2
$\begingroup$

Does there exist a theorem that allow us to say that, if we have an estimate on the Sobolev space $H^s\,,\, s\geq 0$ then we can deduce an estimate on the dual space $H^{-s}$ ?

For instance, assume that the following inequality holds, $$ \sup_{t\in[0,T]}\|u(t,\cdot)\|_{H^s}\leq \int\limits_0^T\|Tu(t,\cdot)\|_{H^s}\, dt\,,$$ where $T$ is an operator, can we deduce a similar inequality with the norm $H^{-s}$ ? (For example, $$ \sup_{t\in[0,T]}\|u(t,\cdot)\|_{H^{-s}}\leq \int\limits_0^T\|T^*u(t,\cdot)\|_{H^{-s}}\, dt\,? )$$

Improve this question
$\endgroup$
2
  • $\begingroup$ What is $T$? An operator in $H^s$ that is independent of t? Then I do not see how such an inequality would ever hold in the first place. $\endgroup$
    – Michael Renardy
    Commented Jan 29, 2022 at 17:27
  • $\begingroup$ $T=\partial_t +a(t,x,\partial_x)$ where $a$ is a differential operator for example. $\endgroup$
    – Niser
    Commented Jan 29, 2022 at 17:48

0

Reset to default

You must log in to answer this question.