Three-dimensional Dirichlet energy-to-mixed-norm estimate

Put $A=I-{\mathrm{\Delta }}_D$ and $E=H_D^1(\mathrm{\Omega })=H_0^1(\mathrm{\Omega })$ with its inherited $H^1$ norm. The scale makes $E$ a complete complex Hilbert space continuously included in $L^2$, with $\mathrm{\parallel }h{\mathrm{\parallel }}_2\le \mathrm{\parallel }h{\mathrm{\parallel }}_E=\mathrm{\parallel }{A}^{1\mathrm{/}2}h{\mathrm{\parallel }}_2$. Since the spectral multipliers $e^{it\lambda }$ and $(1-\lambda {)}^{1\mathrm{/}2}$ commute on the latter's domain, $A^{1\mathrm{/}2}U(t)h=U(t)A^{1\mathrm{/}2}h$. Here the square root is used on the nonpositive spectrum of ${\mathrm{\Delta }}_D$, extending it arbitrarily to a Borel function elsewhere. Thus $\mathrm{\parallel }U(t)h-U(t_0)h{\mathrm{\parallel }}_E=\mathrm{\parallel }U(t)A^{1\mathrm{/}2}h-U(t_0)A^{1\mathrm{/}2}h{\mathrm{\parallel }}_2\to 0$ by strong continuity on $L^2$. Together with F2 this gives a strongly continuous unitary group on $E$, also as a real Banach space.

Fix $0<S<T$, with $S$ finite. Apply F5 on $[0,S]$ to $E$, $W=U$, $r=0$, and the restriction of $F$, using $M=1$ from step 1.1. It establishes strong measurability and Bochner integrability of $s\mapsto U(t-s)F(s)$ on $[0,t]$ for each $t\le S$, and continuity in $E$ of $D(t)={\int }_0^{t}U(t-s)F(s)ds$. The complex and real Bochner integrals have the same simple approximations and norm limits, so they agree. Consequently $u(t)=U(t)f+iD(t)$ is continuous in $E$ and hence in $L^2$, with $\mathrm{\parallel }u(t){\mathrm{\parallel }}_E\le \mathrm{\parallel }f{\mathrm{\parallel }}_E+{\int }_0^t\mathrm{\parallel }F(s){\mathrm{\parallel }}_{E}ds$. Changing a forcing representative on a time-null set changes none of the integrals. The constructions for different $S$ agree on their overlaps because their integrands agree there.

On $[0,S]$, construct finite-range time maps $v_n$ uniformly approximating $u$ in $L^2$ within $2^{-n}$. For each time $a$, continuity gives a relatively open neighborhood on which $\mathrm{\parallel }u(t)-u(a){\mathrm{\parallel }}_2<2^{-n}$. Compactness in F9 gives a finite subcover; assign the sample value from the first neighborhood containing $t$. The resulting cells are Borel, so after choosing measurable spatial representatives of the finitely many sample values, $v_n(t,x)$ is jointly measurable. Choice permits these countably many representative selections. The path $u$ is bounded on this interval by step 2.1, so each $v_n$ belongs to product $L^2$. By F6, $\mathrm{\parallel }{v}_n-v_m{\mathrm{\parallel }}_{L^2((0,S)\times \mathrm{\Omega })}\le S^{1\mathrm{/}2}(2^{-n}+2^{-m})$. F7 therefore supplies a product $L^2$ limit with a jointly measurable representative $w_S$.

Choose a subsequence $v_{n_k}$ with $\mathrm{\parallel }{v}_{n_k}-w_S{\mathrm{\parallel }}_{L^2((0,S)\times \mathrm{\Omega })}^2\le 2^{-k}$. Tonelli, first on time-space and then on counting measure times time, gives $w_S(t,\cdot )\in L^2(\mathrm{\Omega })$ and ${\sum }_k\mathrm{\parallel }{v}_{n_k}(t,\cdot )-w_S(t,\cdot ){\mathrm{\parallel }}_2^2<\mathrm{\infty }$ for almost every $t$. Thus those sections converge in $L^2$ to $w_S(t,\cdot )$; step 3.1 also makes their classes converge to $u(t)$. Uniqueness of a norm limit gives $[w_S(t,\cdot )]=u(t)$ for almost every $t$. Choose finite $0=S_0<S_1<\cdots \uparrow T$ and construct $w_{S_j}$ as above; for $T=\mathrm{\infty }$ take $S_j=j$. On each strip $[S_{j-1},S_j)\times \mathrm{\Omega }$ use $w_{S_j}$. This countable gluing is jointly measurable and represents $u(t)$ in $L^2$ for almost every $t\in (0,T)$. Any two such representatives agree product-almost everywhere by F6 applied to their disagreement set. Hence F3 makes the scalar sectionwise mixed norm meaningful, allowing infinity before applying the estimate. No $L^{\mathrm{\infty }}$-valued strong measurability is needed.

The expression constructed in steps 1.1–2.1 uses exactly the Dirichlet spectral propagator, energy initial datum, and boundary-compatible forcing in the independent estimate F8. Applying that imported bound yields precisely the inequality in the Statement, with its domain-dependent constant independent of $f$, $F$ and $T$. Its global Dirichlet scope includes $T=\mathrm{\infty }$; restriction gives every finite interval. The identification of representatives in step 4.1 makes this the scalar mixed norm specified in F3. Taking $F=0$ gives the stated homogeneous case.