On the absolute integrability of radially symmetric functions

Question

Let $\phi:\mathbb R\to\mathbb R$ be an smooth, even function and $\int_\mathbb R|\phi(t)|^p\,\mathrm dt<\infty$, that is, $\phi$ is pth-power integrable in $\mathbb R$ iff $p\geq p_0$ for $p_0\in\mathbb N$. Now let $\Phi(x):=\phi(|x|)$ with $x\in\mathbb R^n$.

My question is: 1.) Which is the smallest $q$, such that $\int_{\mathbb R^n}|\Phi(x)|^q\, \mathrm dx<\infty$?

I am aware of the fact that $(f\in L^p(\mathbb R^n)) \Rightarrow (f\in L^q(\mathbb R^n))$ for $p\leq q$ might not hold if the function support intervals get smaller and smaller with $|x|\to\infty$, but I'm interested in "non-pathologic" smooth functions, e.g. Bessel functions. I just don't know which function space I should use (just regarding analytic functions seems to restrictive for me).

So the second part of my question is 2.) What is the appropriate function space to formulate question 1.)

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I started my calculations as follows: \begin{align} \int_{\mathbb R^n}|\Phi(x)|dx &\,=\, \int_{\mathbb R^n}|\phi(|x|)|\,dx\\ &\,=\,\int_0^\infty\int_{S^{n-1}}|\phi(r)|\,\mathrm dS\,r^{n-1}\,\mathrm dr\\ &\,=\,\omega_{n-1}\int_0^\infty |\phi(r)|\,r^{n-1}\,\mathrm dr, \end{align} where $\omega_{n-1}$ is the surface of the unit sphere $S^{n-1}$ in $\mathbb R^n$. From this, it seems that the answer to my question is dimension-dependent, and that the needed $q$ will increase with the dimension $n$.

Answer 1

You want to make $|\phi(r)|^q r^{n-1}$ integrable, based on the integrability of $|\phi|^p$.

Smoothness, and even analyticity, do not help. An analytic function $\phi$ could still have a sequence of smooth bumps near points $r_n$, where the sequence $r_n$ grows arbitrarily fast. This can make $|\phi(r)|^q r^{n-1}$ non-integrable, regardless of integrability assumptions on $\phi$.

What is the appropriate function space to formulate question 1.)

The weighted Lebesgue space $L^q_w$ with the weight $w=r^{n-1}$. You have $\phi\in L^q_w$ if and only if $\Phi\in L^q(\mathbb R^n)$.

Equivalent formulation: for $k=1,2,\dots$ let $\phi_k$ be the restriction of $\phi$ to the interval $[2^{k-1},2^{k}]$. Then you want $\sum_{k=1}^\infty 2^{k(n-1)}\|\phi_k\|_{L^q}<\infty$ in order to conclude that $\Phi\in L^q(\mathbb R^n)$.