Physlib.SpaceAndTime.Space.ConstantSliceDist

For any natural numbers $n, m$ and dimension $d$, and for any coordinate index $i \in \{0, \dots, d\}$, there exists an exponent $rt$ such that for every Schwartz function $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$, there exists a constant $k$ satisfying: 1. The function $r \mapsto (1 + |r|)^{-rt}$ is integrable with respect to the volume measure on $\mathbb{R}$. 2. For all $x \in \mathbb{R}^d$ and $r \in \mathbb{R}$, the inequality $$\|p\|^m \cdot \|D^n \eta(p)\| \le k (1 + |r|)^{-rt}$$ holds, where $p = (\text{slice } i)^{-1}(r, x)$ is the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r$ and whose remaining coordinates are given by $x$, and $D^n \eta$ is the $n$-th Fréchet derivative of $\eta$. 3. The constant $k$ is given by $k = 2^{rt+m} \cdot \sup_{(a, b) \le (rt+m, n)} p_{a, b}(\eta)$, where $p_{a, b}(\eta)$ are the standard seminorms of the Schwartz space.

For any natural numbers $n, m$ and dimension $d$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. For any coordinate index $i \in \{0, \dots, d\}$ and any fixed point $x \in \mathbb{R}^d$, the function $$r \mapsto \|p\|^m \cdot \|D^n \eta(p)\|$$ is integrable over $\mathbb{R}$ with respect to the Lebesgue measure, where $p = (\text{slice } i)^{-1}(r, x)$ is the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r$ and whose remaining coordinates are given by $x$, and $D^n \eta$ denotes the $n$-th Fréchet derivative of $\eta$.

For any dimension $d$ and coordinate index $i \in \{0, \dots, d\}$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. For any fixed point $x \in \mathbb{R}^d$, the function $$r \mapsto \eta(p)$$ is integrable over $\mathbb{R}$ with respect to the Lebesgue measure, where $p = (\text{slice } i)^{-1}(r, x)$ is the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r$ and whose remaining $d$ coordinates are given by the vector $x$.

For any dimension $d$, coordinate index $i \in \{0, \dots, d\}$, and Schwartz function $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$, let $(\text{slice } i)^{-1}(r, x)$ denote the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r \in \mathbb{R}$ and whose remaining $d$ coordinates are given by $x \in \mathbb{R}^d$. For a fixed $x$, the Fréchet derivative of the map $x \mapsto \eta((\text{slice } i)^{-1}(r, x))$ evaluated at $x$ is integrable as a function of $r$ over $\mathbb{R}$ with respect to the Lebesgue measure.

For any dimension $d$, coordinate index $i \in \{0, \dots, d\}$, and Schwartz function $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$, let $(\text{slice } i)^{-1}(r, x)$ denote the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r \in \mathbb{R}$ and whose remaining $d$ coordinates are given by $x \in \mathbb{R}^d$. For a fixed point $x$, the derivative of the map $r \mapsto \eta((\text{slice } i)^{-1}(r, x))$ with respect to $r$ is integrable as a function of $r$ over $\mathbb{R}$ with respect to the Lebesgue measure.

For any dimension $d$ and natural number $n$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. For any fixed point $x \in \mathbb{R}^d$ and coordinate index $i \in \{0, \dots, d\}$, the function $$r \mapsto \|D^n \eta(p)\|$$ is integrable over $\mathbb{R}$ with respect to the Lebesgue measure, where $p = (\text{slice } i)^{-1}(r, x)$ is the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r$ and whose remaining $d$ coordinates are given by $x$, and $D^n \eta$ denotes the $n$-th Fréchet derivative of $\eta$.

For any dimension $d$ and natural number $n$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. For any fixed point $x \in \mathbb{R}^d$ and coordinate index $i \in \{0, \dots, d\}$, the function $$r \mapsto D^n \eta(p)$$ is integrable over $\mathbb{R}$ with respect to the Lebesgue measure, where $p = (\text{slice } i)^{-1}(r, x)$ is the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r$ and whose remaining $d$ coordinates are given by $x$, and $D^n \eta$ denotes the $n$-th Fréchet derivative of $\eta$.

