A lemma of J. L. Lions


Let X,Y and Z be three Banach spaces with norms \|\cdot \|_X,\ \|\cdot \|_Y and \|\cdot \|_Z. Assume that X \subset Y with compact injection and that Y\subset Z with continuous injection. Prove that

\forall \varepsilon >0 \exists C_\varepsilon \geq 0 satisfying \|u\|_Y \leq \varepsilon \|u\|_X+C_\varepsilon \|u\|_Z,\ \forall u \in X.

Applications:

  1. Prove that for every \varepsilon >0 there exists C_\varepsilon \geq 0 satisfying\displaystyle \max_{[0,1]}|u| \leq \varepsilon \max_{[0,1]}|u^\prime|+C_\varepsilon\|u\|L^1,\ \forall u \in C^1([0,1]).
  2. Pick p>1. Prove that for every \varepsilon >0 there exists C=C(\varepsilon,p) such that \|u\|_{L^\infty(0,1)} \leq \varepsilon \|u^\prime\|_{L^p(0,1)}+C\|u\|_{L^1(0,1)},\ \forall u \in W^{1,p}(0,1).

Source: Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011

Proof: For the initial lemma, just argue by contradiction. The two application are more or less immediate after using the given lemma.

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