here are $k$ disjoint supported functions $f_0,\cdots, f_k$, such that for all $0\leq i\leq k$, $supp(f_i)\subseteq supp(f)$, and $$\mathcal{R}(f_i)\leq Ck^{p}\bigg(\frac{\mathcal{R}(f)}{\phi(f)}\bigg)^q, \quad \forall\ 0\leq i\leq k,