Joint smoothness of the spherical-mean family of a smooth spacetime function

The setting and the reduction to the open set $\mathrm{\Omega }$. Let $h$ also denote the $C^{\mathrm{\infty }}$ function on the open set ${\mathbb{R}}^3\times J$ whose restriction to ${\mathbb{R}}^3\times [t_0,T]$ is the $h$ of the statement, put $\mathrm{\Omega }:={\mathbb{R}}^3\times \mathbb{R}\times J$, an open subset of ${\mathbb{R}}^5$, and let $S^2=\{n\in {\mathbb{R}}^3:⟨n,n⟩=1\}$ carry the surface measure $d\mathrm{\Omega }$ of total mass $4\pi$ of [F6]. Define $M(x,s,v):=\frac{1}{4\pi }{\int }_{S^2}h(x+sn,v)d\mathrm{\Omega }(n)$ for $(x,s,v)\in \mathrm{\Omega }$; its restriction to the closed set ${\mathbb{R}}^3\times \mathbb{R}\times [t_0,T]$ is the family of the statement, so every assertion proved below on $\mathrm{\Omega }$, or on an open neighbourhood of a closed subset of $\mathrm{\Omega }$, restricts to the corresponding extension-sense assertion of the statement. All surface integrals are read under the ${\mathrm{A}\mathrm{C}}_{\omega }$ convention carried by [F5] and [F6].

The composed integrand. Define $\mathrm{\Phi }:\mathrm{\Omega }\times {\mathbb{R}}^3\to {\mathbb{R}}^4$ by $\mathrm{\Phi }(x,s,v,n):=(x+sn,v)$. Each component of $\mathrm{\Phi }$ is a finite sum of products of coordinate functions, so $\mathrm{\Phi }$ is $C^{\mathrm{\infty }}$ on the open set $\mathrm{\Omega }\times {\mathbb{R}}^3$ by [F3], and $\mathrm{\Phi }$ takes values in ${\mathbb{R}}^3\times J$. Hence $H:=h\circ \mathrm{\Phi }$ is $C^{\mathrm{\infty }}$ on $\mathrm{\Omega }\times {\mathbb{R}}^3$ by the composition clause of [F3], so every word iterated partial derivative $D^{v}H$ exists and is continuous on $\mathrm{\Omega }\times {\mathbb{R}}^3$ by [F1]. At $p=(x,s,v)\in \mathrm{\Omega }$ and $n\in S^2$, the chain rule [F2], total differentiability of $h$ [F4] and ${\mathrm{\partial }}_s\mathrm{\Phi }(p,n)=(n,0)$ give ${\mathrm{\partial }}_{s}H(p,n)=Dh(x+sn,v)(n,0)={\sum }_{j<3}{\mathrm{\partial }}_{j}h(x+sn,v)n_j=⟨{\mathrm{\nabla }}_{x}h(x+sn,v),n⟩$, the directional derivative of $h$ at $(x+sn,v)$ in direction $(n,0)$ [F4]; applying the same computation to the already differentiated integrands shows that the iterated derivatives $D_x^{\alpha }{\mathrm{\partial }}_s^m{\mathrm{\partial }}_v^{l}H(p,n)$ are the corresponding iterated partial derivatives of the composed map $(x,s,v)\mapsto h(x+sn,v)$, which is $H$ itself, and in particular that ${\mathrm{\partial }}_s^{m}H(p,n)$ is the $m$-th iterated derivative at $\lambda =s$ of $\lambda \mapsto h(x+\lambda n,v)$.

Local bounds on compact boxes. Let $K={\prod }_{j<5}[a_j,b_j]$ be a nonempty closed box contained in $\mathrm{\Omega }$. Then $K\times [-1,1{]}^3$ is a nonempty closed box in ${\mathbb{R}}^8$, hence compact by [F9], and by step 1.2 every $\mathrm{\mid }{D}^{v}H\mathrm{\mid }$ is continuous on the open set $\mathrm{\Omega }\times {\mathbb{R}}^3$, which contains $K\times [-1,1{]}^3$; so by [F9] each $\mathrm{\mid }{D}^{v}H\mathrm{\mid }$ attains a maximum there and $C_v:={\max }_{K\times [-1,1{]}^3}\mathrm{\mid }{D}^{v}H\mathrm{\mid }$ is a finite nonnegative real number with $\mathrm{\mid }{D}^{v}H(p,n)\mathrm{\mid }\le C_v$ for all $(p,n)\in K\times S^2$, because $S^2\subseteq [-1,1{]}^3$.