For any dimension $d$ and coordinate index $i \in \{0, \dots, d\}$, let $\eta: \text{Space}(d+1) \to \mathbb{R}$ be a Schwartz function. The function $f: \text{Space } d \to \mathbb{R}$ defined by $$f(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ is continuous, where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\text{Space}(d+1)$ whose $i$-th coordinate is $r$ and whose remaining coordinates are given by $x \in \text{Space } d$.

For any dimension $d$ and coordinate index $i \in \{0, \dots, d\}$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. Define the function $F: \mathbb{R}^d \to \mathbb{R}$ by integrating $\eta$ over the $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r$ and whose remaining $d$ coordinates are given by the vector $x \in \mathbb{R}^d$. Then, for any point $x_0 \in \mathbb{R}^d$, the function $F$ is Fréchet differentiable at $x_0$, and its Fréchet derivative $DF(x_0)$ is equal to the integral of the Fréchet derivative of the integrand with respect to $x$: $$DF(x_0) = \int_{-\infty}^{\infty} \text{D}_x \left( \eta((\text{slice } i)^{-1}(r, x)) \right) \Big|_{x=x_0} \, dr$$

For any dimension $d$ and coordinate index $i \in \{0, \dots, d\}$, let $\eta: \text{Space}(d+1) \to \mathbb{R}$ be a Schwartz function. Define the function $F: \text{Space } d \to \mathbb{R}$ by integrating $\eta$ over the $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\text{Space}(d+1)$ whose $i$-th coordinate is $r$ and whose remaining $d$ coordinates are given by the vector $x \in \text{Space } d$. Then, the function $F$ is differentiable on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$, any natural number $n$, and any coordinate index $i \in \{0, \dots, d\}$, let $\eta: \text{Space}(d+1) \to \mathbb{R}$ be a Schwartz function. The function $F: \text{Space } d \to \mathbb{R}$ defined by integrating $\eta$ over its $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ is $n$-times continuously differentiable ($C^n$), where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\text{Space}(d+1)$ whose $i$-th coordinate is $r \in \mathbb{R}$ and whose remaining $d$ coordinates are given by $x \in \text{Space } d$.

For any dimension $d \in \mathbb{N}$, any natural number $n$, and any coordinate index $i \in \{0, \dots, d\}$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. Define the function $F: \mathbb{R}^d \to \mathbb{R}$ by integrating $\eta$ over its $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r \in \mathbb{R}$ and whose remaining $d$ coordinates are given by $x \in \mathbb{R}^d$. Then, for any $x \in \mathbb{R}^d$ and any sequence of $n$ vectors $y_1, \dots, y_n \in \mathbb{R}^d$, the $n$-th iterated Fréchet derivative of $F$ at $x$ applied to the vectors $y_1, \dots, y_n$ is equal to the integral of the $n$-th iterated Fréchet derivative of $\eta$: $$D^n F(x)(y_1, \dots, y_n) = \int_{-\infty}^{\infty} D^n \eta ((\text{slice } i)^{-1}(r, x)) (Y_1, \dots, Y_n) \, dr$$ where each $Y_j = (\text{slice } i)^{-1}(0, y_j)$ is the vector in $\mathbb{R}^{d+1}$ with $0$ at the $i$-th coordinate and the components of $y_j$ at the other $d$ coordinates.

For any dimension $d \in \mathbb{N}$, natural number $n$, and coordinate index $i \in \{0, \dots, d\}$, let $\eta: \mathbb{R}^{d+1} \to \mathbb{R}$ be a Schwartz function. Let $F: \mathbb{R}^d \to \mathbb{R}$ be the function defined by integrating $\eta$ over its $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r \in \mathbb{R}$ and whose remaining $d$ coordinates are given by $x \in \mathbb{R}^d$. Then, for any $x \in \mathbb{R}^d$, the $n$-th iterated Fréchet derivative of $F$ at $x$ is equal to the integral of the $n$-th iterated Fréchet derivative of $\eta$ composed with a linear embedding: $$D^n F(x) = \left( \int_{-\infty}^{\infty} D^n \eta((\text{slice } i)^{-1}(r, x)) \, dr \right) \circ L$$ where $L: \mathbb{R}^d \to \mathbb{R}^{d+1}$ is the continuous linear map $y \mapsto (\text{slice } i)^{-1}(0, y)$, which inserts $0$ at the $i$-th coordinate and the components of $y$ at the other coordinates. The composition $\circ L$ signifies that each of the $n$ arguments of the multilinear map $D^n \eta$ is pre-composed with $L$.

