Three-dimensional L4 time Linfinity space estimate

PDF p. 7: Proposition 2.4; (2.6)

Assume the Axiom of Choice. Let $\mathrm{\Omega }={\mathbb{R}}^3\setminus \mathrm{\Theta }$ be the connected exterior of a nonempty smooth compact non-trapping obstacle, and put $W(t)=e^{it{\mathrm{\Delta }}_D}$ for its homogeneous Dirichlet evolution. For every $T>0$ there is a constant $C$ depending on the domain such that, for every $u_0\in H_0^1(\mathrm{\Omega })$, $\mathrm{\parallel }W(\cdot )u_0{\mathrm{\parallel }}_{L_t^4{L}_x^{\mathrm{\infty }}((0,T)\times \mathrm{\Omega })}\le C\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H_0^1(\mathrm{\Omega })}\mathrm{.}$ The $H_0^1$ norm is the inherited $H^1$ norm. The mixed norm is the scalar sectionwise norm, with spatial essential supremum; no strong measurability into the nonseparable space $L^{\mathrm{\infty }}(\mathrm{\Omega })$ is asserted. This is the three-dimensional energy estimate attributed to Blair–Smith–Sogge, §5.

The original article’s Proposition2.4, equation(2.6), is the attribution for this item. The mathematical premise is the completed independent BSS import, specialized to zero forcing; its source locator and reading limits are preserved there and no fresh external-text reading is claimed here. The mathematical self-adjoint realization and flow suppliers stored in the Physics Library remain available through the declared scale-flow chain. The existing Dirichlet definition supplies the direct formulation input; no upstream item is changed, and no review or article-defect judgment is asserted.