Integrable functions that decay slower than every polynomially integrable function

Question

What are some examples of some $L^1(\mathbb R^d)$ functions that decay slower than $1/|x|^{1+}$ on $\mathbb R^d$ (at infinity)? That is, they are strictly larger than $1/|x|^{1+}$ for all sufficiently large $|x|$. (Do they exist?) Here, the exponent $1+$ denotes $1+\delta$ for every small $\delta>0$.

Answer 1

Here are some examples (for $d=1$, but which can be easily adapted). Let $L(x):=\max\{\ln x,1\}$ for positive $x$ and $L_k(x)=L\circ L_{k-1}(x)$, $x>0$, $k\geq 2$, $L_1=L$. Let $$ f_n(x):= \frac 1x\left(\prod_{k=1}^n\frac1{L_k(x)}\right)\frac 1{\left(L_{n+1}(x)\right)^{1+\delta}},x\geqslant 1, \delta>0 $$ and $f_n(x)=0$ for $x\leqslant 1$.