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Exterior Strichartz estimates with precise loss

Statement

Assume the Axiom of Choice. Let $Ω$ be the connected smooth exterior of a nonempty compact non-trapping obstacle in $R^{n}$ , $n \geq 2$ , as in Exterior and truncated obstacle domains and Non-trapping obstacle. Let $Δ_{D}$ and $H_{D}^{r}$ be the Dirichlet Laplacian and spectral scale of Dirichlet spectral Sobolev scale and duality. Fix $T > 0$ , $χ \in C_{c}^{\infty} (R^{n})$ equal to one near the obstacle, and $2 < p \leq \infty$ , $2 \leq q < \infty$ with $2 / p + n / q = n / 2$ , taking $1 / \infty = 0$ . For $0 < s < 1$ write $W^{s, q} = [L^{q} (Ω), W^{1, q} (Ω)]_{s}$ in the complex strip convention of An explicit common interpolation model for the fractional Strichartz targets, and set $W^{0, q} = L^{q} (Ω)$ and $W^{1, q} = W^{1, q} (Ω)$ . The time norms are Bochner norms on $(0, T)$ .

For each $0 \leq s \leq 1$ there are constants depending on the fixed data such that $∥ (1 - χ) e^{i t Δ_{D}} u_{0} ∥_{L^{p} (0, T; W^{s, q})} \leq C ∥ u_{0} ∥_{H_{D}^{s}},$ and $∥ e^{i t Δ_{D}} u_{0} ∥_{L^{p} (0, T; W^{s, q})} \leq C ∥ u_{0} ∥_{H_{D}^{s + 1 / p}} .$

External input

For finite $p > 2$ and the hypotheses above, the integer cases $s = 0, 1$ of the two estimates are imported from reference 0, Propositions 2.10 and 2.11. Their input spaces are respectively $H_{D}^{s}$ and $H_{D}^{s + 1 / p}$ , and their target spaces are respectively $L_{t}^{p} L_{x}^{q}$ and $L_{t}^{p} W_{x}^{1, q}$ at $s = 0$ and $s = 1$ . This item does not reconstruct the source proofs of those estimates.

Facts & Assumptions

Given: The hypotheses and notation of the Statement.

[F1]

Exterior and truncated obstacle domains and Non-trapping obstacle specify the exterior and generalized-ray non-trapping hypothesis required by the independent endpoint input.

[F2]

Dirichlet spectral Sobolev scale and duality defines $H_{D}^{r}$ and the unitary Dirichlet group; Exterior Dirichlet realization, H² domain and form domain identifies $H_{D}^{1}$ with $H_{0}^{1}$ and its form norm.

[F3]

Positive spectral Hilbert scale: duality and interpolation gives $[H_{D}^{a}, H_{D}^{b}]_{s} = H_{D}^{(1 - s) a + s b}$ for $0 < s < 1$ , in the stated complex strip convention.

[F4]

Well-defined strip norm and the action of a common endpoint operator gives the exact operator bound $M_{0}^{1 - s} M_{1}^{s}$ for a single complex-linear map between the sums of compatible complex Banach couples with endpoint bounds $M_{0}, M_{1}$ . Separately, An explicit common interpolation model for the fractional Strichartz targets gives, for finite $p$ and $1 < q < \infty$ , the identity $[L_{t}^{p} L_{x}^{q}, L_{t}^{p} W_{x}^{1, q}]_{s} = L_{t}^{p} W_{x}^{s, q}$ . For every $1 \leq p \leq \infty$ , the time spaces use strongly measurable almost-everywhere classes and the Bochner norms of Continuous and mixed time-space function spaces, with essential supremum when $p = \infty$ .

[F5]

Reference 0 supplies the two finite- $p$ integer endpoint estimates described in External input.

[F6]

The Axiom of Choice records the Choice assumption required by the stated spectral and interpolation suppliers.

