Asymptotics for the tail of $L_p$ norms

Question

Let $1<p<\infty$ be fixed and $f\in L^p(\mathbb{R}^d)$. I would like to obtain some asymptotic estimate for the function \begin{equation}\tag{1} F(R)=\lVert f\chi_{\{\lvert x \rvert >R\}}\rVert_{L^p}=\left(\int_{\lvert x\rvert>R}\lvert f(x)\rvert^p\, dx\right)^{\frac{1}{p}}\qquad \text{ as }R\to +\infty, \end{equation} under suitable further assumptions on $f$. Here's a conjecture based on the heuristic that a lower summability index corresponds to a faster decay at infinity:

Question. Assume that $f\in L^p\cap L^1(\mathbb{R}^d)$. Is it true that $$\tag{2}F(R)\le C_f \cdot R^{-\frac{1}{p'}}?$$ If (2) does not hold, can we give some other bound on $F$?

Here $C_f>0$ is a constant that depends on $f$ (probably through one or more of its norms).

Answer 1

I don't think you can get $C_f$ to depend only on the norm of $f$, because $f(\cdot - y)$ has the same $L^1$ and $L^p$ as $f$.

I think you can use this idea to find a counterexample to your conjecture: let $g$ be a function whose support is in the unit ball, and whose $L^1$ and $L^p$ norms are non-zero. Let $a_n > 0$ be a summable sequence. Pick $y_n \in \mathbb R^d$ to be a sequence with $|y_n - y_m| > 2$. Then by adjusting $a_n$ and $y_n$, you should be able to find an $f(x) = \sum_n a_n g(x - y_n)$ such that $$ \sup_{R>0} {\|f \chi_{|x|>R}\|}_p R^{1/p'} = \infty .$$

Answer 2

Claim. The conjecture is false. Indeed, proceeding as in Stephen Montgomery-Smith's answer, one may see that the problem is reduced to the asymptotic estimate of the tails $$S_N^{(p)}=\sum_{n=N}^\infty a_n^p, $$ where $a_n\ge 0$ is a summable sequence (i.e. $\sum_n a_n < \infty$). If the conjecture were true, then $S^{(p)}_N$ should decay as $N^{1-p}$ (or faster). However, we can find examples of sequences $a_n$ such that $S_N^{(p)}$ converges only logarithmically.

Namely, if we take a $\alpha>1$ and we take $$ a_n=\begin{cases} k^{-\alpha}, & n=2^k \\ 0, & \mathrm{otherwise}\end{cases}, $$ we obtain $$ S_N^{(p)}=\sum_{k\ge \left\lfloor \frac{\log N}{\log 2}\right\rfloor}k^{-\alpha p} \ge c \left(\left\lfloor \frac{\log N}{\log 2}\right\rfloor\right)^{1-\alpha p}, $$ for some constant $c>0$. This is enough to disprove the conjecture. $\square$

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EDIT (in response to Stephen Montgomery-Smith's comment below). Here's a more general construction, proving that no asymptotic estimate for $F(R)$ can exist.

Let $\phi\colon (0, \infty)\to (0, \infty)$ be a monotonically increasing function such that $\phi(R)\to \infty$ as $R\to \infty$. We claim that a function $f\in L^1\cap L^p(\mathbb{R}^d)$ exists such that $$\sup_{R>0} \lVert f\chi_{\{\lvert x \rvert > R\}}\rVert_{L^p}\phi(R)=\infty.$$ To build $f$ we first take a nonvanishing bump function $g\ge 0$ supported in the unit ball. Then we set $$ \tilde{\phi}(R)=\phi(R)^{\frac{p}{2}\frac{1}{2p-1}},$$ observing that $\tilde{\phi}(R)$ increases monotonically towards $\infty$, and we set \begin{equation} y_n=[\tilde{\phi}]^{-1}(n) \mathbf{e}_1. \end{equation} Here $\cdot^{-1}$ refers to functional inversion and $\mathbf{e}_1=(1, 0,\ldots, 0)\in \mathbb{R}^d$. Then we set $a_n=n^{-2}$ and we define $f$ to be \begin{equation} f(x)=\sum_{n=1}^\infty a_n g(x-y_n). \end{equation} Since $\lVert f\rVert_{L^1} = C_1\sum_{n} a_n$ and $\lVert f\rVert_{L^p}^p=C_p \sum_n a_n^p$, the function $f$ is in $L^1\cap L^p(\mathbb{R}^d)$. We have $$ \begin{split} \lVert f\chi_{\{\lvert x \rvert > R\}}\rVert_{L^p}^p &\ge C \sum_{n> \tilde{\phi}(R)} n^{-2p} \\ &\ge C \tilde{\phi}(R)^{1-2p}= C\phi(R)^{-\frac{p}{2}}. \end{split}$$ Therefore $$\sup_{R>0} \lVert f\chi_{\{\lvert x \rvert > R\}}\rVert_{L^p}^p\phi(R)^p=\sup_{R>0}\phi^{\frac{p}{2}}(R)=\infty.$$