Continuity of parameter sphere integrals. Claim: if $q:\mathrm{\Omega }\times {\mathbb{R}}^3\to \mathbb{R}$ is continuous with continuous first partial derivatives, $K\subseteq \mathrm{\Omega }$ is a nonempty closed box and $\mathrm{\mid }{\mathrm{\partial }}_{p_j}q(p,n)\mathrm{\mid }\le C_j$ for all $(p,n)\in K\times S^2$ and all $j<5$, then $\mid {\int }_{S^2}q(p,n)d\nu (n)-{\int }_{S^2}q(p_0,n)d\nu (n)\mid \le ({\sum }_{j<5}{C}_j^2{)}^{1\mathrm{/}2}\mathrm{\mid }p-p_0\mathrm{\mid }$ for all $p,p_0\in K$, where $\nu :=d\mathrm{\Omega }\mathrm{/}4\pi$. Indeed, for fixed $n\in S^2$ the function $g(\lambda ):=q(p_0+\lambda u,n)$, $u:=p-p_0$, is defined on $[0,1]$ because the box is convex, and the affine map $\lambda \mapsto (p_0+\lambda u,n)$ has constant partial derivatives, hence is totally differentiable by [F4]; the chain rule [F2] with total differentiability of $q$ [F4] therefore gives $g^{\mathrm{\prime }}(\lambda )=Dq(p_0+\lambda u,n)(u,0)={\sum }_{j<5}{\mathrm{\partial }}_{p_j}q(p_0+\lambda u,n)u_j$ at every $\lambda \in (0,1)$, and the mean value theorem [F8] bounds $\mathrm{\mid }q(p,n)-q(p_0,n)\mathrm{\mid }\le {\sup }_{\lambda }\mid {\sum }_j{\mathrm{\partial }}_{p_j}q(p_0+\lambda u,n)u_j\mid \le ({\sum }_j{C}_j^2{)}^{1\mathrm{/}2}\mathrm{\mid }u\mathrm{\mid }$ by Cauchy-Schwarz [F10] and the local bounds. Integrating this pointwise bound against $\nu$: the signed surface integral is the difference of the integrals of the positive and negative parts [F6], so $\mid {\int }_{S^2}q(p,\cdot )d\nu -{\int }_{S^2}q(p_0,\cdot )d\nu \mid \le {\int }_{S^2}\mathrm{\mid }q(p,n)-q(p_0,n)\mathrm{\mid }d\nu (n)\le ({\sum }_j{C}_j^2{)}^{1\mathrm{/}2}\mathrm{\mid }u\mathrm{\mid }{\int }_{S^2}d\nu =({\sum }_j{C}_j^2{)}^{1\mathrm{/}2}\mathrm{\mid }u\mathrm{\mid }$, where the chart formula of [F6] with the monotonicity and scalar rules of [F7] gives monotonicity, the bound ${\int }_{S^2}cd\nu =c$ for constants $c\ge 0$, and ${\int }_{S^2}d\nu =1$ from the normalisation ${\int }_{S^2}d\mathrm{\Omega }=4\pi$ of [F6].