[F7]

Integer-order Sobolev spaces and their norms defines $W^{j, q} (Ω; C)$ and its weak-derivative norm for $j = 0, 1$ ; in particular $W^{0, q} = L^{q}$ . Its Countable Choice hypothesis follows from the assumed AC.

[F8]

Under Countable Choice, Weak Leibniz rule with a smooth factor gives the weak product formula on any open set for a smooth multiplier, and preserves $W^{k, q}$ when its classical derivatives through order $k$ are bounded. Here apply it on $Ω$ with $k = 1$ , $q = 2$ and multiplier $1 - χ$ ; compact support of the multiplier itself is not required.

[F9]

Integer-order Sobolev spaces are Banach applies under AC on the open set $Ω$ , with complex scalars, $k = 0, 1$ and exponent $q$ , and gives the complex Banach endpoints $L^{q}$ and $W^{1, q}$ . Their weak-derivative norm in [F7] gives $∥ v ∥_{q} \leq ∥ v ∥_{W^{1, q}}$ .

[F10]

Bochner Lp is complete for a Banach value space proves completeness on $(0, T)$ for real Banach values, finite $1 \leq p < \infty$ and Countable Choice. Realification preserves a complex norm and its metric preserves the norm and metric on restricting complex scalars to real scalars. Thus the same completeness theorem applies to the unchanged norm and simple-function classes of a complex Banach value space; complex homogeneity is inherited pointwise and under the integral. AC supplies Countable Choice.

Proof

Given: The hypotheses and notation of the Statement.

[F1] Exterior and truncated obstacle domains and Non-trapping obstacle specify the exterior and generalized-ray non-trapping hypothesis required by the independent endpoint input.

[F2] Dirichlet spectral Sobolev scale and duality defines $H_{D}^{r}$ and the unitary Dirichlet group; Exterior Dirichlet realization, H² domain and form domain identifies $H_{D}^{1}$ with $H_{0}^{1}$ and its form norm.

[F3] Positive spectral Hilbert scale: duality and interpolation gives $[H_{D}^{a}, H_{D}^{b}]_{s} = H_{D}^{(1 - s) a + s b}$ for $0 < s < 1$ , in the stated complex strip convention.

[F4] Well-defined strip norm and the action of a common endpoint operator gives the exact operator bound $M_{0}^{1 - s} M_{1}^{s}$ for a single complex-linear map between the sums of compatible complex Banach couples with endpoint bounds $M_{0}, M_{1}$ . Separately, An explicit common interpolation model for the fractional Strichartz targets gives, for finite $p$ and $1 < q < \infty$ , the identity $[L_{t}^{p} L_{x}^{q}, L_{t}^{p} W_{x}^{1, q}]_{s} = L_{t}^{p} W_{x}^{s, q}$ . For every $1 \leq p \leq \infty$ , the time spaces use strongly measurable almost-everywhere classes and the Bochner norms of Continuous and mixed time-space function spaces, with essential supremum when $p = \infty$ .

[F5] Reference 0 supplies the two finite- $p$ integer endpoint estimates described in External input.

[F6] The Axiom of Choice records the Choice assumption required by the stated spectral and interpolation suppliers.

[F7] Integer-order Sobolev spaces and their norms defines $W^{j, q} (Ω; C)$ and its weak-derivative norm for $j = 0, 1$ ; in particular $W^{0, q} = L^{q}$ . Its Countable Choice hypothesis follows from the assumed AC.

[F8] Under Countable Choice, Weak Leibniz rule with a smooth factor gives the weak product formula on any open set for a smooth multiplier, and preserves $W^{k, q}$ when its classical derivatives through order $k$ are bounded. Here apply it on $Ω$ with $k = 1$ , $q = 2$ and multiplier $1 - χ$ ; compact support of the multiplier itself is not required.

[F9] Integer-order Sobolev spaces are Banach applies under AC on the open set $Ω$ , with complex scalars, $k = 0, 1$ and exponent $q$ , and gives the complex Banach endpoints $L^{q}$ and $W^{1, q}$ . Their weak-derivative norm in [F7] gives $∥ v ∥_{q} \leq ∥ v ∥_{W^{1, q}}$ .

