Physlib.SpaceAndTime.Space.IsDistBounded

Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. The predicate `IsDistBounded f` holds if $f$ is almost everywhere strongly measurable with respect to the volume measure and there exists a finite natural number $n$ such that the norm of $f$ is bounded by a sum of the form: $$\|f(x)\| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$$ where for each $i \in \{1, \dots, n\}$, the coefficient $c_i$ is a non-negative real number ($c_i \ge 0$), $g_i$ is a vector in $\text{Space } d$, and the exponent $p_i$ is an integer satisfying $p_i \ge -(d - 1)$.

Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (satisfies $\text{IsDistBounded } f$), then $f$ is almost everywhere strongly measurable with respect to the volume measure on $\text{Space } d$.

Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (satisfies `IsDistBounded f`), then for any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the function mapping $x \in \text{Space } d$ to the scalar product $\eta(x) \cdot f(x)$ is almost everywhere strongly measurable with respect to the volume measure.

Let $d$ be a natural number and $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (satisfies `IsDistBounded f`), then for any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$ and any vector $y \in \text{Space } d$, the function $x \mapsto (D\eta(x) \cdot y) f(x)$ is almost everywhere strongly measurable with respect to the volume measure, where $D\eta(x) \cdot y$ denotes the Fréchet derivative of $\eta$ at $x$ evaluated in the direction $y$.

Let $d$ and $r$ be natural numbers. Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (meaning it is almost everywhere strongly measurable and its norm is bounded by a finite sum of terms of the form $c_i \|x + g_i\|^{p_i}$), then the function $x \mapsto \frac{1}{(1 + \|x\|)^r} f(x)$ is almost everywhere strongly measurable with respect to the volume measure on $\text{Space } d$.

Let $d$ be a natural number and $f: \text{Space } d \to F$ be a distribution-bounded function mapping into a normed space $F$. For any Schwartz map $\eta \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the function $(t, x) \mapsto \eta(t, x) \cdot f(x)$ is almost everywhere strongly measurable with respect to the volume measure on $\text{Time} \times \text{Space } d$.

Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (satisfies `IsDistBounded f`), then for any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the function $x \mapsto \eta(x) \cdot f(x)$ is integrable on $\text{Space } d$ with respect to the volume measure.

Let $f: \text{Space } d \to \mathbb{R}$ be a real-valued function on a $d$-dimensional Euclidean space. If $f$ is distribution-bounded (satisfies `IsDistBounded f`), then for any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the product function $x \mapsto \eta(x) f(x)$ is integrable with respect to the volume measure.

Let $d$ be a natural number and let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (satisfies `IsDistBounded f`), then for any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$ and any vector $y \in \text{Space } d$, the function $x \mapsto (D\eta(x)y) \cdot f(x)$ is integrable with respect to the volume measure, where $D\eta(x)y$ denotes the Fréchet derivative of $\eta$ at $x$ applied to the vector $y$.

Let $f: \text{Space } d \to \mathbb{R}$ be a distribution-bounded function on a $d$-dimensional Euclidean space. For any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$ and any vector $y \in \text{Space } d$, the function $x \mapsto (D\eta(x)y) \cdot f(x)$ is integrable with respect to the volume measure on $\text{Space } d$, where $D\eta(x)y$ denotes the Fréchet derivative of $\eta$ at $x$ applied to the vector $y$.

Let $D_1$ and $D_2$ be normed additive commutative groups equipped with measurable structures. If $\mu_1$ and $\mu_2$ are measures on $D_1$ and $D_2$ respectively that both have temperate growth, then the product measure $\mu_1 \times \mu_2$ on the product space $D_1 \times D_2$ also has temperate growth, provided that the product space is an opens-measurable space (i.e., its $\sigma$-algebra is generated by its open sets).

Let $d$ be a natural number and $f: \text{Space } d \to F$ be a distribution-bounded function mapping into a normed space $F$. For any Schwartz map $\eta$ in the space $\mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the function $(t, x) \mapsto \eta(t, x) \cdot f(x)$ is integrable over $\text{Time} \times \text{Space } d$ with respect to the volume measure.

