Physlib.SpaceAndTime.Space.Integrals.NormPow

Integrability of norm powers on subsets of Space

i. Overview

This module contains results on the integrability of `x ↦ ‖x‖ᵖ` for real exponents `p` on subsets of `Space d`.

The integrability of `x ↦ ‖x‖ᵖ` on `ball 0 b` and `(ball 0 b)ᶜ` follows from the 1d results `integrableOn_Ioo_rpow_iff` and `integrableOn_Ioi_rpow_iff` after changing to spherical coordinates.

ii. Key results

- `integrableOn_norm_rpow_iff_of_isBounded_nhds`: Necessary and sufficient condition for integrability on a set `s` which is a bounded neighborhood of the origin. - `integrableOn_norm_rpow_of_isBounded_compl_nhds`: Integrability for all bounded sets `s` with `0 ∉ closure s`. - `integrableOn_norm_rpow_of_compl_nhds`: A sufficient condition for integrability on `s` with `0 ∉ closure s`.

iii. Table of contents

iv. References

6 declarations

theorem

$\|x\|^p$ is integrable on $B(0, b)$ iff $d + p > 0$

#integrableOn_norm_rpow_ball_iff

Let $\text{Space } d$ be a $d$ -dimensional real inner product space equipped with the Euclidean norm $\|\cdot\|$ . For any positive integer $d$ , positive radius $b \in \mathbb{R}$ , and exponent $p \in \mathbb{R}$ , the function $x \mapsto \|x\|^p$ is integrable on the open ball $B(0, b) = \{x \in \text{Space } d : \|x\| < b\}$ if and only if $d + p > 0$ .

theorem

$\|x\|^p$ is integrable on $(\text{ball } 0 a)^c \iff d + p < 0$

#integrableOn_norm_rpow_ball_compl_iff

For a positive natural number $d$ and the $d$ -dimensional real inner product space $\text{Space } d$ , let $a$ be a positive real number and $p$ be a real exponent. The function $f(x) = \|x\|^p$ is integrable on the complement of the ball centered at the origin with radius $a$ , $\{x \in \text{Space } d \mid \|x\| \geq a\}$ , if and only if $d + p < 0$ .

theorem

$\|x\|^p$ is integrable on the shell $\{x \mid a \le \|x\| < b\}$

#integrableOn_norm_rpow_shell

Let $\text{Space } d$ be a $d$ -dimensional real inner product space. For any real number $a > 0$ and any real numbers $b$ and $p$ , the function $x \mapsto \|x\|^p$ is integrable on the shell defined by $\{x \in \text{Space } d \mid a \le \|x\| < b\}$ .

theorem

$\|x\|^p$ is integrable on a bounded neighborhood of $0 \iff d + p > 0$

#integrableOn_norm_rpow_iff_of_isBounded_nhds

Let $\text{Space } d$ be a $d$ -dimensional real inner product space with dimension $d > 0$ . Let $s \subseteq \text{Space } d$ be a bounded set that is also a neighborhood of the origin. For any real exponent $p$ , the function $f(x) = \|x\|^p$ is integrable on $s$ if and only if $d + p > 0$ , where $\|\cdot\|$ denotes the Euclidean norm on $\text{Space } d$ .

theorem

$\|x\|^p$ is integrable on bounded sets away from the origin

#integrableOn_norm_rpow_of_isBounded_compl_nhds

Let $\text{Space } d$ be a $d$ -dimensional real inner product space. For any bounded subset $s \subseteq \text{Space } d$ such that the origin is in the exterior of $s$ (i.e., the complement $s^c$ is a neighborhood of $0$ ), and for any real exponent $p$ , the function $x \mapsto \|x\|^p$ is integrable on $s$ .

theorem

$d + p < 0$ implies integrability of $\|x\|^p$ on $s$ when $s^c \in \text{nhds } 0$

#integrableOn_norm_rpow_of_compl_nhds

Let $d$ be a positive natural number and $\text{Space}(d)$ be a $d$ -dimensional real inner product space. Let $s \subseteq \text{Space}(d)$ be a set such that its complement $s^c$ is a neighborhood of the origin (implying the origin is in the exterior of $s$ ). For any real number $p$ , if $d + p < 0$ , then the function $x \mapsto \|x\|^p$ is integrable on $s$ .

Physlib.SpaceAndTime.Space.Integrals.NormPow

Integrability of norm powers on subsets of Space

i. Overview

ii. Key results

iii. Table of contents

iv. References

∥x∥p\|x\|^p∥x∥p is integrable on B(0,b)B(0, b)B(0,b) iff d+p>0d + p > 0d+p>0

∥x∥p\|x\|^p∥x∥p is integrable on (ball 0a)c ⟺ d+p<0(\text{ball } 0 a)^c \iff d + p < 0(ball 0a)c⟺d+p<0

∥x∥p\|x\|^p∥x∥p is integrable on the shell {x∣a≤∥x∥<b}\{x \mid a \le \|x\| < b\}{x∣a≤∥x∥<b}

∥x∥p\|x\|^p∥x∥p is integrable on a bounded neighborhood of 0 ⟺ d+p>00 \iff d + p > 00⟺d+p>0

∥x∥p\|x\|^p∥x∥p is integrable on bounded sets away from the origin

d+p<0d + p < 0d+p<0 implies integrability of ∥x∥p\|x\|^p∥x∥p on sss when sc∈nhds 0s^c \in \text{nhds } 0sc∈nhds 0

$\|x\|^p$ is integrable on $B(0, b)$ iff $d + p > 0$

$\|x\|^p$ is integrable on $(\text{ball } 0 a)^c \iff d + p < 0$

$\|x\|^p$ is integrable on the shell $\{x \mid a \le \|x\| < b\}$

$\|x\|^p$ is integrable on a bounded neighborhood of $0 \iff d + p > 0$

$\|x\|^p$ is integrable on bounded sets away from the origin

$d + p < 0$ implies integrability of $\|x\|^p$ on $s$ when $s^c \in \text{nhds } 0$