[F10] Bochner Lp is complete for a Banach value space proves completeness on $(0, T)$ for real Banach values, finite $1 \leq p < \infty$ and Countable Choice. Realification preserves a complex norm and its metric preserves the norm and metric on restricting complex scalars to real scalars. Thus the same completeness theorem applies to the unchanged norm and simple-function classes of a complex Banach value space; complex homogeneity is inherited pointwise and under the integral. AC supplies Countable Choice.

Given: The data and hypotheses in the Statement.

Goal: Prove both displayed estimates for every stated $p, q, s$ .

1.1F1F2F5F7given

Suppose first $p < \infty$ . The admissibility relation and $p > 2$ imply $2 \leq q < \infty$ , including in dimension two. Define the consistent linear maps $S_{0} u (t) = (1 - χ) e^{i t Δ_{D}} u$ and $S_{1} u (t) = e^{i t Δ_{D}} u$ . The independent input bounds $S_{0}$ from $H_{D}^{j}$ and $S_{1}$ from $H_{D}^{j + 1 / p}$ to $L^{p} (0, T; W^{j, q})$ for $j = 0, 1$ , where $W^{0, q} = L^{q}$ . The operators agree on their nested endpoint domains because both are given by the same spectral evolution and multiplication.

1.2F2F6F7F8given

If $p = \infty$ , admissibility gives $q = 2$ and $1 / p = 0$ . Spectral unitarity bounds $e^{i t Δ_{D}}$ on both $H_{D}^{0} = L^{2}$ and $H_{D}^{1} = H_{0}^{1}$ , uniformly in $t$ . The inclusion $H_{0}^{1} ↪ W^{1, 2}$ is continuous by the form norm. Put $a = (1 - χ) ∣_{Ω}$ . Smoothness and compact support of $χ$ imply that $a$ and each classical first derivative $\partial_{j} a = - \partial_{j} χ$ are bounded on $Ω$ . Apply [F8] on this open set to every $v \in W^{1, 2} (Ω; C)$ : $D_{j} (a v) = a D_{j} v - (\partial_{j} χ) v$ in $L^{2} (Ω)$ . The bounds $∥ a v ∥_{2} \leq ∥ a ∥_{\infty} ∥ v ∥_{2}$ and $∥ D_{j} (a v) ∥_{2} \leq ∥ a ∥_{\infty} ∥ D_{j} v ∥_{2} + ∥ \partial_{j} χ ∥_{\infty} ∥ v ∥_{2}$ , summed over the finitely many derivatives in [F7], show bounded multiplication on $W^{1, 2}$ . The value bound also gives bounded multiplication on $L^{2}$ . Thus both spatial maps have uniform bounds from $H_{D}^{j}$ to $W^{j, 2}$ for $j = 0, 1$ .

2.1F2F3F4F5F6F7F9F10step 1.1

For finite $p$ , the source couples $(H_{D}^{0}, H_{D}^{1})$ and $(H_{D}^{1 / p}, H_{D}^{1 + 1 / p})$ are compatible complex Hilbert couples in their respective lower endpoints, by the scale inclusions and norms in [F2, F3]. By [F9, F10], the target endpoints $Y_{0} = L^{p} (0, T; L^{q})$ and $Y_{1} = L^{p} (0, T; W^{1, q})$ are complex Banach spaces. The inequality in [F9] integrates to a continuous injective inclusion $Y_{1} ↪ Y_{0}$ , so they are a compatible couple in the Hausdorff space $Y_{0}$ . The inclusion respects strongly measurable classes: applying it to their simple approximants proves strong measurability in $L^{q}$ , and vanishing as an $L^{q}$ class implies vanishing as a $W^{1, q}$ class. Each source sum is its lower endpoint; step 1.1 defines a single complex-linear map on that sum with values in $Y_{0} = Y_{0} + Y_{1}$ and the two required endpoint bounds. The exact operator theorem in [F4] therefore applies to each map. By [F3] the sources interpolate to $H_{D}^{s}$ and $H_{D}^{s + 1 / p}$ , and by [F4] the target interpolates to $L^{p} (0, T; W^{s, q})$ . This proves both bounds for $0 < s < 1$ ; $s = 0, 1$ are the imported endpoints.