For any dimension $d \in \mathbb{N}$, natural number $n$, and coordinate index $i \in \{0, \dots, d\}$, let $\eta \in \mathcal{S}(\mathbb{R}^{d+1}, \mathbb{R})$ be a Schwartz function. Let $F: \mathbb{R}^d \to \mathbb{R}$ be the function defined by integrating $\eta$ over its $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta((\text{slice } i)^{-1}(r, x)) \, dr$$ where $(\text{slice } i)^{-1}(r, x)$ denotes the point in $\mathbb{R}^{d+1}$ whose $i$-th coordinate is $r \in \mathbb{R}$ and whose remaining $d$ coordinates are given by $x \in \mathbb{R}^d$. Then, for any $x \in \mathbb{R}^d$, the norm of the $n$-th iterated Fréchet derivative of $F$ at $x$ satisfies the following inequality: $$\|D^n F(x)\| \leq \left( \int_{-\infty}^{\infty} \|D^n \eta((\text{slice } i)^{-1}(r, x))\| \, dr \right) \cdot \|L\|^n$$ where $L: \mathbb{R}^d \to \mathbb{R}^{d+1}$ is the continuous linear map $y \mapsto (\text{slice } i)^{-1}(0, y)$, which inserts $0$ at the $i$-th coordinate and the components of $y$ at the other coordinates.

For any dimension $d \in \mathbb{N}$, natural numbers $n$ and $m$, and coordinate index $i \in \{0, \dots, d\}$, there exists an exponent $rt$ such that for every Schwartz function $\eta \in \mathcal{S}(\mathbb{R}^{d+1}, \mathbb{R})$, the function $F: \mathbb{R}^d \to \mathbb{R}$ defined by integrating $\eta$ over its $i$-th coordinate: $$F(x) = \int_{-\infty}^{\infty} \eta(\sigma_i(r, x)) \, dr$$ (where $\sigma_i(r, x) \in \mathbb{R}^{d+1}$ is the vector with $r$ at the $i$-th coordinate and the components of $x \in \mathbb{R}^d$ at the remaining positions) satisfies the following properties: 1. The function $r \mapsto (1 + |r|)^{-rt}$ is integrable with respect to the Lebesgue measure on $\mathbb{R}$. 2. For all $x \in \mathbb{R}^d$, the weighted norm of the $n$-th Fréchet derivative of $F$ is bounded by: $$\|x\|^m \cdot \|D^n F(x)\| \le \left( \int_{-\infty}^{\infty} (1 + |r|)^{-rt} \, dr \right) \cdot \|L\|^n \cdot 2^{rt+m} \cdot \sup_{(a, b) \le (rt+m, n)} p_{a, b}(\eta)$$ where $L: \mathbb{R}^d \to \mathbb{R}^{d+1}$ is the linear map $y \mapsto \sigma_i(0, y)$, and $p_{a, b}(\eta)$ are the standard seminorms of the Schwartz space $\mathcal{S}(\mathbb{R}^{d+1}, \mathbb{R})$.

For a natural number $d$ and an index $i \in \{0, \dots, d\}$, the continuous linear map $\text{sliceSchwartz}_i$ maps a Schwartz function $\eta \in \mathcal{S}(\text{Space}(d+1), \mathbb{R})$ to a Schwartz function in $\mathcal{S}(\text{Space } d, \mathbb{R})$. For any $x \in \text{Space } d$, the value of the resulting function is defined by the integral of $\eta$ over the $i$-th coordinate: $$(\text{sliceSchwartz}_i \, \eta)(x) = \int_{-\infty}^{\infty} \eta(\sigma_i(r, x)) \, dr$$ where $\sigma_i(r, x)$ denotes the vector in $\text{Space}(d+1)$ obtained by inserting the real value $r$ at the $i$-th coordinate and filling the remaining positions with the components of $x$.

