Let $\phi\in C_c^{\infty}(B_r)$ be a nonnegative function where $B_r=B_r(x_0)$ be a ball of radius $r$ centered at $x_0$. Let $p>2$, then is it possible to write the following inequality:
$$
|\phi^p(x)-\phi^p(y)|\leq C|\phi(x)-\phi(y)|^p
$$
for some positive constant $C$. Here $C_c^{\infty}(B_r)$ denotes the set of infinitely differentiable functions with compact support in $B_r$.
Any help will be very much appreciated.
Thanking you.
Best Answer
This has nothing to do with $C^\infty_c$; the question is just whether $$|s^p-t^p|\le C|s-t|^p$$for $s,t\ge0$.
The answer is no. For small $h>0$ the binomial theorem or just a little calculus shows that$$(1+h)^p-1^p\sim ph >> h^p.$$