For which $p \in [1,\infty]$ does $g, g_{A}$ and $g_{A^{c}} \in \mathcal{L}^p$ hold?

Question

Let $s \in \mathbb R, d \in \mathbb N$, $g:\mathbb R^{d}\to \bar{\mathbb R}, x \mapsto |x|^s$ .

Set $E_{d}:=\{x\in \mathbb R^{d}:|x|\leq 1\}$

Determine $p \in [1,\infty]$ for which $g, g\chi_{E_{d}}$ and $g\chi_{E_{d}^{c}} \in \mathcal{L}^{p}$

Idea: Assume $g \geq 0$ without loss of generality otherwise: $g:=g_{+}-g_{-}$

for $p =1$

$\int_{\mathbb R^{d}}gd\lambda^{d}(x)=\int_{\mathbb R^{d}}x^sd\lambda^{d}(x)=\int_{\mathbb R}...(\int_{\mathbb R}x^sd\lambda)...d\lambda=\int_{\mathbb R}...\frac{x^{s+1}}{s+1}\vert_{-\infty}^{\infty}...d\lambda$ so $\notin \mathcal{L}^{1} $

And if $g\notin \mathcal{L}^{1}$ I want to say $g \notin \mathcal{L}^{p},\forall p\in]1,\infty]$ but this does not seem possible given that fact that $\lambda^{d}(\mathbb R^{d})=\infty$, so this shortcut is not possible.

Next $\int_{E^{d}}gd\lambda^{d}(x)=\int_{\mathbb R}...(\int_{\mathbb R}g\chi_{[-1,1]}d\lambda)...d\lambda=\int_{\mathbb R}...(\int_{\mathbb R}x^{s}\chi_{[-1,1]}d\lambda)...d\lambda=\int_{\mathbb R}...(\int_{\mathbb R}\frac{x^{s+1}}{s+1}|^{1}_{-1}d\lambda)...d\lambda$

and $\frac{x^{s+1}}{s+1}|^{1}_{-1}=\frac{1}{s+1}-\frac{(-1)^{s+1}}{s+1}$ and if $s $ us an odd number then we get $0$, and therefore $\int_{E^{d}}gd\lambda^{d}(x)=0$, I for all other cases of $s \in \mathbb R$, we get $g\chi_{E_{d}}\in \mathcal{L}^1$ but I am unsure on how to reason it precisely.

On $\int_{(E^{d})^{c}}gd\lambda^{d}(x)$, I am lost and for all cases $p \in ]1,\infty]$ I am unsure on what to do for all three function, particulary given:

$(\int_{\mathbb R^{d}}x^{sp}d\lambda^{d}(x))^{\frac{1}{p}}$, where $p \in ]1,\infty]$

Any guidance on this problem is greatly appreciated.

Answer 1

A problem in your approach is that you wrote $x^s$ for an element of $\mathbb R^d$, while we want to work to the norm of $x$ to the power $s$.

Here are some steps to solve the problem:

1. First treat completely the one dimensional case, which consists in studying the convergence of $\int_0^1 x^{ps}\mathrm dx$ and $\int_1^{+\infty} x^{ps}\mathrm dx$.
2. The function $x\mapsto \left\lvert x\right\rvert^{ps}$ is radial, in sense that it depends only on the norm of $x$ hence (by a change in polar coordinates for example) $$\int_{E_d}\left\lvert x\right\rvert^{ps}=c_d \int_{0}^1t^{ps}t^{n-1}\mathrm dt;$$ $$\int_{E_d^c}\left\lvert x\right\rvert^{ps}=c_d \int_{1}^{+\infty}t^{ps}t^{n-1}\mathrm dt.$$