I have to prove the Lemma 4 (Compactness) on page 116 of the article https://arxiv.org/pdf/1411.2567.pdf. In particular, at the beginning of the proof, when the author says "Hence, by the Hölder inequality, $$ \|Du_m\|_{L^m(\Omega)}=E_m(u_m, \Omega)^m\leq E_m(b, \Omega)^m=\|Db\|_{L^m(\Omega)}" $$ (where $E_m(w)=\int\limits_{\Omega}|Dw|^m\ dx$), I don't understand where and how the Hölder inequality is applied. Can you help me? Thanks
2026-03-26 19:19:05.1774552745
Explanation on the use of the Hölder inequality in a proof
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I guess Hölder is not applied in this line, since it seems to me, that just the minimising property of $u_m$ is used. However in the next inequality stated in the paper $$\|D u_m\|_{L^k} \leq \|Du_m\|_{L^m} |\Omega|^{\frac{1}{k}-\frac{1}{m}}$$ Hölder's inequality has been applied and I think this is what the author of the paper meant.