This boils down to a key property of the Bochner integral: If $T \colon X \to Y$ is a bounded operator, and $f$ is Bochner-integrable, then $Tf$ is Bochner-integrable and the integral operator commutes with $T$, i.e. $$\int_X Tf\,\mathrm d\mu = T \int_X f\,d\mu$$

Having a decomposition such as this can be phrased as "$E$ is an L-summand in $\ell^1$". You have a corresponding continuous projection with image $E$ and kernel $F$. For further reading, I recommend this monograph: Harmand, P.; Werner, D.; Werner, W. $M$-ideals in Banach spaces and Banach algebras. Lecture Notes in Mathematics, 1547. Springer-Verlag, Berlin, 1993. viii+387 pp. ISBN: 3-540-56814-X MR1238713

(btw, did you really mean to refer to Ch. 2.8.1 in the beginning? I find a series of inclusions there but it seems the interpolation formula one needs is (8) in 2.4.2 on p.185)

Beautiful. I especially like how this can be decomposed into four parts, each with an obvious purpose and comprehensible on its own. To me, the key insight is Theorem 2 from section 1.16.4 that enables the third step. The second step from section 1.2.4 is one that is equally important, but so abstractly stated that I would have overlooked its reach. Thanks!

@sharpe you might also find the following article interesting: Biegert, Markus. On traces of Sobolev functions on the boundary of extension domains. Proc. Amer. Math. Soc. 137 (2009), no. 12, 4169–4176. MR2538577

@Hannes You're right. Thanks for being thorough!

Does @deMiranda's answer to mathoverflow.net/questions/37924/… answer your question, too?