Let \(E \subset I \subset \mathbb{R}\). Obviously \(\lVert \cdot\rVert_{L^p(I)} \lesssim \lVert \cdot\rVert_{L^p(E)}\) does not hold in general. However, for certain function classes this is true.
- \(p(t)= \sum^n_{k=1} \beta_k \exp(\lambda_k t)\) (Turan Lemma)
- \(\phi\colon D(z_0, 4) \to \mathbb{C}\) analytic
- \(r(t)= \sum_{k=1}^{n} p_k(t) \exp(i\lambda_k t)\)
Remarks
- Usually the inequality is shown with \(p=\infty\). Afterwards it can be generalized for \(p\ge 1\).