I am trying to estimate the following convolution, $$f_\epsilon(x)=\int f(x-y)\rho_\epsilon(y) dy$$ where $f(x)=(1-x^2)^{-1/4}\chi_B$ where $B$ is the unit ball ($|x|<1$) and $\rho_\epsilon=\epsilon^{-1}\rho(x/\epsilon)$ and $\rho$ is the standard approximation to the identity as defined here.
I want to understand how the following norm behaves $\|f_\epsilon\|_\infty$ as a parameter of $\epsilon.$
By standard triangle inequality we have, $$|f_\epsilon|\leq \|\rho_\epsilon\|_\infty\|f\|_1=C\epsilon^{-1}.$$
And so it seems to me that $\|f_\epsilon\|_\infty \leq C \epsilon^{-1}$, but is this correct? I am not sure.