struggling with and I can't find any tips online. We assume that $f \in L^1(\mathbb{R}^n)$.
(d) (Translation invariance) $M(\tau_y f) (x) = (\tau_y Mf) (x)$, where $\tau_y f(x) = f(x +y)$
(e) (Scaling invariance) $ M(\eta_a f)(x) = (\eta_a Mf)(x)$, where $\eta_a f(x) = f(ax)$.
I think I have gotten somewhere with (d) since the ball does not really change no matter what you call it's centre so I tried doing some substitution business, but I could not quite piece it out. Any help would be appreciated. Using this definition: $$Mf(x)=\sup_{r>0}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(t)|dt$$
Let $g_r(x)=\frac{1}{|B(0,r)|}\chi_{B(0,r)}(x)$. Then $$ Mf(x)=\sup_{r>0} M_rf(x),. $$ where $$ M_rf(x)=\int_{\mathbb{R}^d}|g_r(x-t)f(t)|dt. $$
Notice that changing variables $u=t+y$ we get $$ M_r \tau_y f(x)=\int_{\mathbb{R}^d}|g_r(x-t)\tau_yf(t)|dt=\int_{\mathbb{R}^d}|g_r(x-t)f(t+y)|dt=\int_{\mathbb{R}^d}|g_r(x+y-u)f(u)|du=\tau_y M_r f(x). $$ Part d) follows by taking supremum over $r>0$.
A similar argument based on the change of variables $u=at$ gives part e).