Time translations and mollification in Bochner Lp

Let $s(t)={\sum }_{j=1}^m{x}_j{\mathbf{1}}_{E_j}(t)$ be a simple function with disjoint measurable $E_j$. Extend it by zero to $\mathbb{R}$. By [F2], $\mathrm{\parallel }{\mathbf{1}}_{E_j}(\cdot -h)-{\mathbf{1}}_{E_j}{\mathrm{\parallel }}_{L^p(\mathbb{R})}\to 0$ for each $j$; the triangle inequality and the finite sum show $\mathrm{\parallel }s(\cdot -h)-s{\mathrm{\parallel }}_{L^p(\mathbb{R};X)}\to 0$.

For an interval partition $\mathcal{P}$, let $P_{\mathcal{P}}f$ be the average of $f$ on each cell $I$. By [F4], $\mathrm{\parallel }\mathrm{\mid }I{\mathrm{\mid }}^{-1}{\int }_{I}f{\mathrm{\parallel }}_X\le \mathrm{\mid }I{\mathrm{\mid }}^{-1}{\int }_I\mathrm{\parallel }f{\mathrm{\parallel }}_X$. For $p>1$, scalar Hölder gives $\mathrm{\mid }I\mathrm{\mid }\mathrm{\parallel }\mathrm{\mid }I{\mathrm{\mid }}^{-1}{\int }_{I}f{\mathrm{\parallel }}_X^p\le {\int }_I\mathrm{\parallel }f{\mathrm{\parallel }}_X^p$; for $p=1$ the first inequality gives the same assertion directly. Summing proves contraction on $L^p$. To justify density for arbitrary completed Lebesgue level sets, write $E=B\cup N$ with $B$ Borel and $N$ contained in a Borel null set $Z$ by [F3], after intersecting all sets with $(0,T)$. The Borel set $B\cup Z$ differs from $E$ by a null set. Choose an open superset $O$ with excess less than $\eta$. The bounded open set $O\cap (0,T)$ is the union of the countably many rational-endpoint open intervals contained in it. Enumerate those intervals, and let $J_m$ be their first $m$ unions. By continuity from below [F3], $\mathrm{\mid }(O\cap (0,T))\setminus J_m\mathrm{\mid }\to 0$. Thus a finite union of intervals differs from $E$ in measure by less than $2\eta$ for large $m$. Round the finitely many endpoints by continuous linear ramps, whose error is supported in intervals of arbitrarily small total length. It follows that continuous $X$-valued functions are dense in $L^p(0,T;X)$ by applying this construction to the finitely many level sets of a simple approximation [F1]. For continuous $g$, uniform continuity gives $\mathrm{\parallel }{P}_{\mathcal{P}}g-g{\mathrm{\parallel }}_{L^p}\le T^{1\mathrm{/}p}{\sup }_{\mathrm{\mid }s-t\mathrm{\mid }\le \max \mathrm{\mid }I\mathrm{\mid }}\mathrm{\parallel }g(s)-g(t){\mathrm{\parallel }}_X\to 0$. Contraction now yields $\mathrm{\parallel }{P}_{\mathcal{P}}f-f{\mathrm{\parallel }}_{L^p}\le 2\mathrm{\parallel }f-g{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }{P}_{\mathcal{P}}g-g{\mathrm{\parallel }}_{L^p}$, proving the averages assertion.

Given $\epsilon >0$, choose a simple $s$ with $\mathrm{\parallel }f-s{\mathrm{\parallel }}_{L^p(0,T;X)}<\epsilon \mathrm{/}3$ by [F1]. Extend both functions by zero outside $(0,T)$. Translation is an isometry of the whole-line Bochner norm because it preserves Lebesgue measure; step 1.1 therefore gives, for all sufficiently small $h$, $\mathrm{\parallel }f(\cdot -h)-f{\mathrm{\parallel }}_{L^p(\mathbb{R};X)}\le 2\mathrm{\parallel }f-s{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }s(\cdot -h)-s{\mathrm{\parallel }}_{L^p}<\epsilon$. Restriction to compact subintervals proves the translation assertion.

Extend $f$ by zero to $\mathbb{R}$ and use the kernels from [F5]. Replacing the measurable level sets of simple approximants by Borel representatives modulo null sets gives a Borel version by [F3]; composing the simple approximants with $s\mapsto t-s$ shows the Bochner integrand is strongly measurable in $s$. Its $L^p(\mathbb{R})$ section norm is at most $2{\rho }_{\delta }(s)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathbb{R};X)}$ by translation invariance and Minkowski. Scalar integral Minkowski [F4] makes the $s$-integral of this norm finite in $L^p$, so the convolution is defined for a.e. $t$; by linearity of the defining simple-function limit and $\int {\rho }_{\delta }=1$, ${\rho }_{\delta }\ast f(t)-f(t)={\int }_{\mathbb{R}}{\rho }_{\delta }(s)(f(t-s)-f(t))ds$ there. The Bochner norm inequality [F4] then gives $\mathrm{\parallel }{\rho }_{\delta }\ast f-f{\mathrm{\parallel }}_{L^p(\mathbb{R};X)}\le \int {\rho }_{\delta }(s)\mathrm{\parallel }f(\cdot -s)-f{\mathrm{\parallel }}_{L^p(\mathbb{R};X)}ds$. Step 2.1 makes the translation modulus uniformly small for $\mathrm{\mid }s\mathrm{\mid }\le \delta$, so this bound tends to zero; restriction to compact subintervals proves the stated mollification convergence.

Steps 1.2, 2.1, and 3.1 prove convergence of interval averages, translation continuity, and convergence of time mollifiers, respectively.