Lemma 3.8: energy-data time-integrable supremum bound

If $n=2$, the hypotheses, exponent range, evolution and inherited $H^1$ norm agree with the two-dimensional estimate in F2. It gives the required Bochner bound directly.

Suppose $n=3$. By density in F3 and the Axiom of Choice in F1, choose core data $u_j$ with $\mathrm{\parallel }{u}_j-u_0{\mathrm{\parallel }}_{H_0^1}\le 2^{-j}$ for $j\ge 1$. Then ${\sum }_{j\ge 1}\mathrm{\parallel }{u}_{j+1}-u_j{\mathrm{\parallel }}_{H_0^1}<\mathrm{\infty }$. Each $u_j$ lies in $H_D^2$; F3 and F4 therefore make $g_j(t)=W(t)u_j$ continuous from $[0,T]$ into $L^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C})$.

For any continuous $h:[0,T]\to L^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{C})$ and $\epsilon >0$, F6 gives a $\delta >0$ such that $\mathrm{\mid }t-r\mathrm{\mid }<\delta$ implies $\mathrm{\parallel }h(t)-h(r){\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$. Take an integer $m$ with $T\mathrm{/}m<\delta$, and set $S(t)=h(kT\mathrm{/}m)$ for $kT\mathrm{/}m\le t<(k+1)T\mathrm{/}m$, restricted to $(0,T)$, for $0\le k<m$. This is a measurable finite-range simple function and ${\sup }_{0<t<T}\mathrm{\parallel }S(t)-h(t){\mathrm{\parallel }}_{\mathrm{\infty }}<\epsilon$. For fixed $h$, choosing $\epsilon =2^{-k}$ gives simple approximants converging in norm at every time, so $h$ is strongly measurable by F4. Apply the construction to each $g_j$ with $\epsilon =2^{-j}$ and denote the resulting simple map by $s_j$.

Put $a_j(t)=\mathrm{\parallel }{g}_{j+1}(t)-g_j(t){\mathrm{\parallel }}_{\mathrm{\infty }}$. These are continuous nonnegative real functions. The scalar estimate in F2, with the representative convention in F4, gives $\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{L^4(0,T)}\le C\mathrm{\parallel }{u}_{j+1}-u_j{\mathrm{\parallel }}_{H_0^1}$. Hölder in F5, applied to $a_j$ and the constant function $1$ with conjugate exponents $(4,4\mathrm{/}3)$, gives ${\int }_0^T{a}_j(t)dt\le T^{3\mathrm{/}4}\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{L^4(0,T)}$. Consequently Tonelli in F4, for Lebesgue time measure and counting measure on the positive integers, gives ${\int }_0^T{\sum }_{j\ge 1}{a}_j(t)dt={\sum }_{j\ge 1}{\int }_0^T{a}_j(t)dt<\mathrm{\infty }$. Thus the sum is finite outside a measurable null set $E$. At every $t\mathrm{\notin }E$, the norm triangle inequality makes $(g_j(t))$ Cauchy in complex $L^{\mathrm{\infty }}$, whose completeness is supplied in F6. Let $v(t)$ be its norm limit there and set $v(t)=0$ on $E$.

The finite-range simple functions in step 2.1 satisfy $\mathrm{\parallel }{s}_j(t)-v(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le 2^{-j}+\mathrm{\parallel }{g}_j(t)-v(t){\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$ for every $t\mathrm{\notin }E$. They therefore witness strong measurability of $v$ directly by the definition in F4, without assuming a general closure theorem for maps into a nonseparable Banach space.

Unitarity on $L^2$ makes $W(t)u_j\to W(t)u_0$ in $L^2(\mathrm{\Omega })$ for every $t$. At each $t\mathrm{\notin }E$, the same sequence converges to $v(t)$ in $L^{\mathrm{\infty }}(\mathrm{\Omega })$ and hence in $L^2(\mathrm{\Omega }\cap B_R)$ for every positive integer $R$, since the local $L^2$ distance is at most $\mathrm{\mid }\mathrm{\Omega }\cap B_R{\mathrm{\mid }}^{1\mathrm{/}2}$ times the $L^{\mathrm{\infty }}$ distance. The norm triangle inequality identifies the two local $L^2$ limits. The countable balls cover $\mathrm{\Omega }$, so $v(t)$ equals the spatial class of $W(t)u_0$ for almost every time. With the product and section conventions in F4, $v$ is a strongly measurable representative of the evolution.

The scalar estimate gives ${\int }_0^T\mathrm{\parallel }{g}_j(t){\mathrm{\parallel }}_{\mathrm{\infty }}^{4}dt\le C^4\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{H_0^1}^4$. The norm convergence outside $E$ in step 3.1 and Fatou's lemma in F5, applied to the nonnegative measurable functions $\mathrm{\parallel }{g}_j(t){\mathrm{\parallel }}_{\mathrm{\infty }}^4$, give ${\int }_0^T\mathrm{\parallel }v(t){\mathrm{\parallel }}_{\mathrm{\infty }}^{4}dt\le C^4\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H_0^1}^4$, because $u_j\to u_0$ in $H_0^1$. Taking fourth roots and using steps 4.1 and 5.1 proves the Bochner bound in dimension three. Together with step 1.1 this proves the assertion in both dimensions.