that $f : G \to \mathbb{C}$ is continuous and that $f \in \mathcal{M}^\rho_{Q_0}(L_{p,w})$ for some compact unit neighborhood $Q_0$ with $Q_0 = \overline{\operatorname{int}
    Q_0}$. Then the following hold:

i)  $\lVert{\mathrm{osc}_{Q_0} f}\rVert_{L_{p,w}} < \infty$.

ii) For arbitrary $\varepsilon > 0$, there is a unit neighborhood $Q_\varepsilon \subset { Q_0}$ such that for each unit neighborhood $Q \subset Q_\varepsilon$, we have $\lVert{\mathrm{osc}_Q f}\rVert_{L_{p,w}} < \varepsilon$. Put briefly, $$\begin{aligned}
                  \lim_{Q \to \{e\}} \lVert{\mathrm{osc}_Q f}\rVert_{L_{p,w}} = 0.
                \end{aligned}$$

An Assumption on the Kernel
---------------------------

From now on we make the following assumption on the reproducing kernel space.

\[assumption1\] Assumption \[assume:KernelAlmostIntegrable\] is satisfied, and $\operatorname{span} \{\lambda(x) K\}_{x\in G}$ is dense in $\mathcal{M}_{r,m}$.

This assumption is similar to the density of $\mathrm{span} \, \{\pi(x) K\}_{x \in G}$ in $\mathcal{H}$—which is equivalent to $K$ being a cyclic vector for the representation $\pi$ on $\mathcal{H}$—and in $\mathcal{M}^{\mathcal{T}_w}$, which is Assumption 3 of [@dadedelastte17] and fulfilled in our setting, as can be seen by combining Theorems \[intersections\] and \[intersections-1\]

In the following we will denote with $RC_K$ the right convolution operator $RC_K f := f\ast K$, where the space on which $RC_K$ acts may vary depending of the context.

Before we provide a sufficient condition under which Assumption \[assumption1\] 