2.2F2F3F4F7F9step 1.2

Now let $p = \infty$ , so $q = 2$ , and write $U (t) = e^{i t Δ_{D}}$ . Apply the exact operator theorem in [F4] to the fixed identity inclusion and the fixed multiplier $M_{a} : v \mapsto (1 - χ) v$ , using their $H_{D}^{j} \to W^{j, 2}$ bounds from step 1.2 at $t = 0$ , $j = 0, 1$ . The source couple is $(H_{D}^{0}, H_{D}^{1})$ in $L^{2}$ ; the target couple is $(L^{2}, W^{1, 2})$ in $L^{2}$ . Both are compatible complex Banach couples by [F2, F3, F7, F9], with continuous endpoint inclusions, and each map is defined consistently on the source sum $L^{2}$ . Thus for every $0 < s < 1$ these fixed maps restrict to bounded maps $B_{s}^{0} : H_{D}^{s} \to W^{s, 2}$ and $B_{s}^{a} : H_{D}^{s} \to W^{s, 2}$ , respectively. The same notation at $s = 0, 1$ denotes their endpoint restrictions. Unitarity on $H_{D}^{s}$ gives $∥ B_{s}^{ν} U (t) u ∥_{W^{s, 2}} \leq ∥ B_{s}^{ν} ∥ ∥ u ∥_{H_{D}^{s}}$ for $ν = 0, a$ , uniformly in $t$ .

3.1F2F3F4step 2.1step 2.2∎

Strong continuity of the spectral group on $H_{D}^{s}$ and boundedness of the fixed maps $B_{s}^{0}, B_{s}^{a}$ give the required measurable representatives of $t \mapsto B_{s}^{ν} U (t) u_{0}$ , so their essential time suprema obey the uniform bounds in step 2.2. These are the whole-domain and cutoff paths. Since $1 / p = 0$ , both estimates follow for $p = \infty$ ; together with step 2.1 this completes the proof.

Source notes

The finite-time integer estimates are independent external premises. The fractional conclusion uses the explicitly stated complex interpolation model; it does not assert that the source uniquely specified that model. The $p = \infty$ case is derived from spectral unitarity and spatial interpolation, without an infinite-exponent Bochner interpolation identity.

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External dependencies

On nonlinear Schrödinger equations in exterior domains · Section 2.4, Propositions 2.10 and 2.11, equations (2.17), (2.18), (2.22), printed pages 304–306, PDF pages 10–12

For the stated exterior problem and admissible exponents with p>2, Proposition 2.10 gives the cutoff-away estimate and Proposition 2.11 gives the whole-domain estimate with loss 1/p. Their integer endpoints s=0 and s=1 are the inputs used here.

Hypotheses: Dimension n at least 2; a nonempty compact smooth non-trapping obstacle; Dirichlet Laplacian; finite T>0; smooth compactly supported cutoff equal to one near the obstacle; 2/p+n/q=n/2 with p>2.

Reading limits: This adjudication read the independent BGT web-extracted PDF text: the geometric hypotheses in the introduction (printed pp.295–297), time-space and spectral conventions (printed pp.298–299), and Section 2.4, Propositions 2.10 and 2.11 including their integer-endpoint passages (printed pp.304–306, PDF pp.10–12). Only the finite-p integer estimates are imported. Their proofs and cited inputs are not reconstructed. Fractional target interpolation and p=infinity are local deductions. No p=2 endpoint or loss-free whole-domain estimate is claimed.

Sources

On nonlinear Schrödinger equations in exterior domains · Section 2.4, Propositions 2.10 and 2.11, equations (2.17), (2.18), (2.22), printed pages 304–306, PDF pages 10–12