Induction: every word derivative of $M$ exists with the differentiated-integral formula, continuously. Claim: for every word $v$ in the five coordinate directions of $\mathrm{\Omega }$, the iterated derivative $D^{v}M$ exists on $\mathrm{\Omega }$ and equals ${\int }_{S^2}{D}^{v}H(p,n)d\nu (n)$, and this integral is continuous on $\mathrm{\Omega }$. Prove the claim by induction on the length $r$ of the word. For $r=0$ it is the definition of $M$ in step 1.1 together with $\nu =d\mathrm{\Omega }\mathrm{/}4\pi$, and continuity is step 2.2 applied with $q:=H$, whose partial derivatives $D^{(j)}H$ are continuous with bounds $C_{(j)}$ on $K\times S^2$ by steps 1.2 and 2.1. Assume the claim for all words of length $r$; let $v$ be such a word and $j$ a coordinate direction, and fix $p_0\in \mathrm{\Omega }$, a nonempty closed box $K\subseteq \mathrm{\Omega }$ with $p_0$ in its interior and coordinate intervals $[a_j,b_j]$. For fixed coordinates other than the $j$-th and $t\in [a_j,b_j]$, put $Q(n,t):=D^{v}H(p(t),n)$, where $p(t)$ is the point of the $j$-th coordinate line with $j$-th coordinate $t$; then both $Q$ and ${\mathrm{\partial }}_{t}Q=D^{v\cdot j}H(p(t),n)$ are continuous on an open neighbourhood of $S^2\times [a_j,b_j]$ in ${\mathbb{R}}^3\times \mathbb{R}$, being restrictions of the continuous $D^{v}H$ and $D^{v\cdot j}H$ of step 1.2. By the surface clause of [F5], $t\mapsto {\int }_{S^2}Q(n,t)d\mathrm{\Omega }(n)$ is differentiable on $[a_j,b_j]$ with derivative ${\int }_{S^2}{\mathrm{\partial }}_{t}Q(n,t)d\mathrm{\Omega }(n)$; multiplying by the constant $1\mathrm{/}4\pi$ (the scalar multiple rule for one-variable derivatives, [F3]) gives ${\mathrm{\partial }}_j{\int }_{S^2}{D}^{v}Hd\nu ={\int }_{S^2}{D}^{v\cdot j}Hd\nu$ on the line through $p_0$; with the induction hypothesis this shows $D^{v\cdot j}M={\int }_{S^2}{D}^{v\cdot j}Hd\nu$ on that line, hence on all of $\mathrm{\Omega }$ since $p_0$ and $j$ are arbitrary, and this integral is continuous on $\mathrm{\Omega }$ by step 2.2 applied to $q:=D^{v\cdot j}H$, whose partial derivatives $D^{(v\cdot j)\cdot i}H$ are continuous by step 1.2 with bounds $C_{(v\cdot j)\cdot i}$ on $K\times S^2$ by step 2.1. This closes the induction.

$M$ is $C^{\mathrm{\infty }}$ and satisfies the displayed formula. By step 3.1 every word iterated partial derivative of $M$ exists and is continuous on the open set $\mathrm{\Omega }$, so $M$ is of class $C^{\mathrm{\infty }}$ on $\mathrm{\Omega }$ by [F1]; in particular its restriction to ${\mathbb{R}}^3\times \mathbb{R}\times [t_0,T]$ is $C^{\mathrm{\infty }}$ in the extension sense of the statement. Every canonical-order derivative $D_x^{\alpha }{\mathrm{\partial }}_s^m{\mathrm{\partial }}_v^{l}M$ is one of the word derivatives covered by step 3.1, so it equals the sphere integral of the corresponding word derivative $D_x^{\alpha }{\mathrm{\partial }}_s^m{\mathrm{\partial }}_v^{l}H$ of the composed map, and step 1.2 identifies those integrands as the corresponding iterated partial derivatives of $(x,s,v)\mapsto h(x+sn,v)$; the asserted $s$-derivative statements are those of step 1.2.

The composed map $\tilde{H}$ and the local nature of the argument. Let $c>0$ and define $\tilde{H}(x,t,v):=(t-v)M(x,c(t-v),v)$ on the open set ${\mathbb{R}}^3\times \mathbb{R}\times J$. The map $\mathrm{\Psi }(x,t,v):=(x,c(t-v),v)$ has components that are finite sums of products of coordinate functions, so it is $C^{\mathrm{\infty }}$ by [F3] and takes values in $\mathrm{\Omega }$; hence $(x,t,v)\mapsto M(\mathrm{\Psi }(x,t,v))$ is $C^{\mathrm{\infty }}$ by composition [F3] with step 4.1, and multiplying by the $C^{\mathrm{\infty }}$ function $t-v$ gives that $\tilde{H}$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times \mathbb{R}\times J$ by [F3]. Since $[t_0,T]\subset J$ and the closed triangle $\mathrm{\Delta }=\{(t,v):t_0\le v\le t\le T\}$ is contained in $[t_0,T]\times [t_0,T]\subset J\times J$, the restriction shows that $\tilde{H}$ is $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times [t_0,T]\times [t_0,T]$ in the extension sense, and for each fixed $x$ the map $(t,v)\mapsto \tilde{H}(x,t,v)$ is $C^{\mathrm{\infty }}$ on the open neighbourhood $J\times J$ of $\mathrm{\Delta }$. The whole argument used only the bounds of step 2.1 on compact parameter boxes and the compactness of $S^2$; no boundedness, decay or compact spatial support of $h$ was assumed or used.