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I think a nice example are Lieb--Thirring inequalities. Consider the Schroedinger operator $H = \Delta + V$ with potential $V \in L^{\gamma + d/2}(\mathbb{R}^{d})$, where $d \geq 3$. Then H defines an (unbounded) operator on $L^2(\mathbb{R}^d)$, whose essential spectrum is $[0,\infty)$ and which has negative eigenvalues $E_j$ (countably many). The Lieb--Thirring inequality then tells us

$$\sum |E_j|^{\gamma} \leq const \|f\|_{L^{\gamma |V\|_{L^{\gamma + d/2}(\mathbb{R}^{d})}.$$

This inequalities requires $L^p$ for $p \in (0,\infty)$.

There are other examples, but they are somewhat more technical to state ...

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I think a nice example are Lieb--Thirring inequalities. Consider the Schroedinger operator $H = \Delta + V$ with $V \in L^{\gamma + d/2}(\mathbb{R}^{d})$, where $d \geq 3$. Then H defines an (unbounded) operator on $L^2(\mathbb{R}^d)$, whose essential spectrum is $[0,\infty)$ and which has negative eigenvalues $E_j$ (countably many). The Lieb--Thirring inequality then tells us

$$\sum |E_j|^{\gamma} \leq const \|f\|_{L^{\gamma + d/2}(\mathbb{R}^{d})}.$$

This inequalities requires $L^p$ for $p \in (0,\infty)$.

There are other examples, but they are somewhat more technical to state ...