Let $d$ be a natural number and $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (i.e., it is almost everywhere strongly measurable and its norm is bounded by a finite sum of terms of the form $c_i \|x + g_i\|^{p_i}$ where $p_i \ge -(d - 1)$), then there exists a natural number $r$ such that the function $$x \mapsto \frac{1}{(1 + \|x\|)^r} f(x)$$ is integrable with respect to the volume measure on $\text{Space } d$.

Let $f: \text{Space } d \to F$ be a distribution-bounded function from a $d$-dimensional Euclidean space to a normed space $F$. Then there exist a finite set of indices $s \subset \mathbb{N} \times \mathbb{N}$ and a non-negative real constant $C \ge 0$ such that for any Schwartz map $\eta \in \mathcal{S}(\text{Space } d, \mathbb{R})$, the norm of the integral of $f$ against $\eta$ is bounded by the supremum of a finite family of Schwartz seminorms: $$\left\| \int_{\text{Space } d} \eta(x) f(x) \, dx \right\| \le C \cdot \max_{(k, \ell) \in s} p_{k, \ell}(\eta)$$ where $p_{k, \ell}$ denotes the $(k, \ell)$-th seminorm in the standard family of seminorms for the Schwartz space $\mathcal{S}(\text{Space } d, \mathbb{R})$.

For any dimension $d$, the constant zero function $f: \text{Space } d \to F$ (where $f(x) = 0$ for all $x \in \text{Space } d$) is distribution-bounded.

Let $f, g: \text{Space } d \to F$ be two functions from a $d$-dimensional Euclidean space to a normed space $F$. If both $f$ and $g$ are distribution-bounded (satisfying the predicate `IsDistBounded`), then their sum $f + g$ is also distribution-bounded.

Let $f, g: \text{Space } d \to F$ be functions from a $d$-dimensional Euclidean space to a normed space $F$. If both $f$ and $g$ are distribution-bounded (satisfying the predicate `IsDistBounded`), then their pointwise sum $x \mapsto f(x) + g(x)$ is also distribution-bounded.

Let $d$ be a natural number and $s$ be a finite set of indices. Let $f_i: \text{Space } d \to F$ be a family of functions from a $d$-dimensional Euclidean space to a normed space $F$ indexed by $i \in s$. If for every $i \in s$, the function $f_i$ is distribution-bounded, then their sum $\sum_{i \in s} f_i$ is also distribution-bounded. A function $f$ is distribution-bounded if it is almost everywhere strongly measurable and there exist constants $c_j \ge 0$, vectors $g_j \in \text{Space } d$, and integers $p_j \ge -(d - 1)$ such that $\|f(x)\| \le \sum_{j=1}^n c_j \|x + g_j\|^{p_j}$.

Let $d$ be a natural number and $s$ be a finite set of indices. Let $f_i: \text{Space } d \to F$ be a family of functions from a $d$-dimensional Euclidean space to a normed space $F$ indexed by $i \in s$. If for every $i \in s$, the function $f_i$ is distribution-bounded, then their pointwise sum $x \mapsto \sum_{i \in s} f_i(x)$ is also distribution-bounded. A function $f$ is distribution-bounded if it is almost everywhere strongly measurable and there exist a finite number of constants $c_j \ge 0$, vectors $g_j \in \text{Space } d$, and integers $p_j \ge -(d - 1)$ such that: $$\|f(x)\| \le \sum_{j=1}^n c_j \|x + g_j\|^{p_j}$$

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. If a function $f : \text{Space } d \to F$ is distribution-bounded, then for any constant $c \in \mathbb{R}$, the scalar multiple $c \cdot f$ is also distribution-bounded. A function is distribution-bounded if it is almost everywhere strongly measurable and there exist constants $c_i \ge 0$, vectors $g_i \in \text{Space } d$, and integers $p_i \ge -(d - 1)$ such that its norm is bounded by a finite sum: $$\|f(x)\| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$$

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. If a function $f : \text{Space } d \to F$ is distribution-bounded, then its negation $x \mapsto -f(x)$ is also distribution-bounded. A function is distribution-bounded if it is almost everywhere strongly measurable and there exist constants $c_i \ge 0$, vectors $g_i \in \text{Space } d$, and integers $p_i \ge -(d - 1)$ such that its norm is bounded by a finite sum: $$\|f(x)\| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$$

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. If a function $f : \text{Space } d \to F$ is distribution-bounded, then for any constant $c \in \mathbb{R}$, the function $x \mapsto c \cdot f(x)$ is also distribution-bounded. A function $f$ is distribution-bounded if it is almost everywhere strongly measurable and there exists a finite sum such that $\|f(x)\| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$, where $c_i \ge 0$, $g_i \in \text{Space } d$, and $p_i \ge -(d - 1)$.

