Prove that $ \Vert (\sum_{j \in \mathbb{Z}} |\phi_{2^j} * f|^2)^{1/2}\ \Vert_p \le C\Vert f\Vert_p $

Question

I was trying to prove the following inequality: Given $\phi$ a Schwartz function in $\mathbb{R}^d$ such that $\int\phi=0$. Let $\phi_t(x)=t^{-d}\phi(xt^{-1})$. I need to prove that: $$ \Vert (\sum_{j \in \mathbb{Z}} |\phi_{2^j} * f|^2)^{1/2}\ \Vert_p \le C\Vert f\Vert_p $$ for all $f \in L^p(\mathbb{R}^d)$. I think that the idea is to prove some weak estimate and conclude by interpolation theorem, but I do not know how to do it.