Suppose $\phi(x)\in C_c^\infty(\mathbb R^n)$ and its support does not contain $0$.
Is the linear map $h(t)\mapsto\int_0^\infty \frac{\phi(sx)h(s)}{\sqrt s}ds$ bounded from $L^2(\mathbb R^+_t)$ to $L^2(\mathbb R^n_x)$?
My original goal is to prove $g(t,x)\mapsto\int_0^\infty t^{-(n+1/2)}(\psi(\frac\cdot t)\ast g(t,\cdot))(x)dt$ is bounded from $L^4(\mathbb R^n_x;L^2(\mathbb R^+_t))$ to $L^4(\mathbb R^n_x)$, where $\hat \psi\in C_c^\infty$ and support does not contain $0$.