Let $d$ be a natural number. If a function $f: \text{Space } d \to \mathbb{R}$ is distribution-bounded, then for any constant $c \in \mathbb{R}$, the function $x \mapsto c f(x)$ is also distribution-bounded. A function $f$ is distribution-bounded if it is almost everywhere strongly measurable and there exists a finite natural number $n$ such that its norm is bounded by a sum of the form: $$\|f(x)\| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$$ where for each $i \in \{1, \dots, n\}$, the coefficient $c_i$ is a non-negative real number ($c_i \ge 0$), $g_i$ is a vector in $\text{Space } d$, and the exponent $p_i$ is an integer satisfying $p_i \ge -(d - 1)$.

Let $d$ be a natural number and $f: \text{Space } d \to \mathbb{R}$ be a function. If $f$ is distribution-bounded, then for any constant $c \in \mathbb{R}$, the function $x \mapsto f(x) \cdot c$ is also distribution-bounded.

Let $f: \text{Space } d \to \mathbb{R}^n$ be a function from a $d$-dimensional space to an $n$-dimensional Euclidean space. If $f$ is distribution-bounded, then for any index $j \in \{1, \dots, n\}$, the $j$-th component function $x \mapsto f(x)_j$ is also distribution-bounded.

Let $f: \text{Space } d \to \text{Vector}_n$ be a function from a $d$-dimensional Euclidean space to the space of $n$-dimensional Lorentz vectors. If $f$ is distribution-bounded, then for any index $j \in \text{Fin } 1 \oplus \text{Fin } n$, the scalar-valued function $x \mapsto f(x)_j$ (the $j$-th component of $f(x)$) is also distribution-bounded.

Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional space to a normed space $F$. If $f$ is distribution-bounded (satisfies the predicate `IsDistBounded f`), then for any vector $c \in \text{Space } d$, the function $x \mapsto f(x + c)$ is also distribution-bounded.

Let $f: \text{Space } d \to F$ be a function from a $d$-dimensional Euclidean space to a normed space $F$. If $f$ is distribution-bounded (satisfies the predicate `IsDistBounded f`), then for any vector $c \in \text{Space } d$, the shifted function $x \mapsto f(x - c)$ is also distribution-bounded.

Let $f: \text{Space } d \to F$ be a distribution-bounded function and $g: \text{Space } d \to F'$ be an almost everywhere strongly measurable function, where $F$ and $F'$ are normed spaces. If $\|g(x)\| = \|f(x)\|$ for all $x \in \text{Space } d$, then $g$ is distribution-bounded.

Let $d$ be a natural number, and let $f: \text{Space } d \to F$ and $g: \text{Space } d \to F'$ be functions mapping from a $d$-dimensional Euclidean space into normed spaces $F$ and $F'$, respectively. If $f$ is distribution-bounded, $g$ is almost everywhere strongly measurable, and the norm of $g$ is point-wise bounded by the norm of $f$ (i.e., $\|g(x)\| \le \|f(x)\|$ for all $x \in \text{Space } d$), then $g$ is also distribution-bounded.

Let $f: \text{Space } d \to \mathbb{R}^n$ be a distribution-bounded function. This means that $f$ is almost everywhere strongly measurable and its norm is bounded by a finite sum of the form $$\|f(x)\| \le \sum_{i=1}^k c_i \|x + g_i\|^{p_i}$$ where $c_i \ge 0$, $g_i \in \text{Space } d$, and the integers $p_i$ satisfy $p_i \ge -(d - 1)$. For any constant vector $y \in \mathbb{R}^n$, the scalar-valued function mapping $x$ to the real inner product $\langle f(x), y \rangle$ is also distribution-bounded.

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. If $c: \text{Space } d \to \mathbb{R}$ is a distribution-bounded function, then for any constant vector $f \in F$, the function mapping $x \in \text{Space } d$ to the scalar product $c(x) \cdot f$ is also distribution-bounded.

