Completeness of weighted $L^p$ spaces

Question

For $\infty >p>1$ consider the weighted $L^p$ spaces $(L^p(\mathbb{R}^n),\omega dx)$ where $\omega$ is some nonnegative weight. Is it true that $(L^p(\mathbb{R}^n),\omega dx)$ is complete iff $\omega$ is a Muckenhoupt weight? I would also be glad about some nice reference on this subject

Answer 1

When you write $(L^p(\mathbb{R}^n),\omega\,dx)$ I would guess you mean the space of all measurable functions $f$ such that $|f|^p\omega\in L^1(\mathbb{R}^n)$? Probably more common notations are $L^p(\mathbb{R}^n,\omega)$ or $L^p_\omega(\mathbb{R}^n)$, with the norm given by $$\|f\|_{p,\omega} = \|f\omega^{1/p}\|_{L^p}=\left(\int|f(x)|^p\omega(x)\,dx\right)^{1/p}$$

Here are some hints:

• What does completeness of $L^p_\omega(\mathbb{R}^n)$ mean in terms of Cauchy sequences?
• Can you rephrase that into a problem in $L^p(\mathbb{R^n})$?

Now, what do you know about Muckenhoupt weights?