\psi_{{\mathbf{t}}}) \subset H_{{\mathbf{t}}}(\psi^{*}_{{\mathbf{t}}}).$$

The contraction property {#the-contraction-property .unnumbered}
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Let $\mu$ be a non-atomic finite doubling measure supported on a bounded subset $\mathbf{S}$ of $\Omega$. We recall that $\mu$ is doubling if there is a constant $\lambda > 1$ such that, for any ball $B$ with centre in ${\mathbf{S}}$, we have $$\mu(2B) \leq \lambda \mu(B),$$ where, for a ball $B$ of radius $r$, we denote by $cB$ the ball with the same centre and radius $cr$. We say that $\mu$ is contracting with respect to $(I, \Psi)$ if, for any $\psi \in \Psi$, there exists $\psi^{+}\in \Psi$ and a sequence of positive numbers $\{k_{{\mathbf{t}}}\}_{{\mathbf{t}}\in {\mathbf{T}}}$ satisfying $$\label{conv}
\sum_{{\mathbf{t}}\in {\mathbf{T}}}k_{{\mathbf{t}}} < \infty,$$ such that, for all but finitely ${\mathbf{t}}\in {\mathbf{T}}$ and all $\alpha \in {\mathcal{A}}$, there exists a collection $C_{{\mathbf{t}}, \alpha}$ of balls $B$ centred at $\mathbf{S}$ satisfying the following conditions: $$\label{inter1}
{\mathbf{S}}\cap I_{{\mathbf{t}}}(\alpha, \psi_{{\mathbf{t}}}) \subset \bigcup_{B \in C_{{\mathbf{t}}, \alpha}} B$$

$$\label{inter2}
{\mathbf{S}}\cap \bigcup_{B \in C_{{\mathbf{t}}, \alpha}} B \subset  I_{{\mathbf{t}}}(\alpha, \psi^{+}_{{\mathbf{t}}})$$

and

$$\label{inter3}
\mu(5B \cap  I_{{\mathbf{t}}}(\alpha, \psi_{{\mathbf{t}}})) \leq k_{{\mathbf{t}}} \mu(5B).$$

We are now in a position to state Theorem $5$ from [@BeVe]

\[transfer\] Suppose that $(H, I, \Psi)$ satisfies the intersection property and that $\mu$ is contracting with respect to $(I, 