Let $d$ be a natural number and $F$ be a normed space. For any element $f \in F$, the constant function $c: \text{Space } d \to F$ defined by $c(x) = f$ for all $x \in \text{Space } d$ is distribution-bounded (satisfies `IsDistBounded`).

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any integer $n$ such that $n \ge -(d - 1)$, the function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = \|x\|^n$ is distribution-bounded.

For any dimension $d \in \mathbb{N}$, integer exponent $n \in \mathbb{Z}$ such that $n \ge -(d - 1)$, and any vector $g \in \text{Space } d$, the function mapping $x$ to $\|x - g\|^n$ is distribution-bounded (satisfies `IsDistBounded`).

For any dimension $d \ge 2$ and any vector $g \in \text{Space } d$, the function mapping $x \in \text{Space } d$ to the inverse Euclidean norm $\|x - g\|^{-1}$ is distribution-bounded (satisfies the property `IsDistBounded`).

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any natural number $n \in \mathbb{N}$, the function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = \|x\|^n$ is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any natural number $n \in \mathbb{N}$ and any real number $a \in \mathbb{R}$, the function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = (\|x\| + a)^n$ is distribution-bounded (satisfies the predicate `IsDistBounded`). Here, $\|\cdot\|$ denotes the Euclidean norm on $\text{Space } d$.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any integer $n \in \mathbb{Z}$ and any real number $a > 0$, the function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = (\|x\| + a)^n$ is distribution-bounded (satisfies the predicate `IsDistBounded`). Here, $\|\cdot\|$ denotes the Euclidean norm on $\text{Space } d$.

For any dimension $d \in \mathbb{N}$, natural number exponent $n \in \mathbb{N}$, and vector $g \in \text{Space } d$, the function mapping $x$ to $\|x - g\|^n$ is distribution-bounded (satisfies the predicate `IsDistBounded`).

For any dimension $d \in \mathbb{N}$ and any vector $g \in \text{Space } d$, the function mapping $x$ to the Euclidean norm $\|x - g\|$ is distribution-bounded.

For any dimension $d \in \mathbb{N}$ and any vector $g \in \text{Space } d$, the function mapping $x$ to the Euclidean norm $\|x + g\|$ is distribution-bounded (satisfies the predicate `IsDistBounded`).

For any natural number $n$, let $d = n + 2$ be the dimension of the Euclidean space $\text{Space } d$. The function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = \|x\|^{-1}$ is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. The function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = \|x\|$, which maps each vector $x$ to its Euclidean norm, is distribution-bounded.

For any natural number $d$, let $D = d + 2$ be the dimension of the Euclidean space $\text{Space } D$. The function $f: \text{Space } D \to \mathbb{R}$ defined by $f(x) = \log \|x\|$ is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any integer $n$ such that $n \ge -(d-1)-1$, the function $f: \text{Space } d \to \text{Space } d$ defined by $f(x) = \|x\|^n x$ is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space and let $\text{repr}(x)$ denote the coordinate representation of a vector $x$ with respect to the standard orthonormal basis. For any integer $n$ such that $n \ge -(d - 1) - 1$, the function $f(x) = \|x\|^n \cdot \text{repr}(x)$ is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any vector $y \in \text{Space } d$ and any integer $n$ such that $n \ge -(d - 1) - 1$, the function mapping $x \in \text{Space } d$ to $\|x - y\|^n \cdot \text{repr}(x - y)$ is distribution-bounded. Here, $\text{repr}(v)$ denotes the coordinate representation of a vector $v$ with respect to the standard orthonormal basis of $\text{Space } d$.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any natural number $n$ such that $-n \ge -(d - 1) - 1$, the function $f: \text{Space } d \to \text{Space } d$ defined by $f(x) = \|x\|^{-n} x$ is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any natural number $n$ such that $-n \ge -(d - 1) - 1$, the function $f(x) = \|x\|^{-n} \cdot \text{basis.repr}(x)$, where $\text{basis.repr}(x)$ denotes the coordinate representation of the vector $x$ with respect to the standard orthonormal basis, is distribution-bounded.

