$\|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $ holds?

Question

I want to find the relation of $p$ and $k$ such that the inequality $$ \|f(x)\|_{L^p(\mathbb R)} \le C_{>0} \|(1+x^2)^{k/2} f(x)\|_{L^\infty(\mathbb R)} $$ holds when r.h.s $<\infty$.

Here $f$ is a real-valued function of $x\in\mathbb R$. It seems like $1\le p\le (\text{a function of }k)$.

Answer 1

Given $p$ and $k$, due to the Holder's inequality you can take $C$ as: $$ C=\left\|\frac{1}{(1+x^2)^{kp/2}}\right\|_q^{1/p},$$ or: $$ C=\left(\frac{\sqrt{\pi}\,\Gamma(\frac{kpq-1}{2})}{\Gamma(\frac{kpq}{2})}\right)^{\frac{1}{pq}}.$$

Answer 2

The answer is $kp>1$. This yields the inequality with $$ C=\left(\int_\mathbb{R}\frac{dx}{(1+x^2)^{pk/2}}\right)^{1/p} $$