Sobolev bound on convolution function

Question

Let $v:\mathbb R^d\to \mathbb R$ be smooth enough and compactly supported on $B(0,1)$. Let $\delta>0$ be a scaling parameter. Assume $v_\delta=v(\cdot/\delta)$. Let $u$ be smooth enough too. I want to prove the following inequality $$ \|u*v_\delta\|_{H^{\beta}(\mathbb R^d)}\leq C\delta^\alpha \|u*v_\delta\|_{H^{\beta+\alpha}(\mathbb R^d)} $$ I am not in general sure that this inequality is true or not! We can also assume that $v$ is positive definite on $\mathbb R^d$, if it is required.