For any dimension $d \in \mathbb{N}$ and index $i \in \{0, \dots, d\}$, let $\eta \in \mathcal{S}(\text{Space}(d+1), \mathbb{R})$ be a Schwartz function. For any $x \in \text{Space } d$, the value of the integrated function $(\text{sliceSchwartz}_i \, \eta)$ at $x$ is given by the integral of $\eta$ over the $i$-th coordinate: $$(\text{sliceSchwartz}_i \, \eta)(x) = \int_{-\infty}^{\infty} \eta(\sigma_i(r, x)) \, dr$$ where $\sigma_i(r, x)$ denotes the vector in $\text{Space}(d+1)$ obtained by inserting the real value $r$ at the $i$-th coordinate and filling the remaining positions with the components of $x$.

Given a dimension $d \in \mathbb{N}$ and an index $i \in \{0, \dots, d\}$, the linear map $\text{constantSliceDist}_i$ transforms an $M$-valued distribution $f$ on $\text{Space } d$ into an $M$-valued distribution on $\text{Space } d+1$. The resulting distribution is constant along the $i$-th direction. For any Schwartz test function $\eta \in \mathcal{S}(\text{Space } d+1, \mathbb{R})$, the action of the lifted distribution is defined by applying $f$ to the integral of $\eta$ over its $i$-th coordinate: $$(\text{constantSliceDist}_i \, f)(\eta) = f(\text{sliceSchwartz}_i \, \eta)$$ where $(\text{sliceSchwartz}_i \, \eta)(x) = \int_{-\infty}^{\infty} \eta(\sigma_i(r, x)) \, dr$ for $x \in \text{Space } d$, and $\sigma_i(r, x)$ is the vector in $\text{Space } d+1$ obtained by inserting the real value $r$ at the $i$-th coordinate of $x$.

For any dimension $d \in \mathbb{N}$ and index $i \in \{0, \dots, d\}$, let $f$ be an $M$-valued distribution on $\text{Space } d$, where $M$ is a real normed vector space. The action of the lifted distribution $\text{constantSliceDist}_i \, f$ (which is a distribution on $\text{Space } (d+1)$ constant along the $i$-th direction) on a Schwartz test function $\eta \in \mathcal{S}(\text{Space } (d+1), \mathbb{R})$ is evaluated as: $$(\text{constantSliceDist}_i \, f)(\eta) = f(\text{sliceSchwartz}_i \, \eta)$$ where $\text{sliceSchwartz}_i \, \eta$ is the Schwartz function on $\text{Space } d$ obtained by integrating $\eta$ over the $i$-th coordinate: $$(\text{sliceSchwartz}_i \, \eta)(x) = \int_{-\infty}^{\infty} \eta(\sigma_i(r, x)) \, dr$$ Here, $x \in \text{Space } d$ and $\sigma_i(r, x)$ denotes the vector in $\text{Space } (d+1)$ formed by inserting the real value $r$ at the $i$-th position of $x$.

Let $M$ be a real normed vector space and $d$ be a natural number. For any coordinate index $i \in \{0, \dots, d\}$ and any $M$-valued distribution $f$ on $\text{Space } d$, let $\text{constantSliceDist}_i f$ be the distribution on $\text{Space } (d+1)$ formed by lifting $f$ such that it is translationally invariant (constant) in the $i$-th coordinate direction. Then the distributional derivative of this lifted distribution with respect to the $i$-th coordinate is zero: $$\partial_i (\text{constantSliceDist}_i f) = 0$$ where $\partial_i$ denotes the partial derivative in the sense of distributions along the $i$-th standard basis vector.

Let $d$ be a natural number and $M$ be a real normed vector space. Let $f$ be an $M$-valued distribution on $\text{Space } d$. For any index $i \in \{0, \dots, d\}$, let $\text{constantSliceDist}_i f$ be the distribution on $\text{Space } (d+1)$ that is constant along the $i$-th coordinate direction. For any index $j \in \{0, \dots, d-1\}$, let $i.\text{succAbove}(j)$ be the index in $\{0, \dots, d\}$ that maps $j$ to the corresponding coordinate in the higher-dimensional space while skipping $i$. Then, the partial derivative of the lifted distribution in the $i.\text{succAbove}(j)$-th direction is equal to the lift of the partial derivative of $f$ in the $j$-th direction: $$\partial_{i.\text{succAbove}(j)} (\text{constantSliceDist}_i f) = \text{constantSliceDist}_i (\partial_j f)$$ where $\partial_k$ denotes the distributional partial derivative in the direction of the $k$-th standard basis vector.