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Transfer of a compatible spacetime norm through the free forcing map

Statement

Use the free-convolution, global-evolution and Bochner variation-of-constants interfaces on a prescribed forward interval $I = [a, b] \subset [0, T]$ , retaining their group and perturbation hypotheses, including Countable Choice. Regard $E$ , $C (I; E)$ and $L^{1} (I; E)$ as real spaces. Let $Y_{I}$ be a real Banach space of time-dependent functions or distributions, with $Y_{I}$ and $C (I; E)$ injected by real-linear maps into the same Hausdorff real topological vector space $A_{I}$ . Equality in $A_{I}$ is the compatibility convention for their representatives. Assume the actual free orbit $H_{a} x (t) = W (t - a) x$ and the actual retarded operator $D_{a} g (t) = \int_{a}^{t} W (t - s) g (s) d s$ define bounded real-linear maps into $Y_{I}$ with norms at most $A_{I}$ and $B_{I}$ , respectively, for every $x \in E$ and every $g \in L^{1} (I; E)$ . Then the forced solution $u (t) = S (t, a) x + \int_{a}^{t} S (t, s) f (s) d s$ lies in $Y_{I}$ and $∥ u ∥_{Y_{I}} \leq A_{I} ∥ x ∥ + B_{I} ((b - a) K C_{I} (∥ x ∥ + ∥ f ∥_{1}) + ∥ f ∥_{1})$ , where $C_{I} = M \exp (M K (b - a))$ . Bounds on subintervals are asserted with their stated constants; uniform bounds require uniform $A_{I}, B_{I}$ . Backward spacetime bounds hold if the same orbit and forcing-map hypotheses are supplied after time reflection, with group $W (- t)$ and the reflected coefficient $- P$ .

Statement references

Continuous free convolution on a finite time interval

Global real evolution, cocycle and inverse

Bochner variation of constants for real evolution

Banach space

A bounded linear operator between normed spaces

Topological vector spaces over the real and complex fields

Facts & Assumptions

Given: The compatible injections, the two bounded free maps on the named interval, and $L^{1} (I; E)$ forcing.

Goal: Obtain the additional spacetime estimate by applying a bounded map already supplied, without integrating into $Y_{I}$ .

[F1]

Continuous free convolution on a finite time interval defines the actual retarded free forcing map in $C (I; E)$ , while the hypotheses here supply its compatible $Y_{I}$ value for the same input.

[F2]

Global real evolution, cocycle and inverse and Bochner variation of constants for real evolution give the solution and the estimate $∥ u ∥_{C (I; E)} \leq C_{I} (∥ x ∥ + ∥ f ∥_{1})$ with initial time $a$ .

[F3]

The real Banach-space convention and its norm structure are supplied by Banach space. The bounds for the real-linear maps $H_{a} : E \to Y_{I}$ and $D_{a} : L^{1} (I; E) \to Y_{I}$ have the meaning of A bounded linear operator between normed spaces with common scalar field $R$ . The ambient space is a Hausdorff TVS over $R$ in Topological vector spaces over the real and complex fields; the injections are real-linear, and equality of their images is the stated compatibility convention.

Proof

Given: The compatible injections, the two bounded free maps on the named interval, and $L^{1} (I; E)$ forcing.

Goal: Obtain the additional spacetime estimate by applying a bounded map already supplied, without integrating into $Y_{I}$ .

1.1F1F2F3given

The continuous function $P (t) u (t)$ has $L^{1} (I; E)$ norm at most $(b - a) K ∥ u ∥_{C}$ . Hence $g = P u + f$ belongs to $L^{1} (I; E)$ . The solution identity is $u = H_{a} x + D_{a} g$ in $C (I; E)$ by [F2]. Both terms on the right have compatible images in $Y_{I}$ by the two map hypotheses. Their sum therefore represents $u$ in $A_{I}$ , proving membership in $Y_{I}$ with the declared compatibility.

2.1F1F2F3step 1.1given

The triangle inequality and the assumed map bounds give $∥ u ∥_{Y_{I}} \leq A_{I} ∥ x ∥ + B_{I} ((b - a) K ∥ u ∥_{C} + ∥ f ∥_{1})$ . Substituting [F2] proves the displayed estimate. This uses no $Y_{I}$ -valued Bochner integral; all integrals used to identify the free map were taken in $E$ . Thus a nonseparable spatial factor in a chosen spacetime norm introduces no additional integration claim.

3.1F1F2step 1.1step 2.1∎

Restricting to a shorter interval gives exactly its stated free-map constants, so uniformity cannot be inferred without their uniform bounds. For a backward interval ending at $b$ , set $τ = b - t$ and $σ = b - s$ . Changing variables in the oriented Volterra equation replaces $W (t - s)$ by $W (- τ + σ)$ , the perturbation by $- P (b - σ)$ and the forcing by $- f (b - σ)$ . This is the same forward problem with reflected group $\tilde{W} (τ) = W (- τ)$ and coefficient $\tilde{P} (σ) = - P (b - σ)$ . Applying steps 1.1–2.1 under its supplied free-map hypotheses proves the final assertion.

Depends on

Used by · 1

Bounded real-linear perturbations preserve solution estimates

Dependency tree · 2 levels

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