Definitions of fractional Sobolev Spaces

Question

In a paper I read that for a bounded domain $W^{s,p}(\Omega)=\{ u \in L^p(\Omega), (id-\Delta)^{s/2}u \in L^p(\Omega) \}$ and $s$ is not assumed to be an integer, $p \neq 2$ in general.

If $p=2$ the borelian calculus in Hilbert spaces allow to define $id-\Delta)^{s/2}$ so that I understant the definition of $W^{s,2}$.

How one can define the operator $(id-\Delta)^{s/2}$ when $p \neq 2$ ? Is there a borelian calculus in reflexive Banach spaces?

Answer 1

Respect to your question. No, i do not think there is an analogue of the continuous functional calculus in non-Hilbert spaces. Nevertheless you can still use holomorphic functional calculus. I think the most straightforward way of showing that $(id - \Delta)^{-\frac{s}{2}}$ is globally defined for $L^p(\Omega)$ is to use the following ''trick'':

1. The heat semigroup $e^{t \Delta}$ is a globally defined contraction in every $L^p(\Omega)$
2. You can express $(id - \Delta)^{-\frac{s}{2}}$ as a formal integral of the elements $e^{t \Delta}$, i.e.: $$ (id - \Delta)^{-\frac{s}{2}} = \frac{1}{\Gamma(s/2)} \int_0^\infty t^{s/2} e^{-t} e^{t \Delta}\, \frac{d \, t}{t}. $$

You can check that the formula above holds in $L^2(\Omega)$ using functional calculus. It extends to $L^2 \cap L^p$ and, since $L^2 \cap L^p$ is dense in $L^p$, and the formula above define a bounded function in $L^2 \cap L^p$, it extends to he whole $L^p$.