3
$\begingroup$

Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and $\Lambda = A^{\frac 12}$ with $A$ a self-adjoint positive operator.

I'm looking for trace theorems regarding functions belonging to the space $W:= \{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$

The norm in this space $W$ is $$\lVert u \rVert_W^2 = \int_0^\infty t^{2s} \lVert u(t) \rVert_{X}^2 + t^{2s}\lVert u_t(t) \rVert_{Y}^2.$$

I am looking for a result that says that the trace $u|_{t=0}$ for $u \in W$ belongs to the space $(X,Y)_{\frac 12 + s} = D(\Lambda^{\frac 12 - s})$. I want boundedness of the trace operator.

I have seen Lions address this kind of issue in his book with Dautray but his assumptions on $s$ is too restrictive (needs to be positive). Also the weight here ($t^s$) is of Muckenhoupt type so I expect a nice result.

$\endgroup$
2

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.