Here they are: $$f_{\infty} \leq C f_q^{\frac {qk} {n+kq}} \left( \sum_{\mu=k} D^\mu f_{BMO} \right)^{ \frac n {n+kq}}$$ and $$f_{Lip_\alpha} \leq C f_q^{\frac {qk} {n+kq} \frac {k\alpha} k} \left( \sum_{\mu=k} D^\mu f_{BMO} \right)^{ \frac n {n+kq} \frac {k\alpha} k + \frac \alpha k};$$ also $$f_{\infty} \leq C f_q^{\frac q r \left( \frac n {rkn} + \frac q r \right)^{1}} \nabla^k f_r^{\frac n {rkn} \left( \frac n {rkn} + \frac q r \right)^{1}} \quad (rk>n)$$ and $$f_{Lip_\alpha} \leq C f_q^{\frac q r \left( \frac n {rkn} + \frac q r \right)^{1} (1  \frac {\alpha r } {rk  n})} \nabla^kf_r^{\frac n {rkn} \left( \frac n {rkn} + \frac q r \right)^{1}(1  \frac {\alpha r } {rk  n}) + \frac {\alpha r } {rk  n}},$$ here $0<\alpha \leq \frac {rkn} r$ and $rk>n$.
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