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Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...

Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below). Given $T>0$ and $n \in \bf Z$, consider the following ...

If $u : \mathbb R^n \to \mathbb R$ is a smooth enough function then on any Euclidean $n$-ball $B_R$ of radius $R$ we have the very well-known Poincaré inequality $$ \int_{B_R} |u - \bar u|^2 \le C(R,...