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any natural number $p \in \mathbb{N}$ and any vector $c \in \text{Space } d$, the function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = \|x\| \cdot \|x + c\|^p$ is distribution-bounded. (A function is distribution-bounded if it is almost everywhere strongly measurable and its norm is bounded by a finite sum of terms of the form $c_i \|x + g_i\|^{p_i}$ where $p_i \ge -(d - 1)$).

Let $\text{Space } d$ be a $d$-dimensional Euclidean space. For any vector $c \in \text{Space } d$ and any integer $p$ such that $p \ge -(d - 1)$, the function $f: \text{Space } d \to \mathbb{R}$ defined by $f(x) = \|x\| \cdot \|x + c\|^p$ is distribution-bounded. (A function is distribution-bounded if it is almost everywhere strongly measurable and its norm is bounded by a finite sum of terms of the form $c_i \|x + g_i\|^{p_i}$ where $p_i \ge -(d - 1)$).

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. If a function $f: \text{Space } d \to F$ is distribution-bounded, then the function mapping $x \in \text{Space } d$ to $\|x\| f(x)$ is also distribution-bounded. A function $f$ is distribution-bounded if it is almost everywhere strongly measurable and there exists a finite natural number $n$ such that its norm is bounded by a sum of the form: $$\|f(x)\| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$$ where for each $i \in \{1, \dots, n\}$, the coefficient $c_i$ is a non-negative real number, $g_i$ is a vector in $\text{Space } d$, and the exponent $p_i$ is an integer satisfying $p_i \ge -(d - 1)$.

Let $d$ be a natural number and $f: \text{Space } d \to \mathbb{R}$ be a real-valued function. If $f$ is distribution-bounded, then the function mapping $x \in \text{Space } d$ to $\|x\| f(x)$ is also distribution-bounded. A function $f$ is distribution-bounded if it is almost everywhere strongly measurable and there exists a finite natural number $n$ such that its absolute value is bounded by a sum of the form: $$|f(x)| \le \sum_{i=1}^n c_i \|x + g_i\|^{p_i}$$ where for each $i \in \{1, \dots, n\}$, the coefficient $c_i$ is a non-negative real number, $g_i$ is a vector in $\text{Space } d$, and the exponent $p_i$ is an integer satisfying $p_i \ge -(d - 1)$.

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. If a function $f: \text{Space } d \to F$ is distribution-bounded, then for any coordinate index $i \in \{0, \dots, d-1\}$, the function mapping $x \in \text{Space } d$ to the scalar product of its $i$-th coordinate $x_i$ and the value $f(x)$, given by $x \mapsto x_i \cdot f(x)$, is also distribution-bounded.

Let $d$ be a natural number and let $f: \text{Space } d \to \mathbb{R}$ be a real-valued function. If $f$ is distribution-bounded, then for any coordinate index $i \in \{0, \dots, d-1\}$, the function mapping $x$ to $x_i f(x)$ is also distribution-bounded, where $x_i$ denotes the $i$-th coordinate of the vector $x$.

Let $d$ be a natural number and $f: \text{Space } d \to \mathbb{R}$ be a real-valued function. If $f$ is distribution-bounded, then the vector-valued function mapping $x \in \text{Space } d$ to the scalar multiplication $f(x)x$ is also distribution-bounded.

Let $d$ be a natural number and $f: \text{Space } d \to \mathbb{R}$ be a distribution-bounded function. Then the function mapping $x \in \text{Space } d$ to the scalar-vector product $f(x) \cdot \text{repr}(x)$ is also distribution-bounded, where $\text{repr}(x)$ denotes the coordinate representation of $x$ with respect to the standard orthonormal basis of $\text{Space } d$.

Let $d$ be a natural number and $F$ be a normed space over $\mathbb{R}$. Suppose $f: \text{Space } d \to F$ is a distribution-bounded function. For any fixed vector $y \in \text{Space } d$, the function $x \mapsto \langle y, x \rangle f(x)$, formed by scaling $f(x)$ by the inner product of $y$ and $x$, is also distribution-bounded. Recall that a function is distribution-bounded if it is almost everywhere strongly measurable and its norm is bounded by a finite sum of terms of the form $c_i \|x + g_i\|^{p_i}$, where $c_i \ge 0$, $g_i \in \text{Space } d$, and $p_i \ge -(d - 1)$.