Analysing a Lebesgue integral inequality for $|t^{-n} \phi(x/t)|$, where $\phi \in C_c^\infty \cap L^1$ with $\| \phi \|_1 = 1$.

Context. Let $C^k(\mathbb R^n)$ denote the space of functions defined on $\mathbb R^n$ that are $k$ times continuously differentiable, where $k \geqslant 1$ is an integer. As usual, define $C^\infty(\mathbb R^n)$ to be the intersection of all $C^k(\mathbb R^n)$ and let $C_c(\mathbb R^n)$ denote the space of continuous functions on $\mathbb R^n$ with compact support. Furthermore, let $C_c^\infty(\mathbb R^n) = C_c(\mathbb R^n) \cap C^\infty(\mathbb R^n).$

Consider the usual Euclidean space $\mathbb R^n$ equipped with the Lebesgue measure. Furthermore, let $\phi \in C_c^\infty(\mathbb R^n)$ be a kernel with unitary norm, that is, $\| \phi \|_1 = 1$ (here, $\| \phi \|_1$ stands for the $L^1$-norm of $\phi$) and define the dillations

$$ \phi_t(x) = t^{-n} \phi\left( \frac{x}{t} \right), $$

for every $x \in \mathbb R^n$ and $t > 0$. Finally, let $\delta > 0$ be an arbitrary fixed constant. In an article I am reading, the authors claim the following.

Now, we use the very well-known inequality$$ \int_{|z| \geqslant \delta}|\phi_t(z)| \, dz \leqslant c \, \int_{|z| \geqslant \delta} \frac{t}{|z|^{n+1}} \, dz \leqslant c \, t, $$where the constant $c$ is independet of $t$.

Basically, I am wondering where this inequality comes from and if anyone has a reference that approaches this result.

For what it's worth, under these conditions I am already aware that $\phi_t \in C_c^\infty(\mathbb R^n)$ with $\| \phi_t \|_1 = 1$ (for details about this, see this post).

Thanks for any help in advance.