# Trace of a function

Let $$T,L> 0$$ two real numbers and we consider the Sobolev space $$X := L^2(0,T; H^1(0,L))\cap H^{1}(0,T;H^{-1}(0,L))$$. My question is:

Given $$f \in X$$, the trace $$t \mapsto f(t,L)$$ belongs to what space? Could someone indicate me a reference?

The spatial evaluation (or trace) operator $$\mathrm{tr}_L$$ at $$L$$ is well defined and continuous on $$H^s(0,L)$$ for $$s>1/2$$; a classic reference is [LM, Chapter 1.9]. (Of course the range $$s>1/2$$ is exactly the one for which $$H^s(0,L)$$ is embedded into $$C([0,L])$$.)
Thus, an easy immediate answer would be to apply $$\mathrm{tr}_L$$ to your function $$f \in L^2(0,T;H^1(0,L))$$ in the sense of $$f(t,L) = \mathrm{tr}_L[f(t)]$$ which would give $$\bigl[t \mapsto f(t,L)\bigr] \in L^2(0,T).$$
If you want to squeeze a bit more: It is a classical result that $$X \hookrightarrow C([0,T];L^2(0,L))$$, see e.g. [LM, Chapter 1.3.1]. In particular, $$X \hookrightarrow L^p(0,T;L^2(0,L))$$ for every $$p \in [1,\infty)$$; so, if $$f \in X$$, then $$f \in L^p(0,T;L^2(0,L)) \cap L^2(0,T;H^1(0,L)) \hookrightarrow \Bigl[L^p(0,T;L^2(0,L)),L^2(0,T;H^1(0,L))\Bigr]_s$$ for all $$s\in (0,1)$$ and $$p \in [1,\infty)$$, where the space $$Y_s$$ on the right is the complex interpolation space. Since the (Bochner) Lebesgue spaces are compatible with complex interpolation as in [BL, Theorem 5.1.2] and $$H^s(0,L) = [L^2(0,L),H^1(0,L)]_s$$ (see again [LM, Chapter 1.9]), the interpolation space $$Y_s$$ is $$Y_s= L^q\bigl(0,T;[L^2(0,L),H^1(0,L)]_s\bigr) = L^q(0,T;H^s(0,L)), \quad \frac1q = \frac{1-s}p + \frac{s}2.$$
As noted above, $$\mathrm{tr}_L$$ is well defined (and continuous) on $$H^s(0,L)$$ for $$s > 1/2$$, so by sending $$p \to \infty$$ and $$s \searrow \frac12$$, we obtain that $$\bigl[t \mapsto f(t,L)\bigr] = \bigl[t \mapsto \mathrm{tr}_L [f(t)]\bigr] \in L^{4-\varepsilon}(0,T)$$ for every $$\varepsilon > 0$$.
[BL] Bergh, Jöran; Löfström, Jörgen, Interpolation spaces. An introduction, Grundlehren der mathematischen Wissenschaften. 223. Berlin-Heidelberg-New York: Springer-Verlag. X, 207 p. with 5 figs. DM 60.00; $24.60 (1976). ZBL0344.46071. [LM] Lions, J. L.; Magenes, E., Non-homogeneous boundary value problems and applications. Vol. I. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften. Band 181. Berlin-Heidelberg-New York: Springer-Verlag. XVI,357 p. DM 78.00 (1972). ZBL0223.35039. • My interpolation theory is a bit rusty and my copy of B&L is in my office: is it necessary to go through the intermediate interpolation for$X\hookrightarrow C([0,T]; L^2)$? Can one not directly take the complex interpolant of$[L^2_t H^1_x, H^1_t H^{-1}_x]_s$? – Willie Wong Jan 25 at 14:47 • You are right, it is not really necessary. I did try to rely on the "best known" results and not open the can of worms of vector-valued interpolation in the differentiability index (and embeddings). I thought one did not lose anything there since the$\varepsilon$missing at the end would occur anyway, but I just realized that seemingly may not occur in the Hilbert space case according to the Lions/Magenes book. I will check further. – Hannes Jan 25 at 15:13 • Please do! I don't have a copy of Lions--Magenes and would love to know if there's a strengthening. (But yes, I entirely agree that vector valued interpolation with differentiability index is a big can of worms. Didn't B&L deliberately omit all but the simplest cases?) – Willie Wong Jan 25 at 15:19 • I guess all the vector valued stuff is not enough as long as one needs$H^s(0,L)$with$s>1/2$for the trace; even if we could interpolate the vector-valued function spaces in an optimal way, the obstruction$s>1/2$will still produce the$4-\varepsilon\$. – Hannes Jan 25 at 15:36