Distributional time derivatives with values in a Banach space

Assume Countable Choice. Fix $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, a Banach space $Y$ over $\mathbb{K}$, and $I=(0,T)$ with $0<T<\mathrm{\infty }$. Equip $I$ with the restricted Lebesgue sigma-algebra $\mathcal{L}(I)=\{E\cap I:E\in \mathcal{L}(\mathbb{R})\}$ and restricted Lebesgue measure $\lambda {\mathrm{\mid }}_I$, denoted $dt$ in integrals. All null sets and almost-everywhere classes below refer to this measure.

Let $u,g:I\to Y$ be strongly measurable with ${\int }_I(\mathrm{\parallel }u(t){\mathrm{\parallel }}_Y+\mathrm{\parallel }g(t){\mathrm{\parallel }}_Y)dt<\mathrm{\infty }$. Their integrals are Bochner integrals. Write ${\mathcal{D}}_{\mathbb{K}}(I)=C_c^{\mathrm{\infty }}(I;\mathbb{K})$: for $\mathbb{K}=\mathbb{C}$ this is the complex test-function space $\mathcal{D}(I)$; for $\mathbb{K}=\mathbb{R}$ it is its subspace of real-valued functions, with the restricted test-function topology. We say that $g$ represents the distributional time derivative ${\mathrm{\partial }}_{t}u$ if ${\int }_{I}u(t){\phi }^{\mathrm{\prime }}(t)dt=-{\int }_{I}g(t)\phi (t)dt\text{in }Y\text{for every }\phi \in {\mathcal{D}}_{\mathbb{K}}(I)\mathrm{.}$ The sign is the scalar distribution convention; the pairing is bilinear, without conjugation of the test function.

The representing $g$ is unique as an almost-everywhere class. Its derivative bound is the Bochner norm $\mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(I;Y)}={\int }_I\mathrm{\parallel }g(t){\mathrm{\parallel }}_{Y}dt$, not the supremum norm of a scalar pairing. The two maps on tests defined by the displayed integrals are continuous for the test-function topology. If $Z$ is a Banach space over the same scalar field $\mathbb{K}$ and $A:Y\to Z$ is bounded and $\mathbb{K}$-linear, then these identities transport to ${\mathrm{\partial }}_t(Au)=Ag$ in the same sense. This includes bounded scalar-valued linear functionals and does not assert norm differentiability of $u(t)$.

The Axiom of Countable Choice (AC_ω)

Time translations and mollification in Bochner $L^p$

Realification preserves a complex norm and its metric

Bochner-integrable function

Bochner integrability criterion

Bochner integral norm inequality

Bounded linear maps commute with Bochner integration

Test function space d of an open set

Distributional derivative

Banach space

Strongly measurable Banach-valued function

Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ

A bounded linear operator between normed spaces

Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume

Fixed support test function frechet space

Test function topology

Linearity of the Bochner integral

Given: The scalar field, Banach spaces, restricted Lebesgue measure, Countable Choice, and integrability hypotheses in the Definition, with tests valued in that scalar field.

1.1 For $v=u$ or $v=g$ and $a=\phi$ or $a={\phi }^{\mathrm{\prime }}$, the scalar factor $a$ is bounded and measurable. Take measurable scalar simple approximants $a_n\to a$ bounded by $\mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}+1$ and the strongly measurable simple approximants $v_n\to v$ off a null set. Then $a_n{v}_n$ are measurable $Y$-valued simple functions converging there to $av$, since all scalars lie in $\mathbb{K}$. Thus $av$ is strongly measurable, and $\mathrm{\parallel }av{\mathrm{\parallel }}_Y\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }v{\mathrm{\parallel }}_Y$ is integrable. The Bochner criterion gives all the integrals in the definition. Changing $u$ or $g$ on a null set changes no integral, by the same criterion and the norm inequality. [given, F2, F3]

2.1 The norm inequality gives $\mathrm{\parallel }{\int }_{I}u\phi dt{\mathrm{\parallel }}_Y\le \mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}{\int }_I\mathrm{\parallel }u{\mathrm{\parallel }}_{Y}dt$ and $\mathrm{\parallel }{\int }_{I}u{\phi }^{\mathrm{\prime }}dt{\mathrm{\parallel }}_Y\le \mathrm{\parallel }{\phi }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}{\int }_I\mathrm{\parallel }u{\mathrm{\parallel }}_{Y}dt$, with the analogous estimate for $g$. The seminorm $p_1(\phi )=\max (\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }},\mathrm{\parallel }{\phi }^{\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }})$ is continuous on every fixed-support space, hence continuous in the test-function topology by its defining family of seminorms. Its restriction to real tests is continuous as well. These bounds prove continuity of the integral maps and give a well-defined vector-valued version of the scalar derivative sign rule. [F2, F3, step 1.1]

2.2 For $Z$ Banach over $\mathbb{K}$ and $A:Y\to Z$ bounded and $\mathbb{K}$-linear, applying $A$ preserves Bochner integrability and commutes with each integral. In particular, $Au,Ag$ satisfy the required norm-integrability hypotheses; their strong measurability also follows by applying $A$ to simple approximants. Since $A(u{\phi }^{\mathrm{\prime }})=(Au){\phi }^{\mathrm{\prime }}$ and $A(g\phi )=(Ag)\phi$, applying $A$ to the defining identity proves ${\mathrm{\partial }}_t(Au)=Ag$ with the same $\mathbb{K}$-valued tests. [given, F4, step 1.1]

3.1 If $g_1,g_2$ represent the same derivative, then $h=g_1-g_2$ is strongly measurable and integrable, and linearity gives ${\int }_{I}h\phi dt=0$ for every admissible test. Extend $h$ by zero outside $I$ and use the real smooth approximate identity ${\rho }_{\epsilon }$ from [F6]. For $t$ in a fixed compact subinterval $J\subset I$ and sufficiently small $\epsilon$, the real function $s\mapsto {\rho }_{\epsilon }(t-s)$ is an admissible test in both scalar-field cases, so $({\rho }_{\epsilon }\ast h)(t)=0$ on $J$. For real $Y$, [F6] gives convergence to $h$ in $L^1(J;Y)$. For complex $Y$, apply that result to its realification, which is Banach with the same norm and metric by [F1] and [F6]; the convolution integrals agree because their defining simple integrals use only real scalar coefficients and the same norm limit. Hence $\mathrm{\parallel }h{\mathrm{\parallel }}_{L^1(J;Y)}=0$ in either case. A countable compact-interval exhaustion of $I$ proves almost-everywhere uniqueness. Therefore the derivative norm is independent of the representative. Countable Choice is used by the cited mollification and Lebesgue-measure suppliers. [F1, F2, F5, F6, step 1.1] ∎