# Reference request on weighted $L^2$ spaces

I'm looking for reference (preferably a textbook) that covers weighted $L^2$ spaces. I've done some googling and have yet come up with anything.

In particular I want to see if, given a sequence $\{\omega_n\}_{n=1}^{\infty}$ of weights on a domain $D\subset\mathbb{R}^n$ (with Lebesgue measure) that converges to a weight $\omega$ on $D$ in, say the $L^2$-norm (or maybe in measure), we will have "convergence" of the corresponding $L^2$-spaces in some reasonable sense. (For example, if given a a $f\in L^2(D,\omega)$ is it the case that there exists a $N\in\mathbb{N}$ such that $f\in L^2(D,\omega_n)$ for all $n\geq N$?) This seems like a reasonable question to ask, so I assume mathematicians in the past have attempted to produce results related to this, but since my searching has come up with nothing I fear that there is not much to say here.

• I have found the papers by Gurka-Opic to be very useful on weighted $L^p$ and Sobolev spaces. However, I can't remember if they address your specific problems well enough. I used it mostly to prove imbedding results. Aug 3, 2017 at 14:11
• @DominikS Thank you! I'll take a look. At the very last they may give me some insight. Aug 3, 2017 at 14:14
• "Muckenhoupt weight" is a keyword you could try to look for. Those things are of interest to the "function spaces community". Opic is definitely one of the guys in that community. Aug 3, 2017 at 14:33

If $f\in L^2(\omega)$, when is $f\in L^2(\nu)$? (I have replaced $\omega_n$ by any weight function $\nu$, the index does not play a role in the following.)

We find

$$\|f\|_\nu^2 = \int f^2(x)\nu(x) dx = \int f^2(x)\omega(x)\frac{\nu(x)}{\omega(x)}dx \le \|f\|_\omega^2\cdot \left\|\frac{\nu}{\omega}\right\|_{L^\infty},$$ where we obtain the estimate by using Hölder's inequality with $p=1$ and $q = \infty$. This gives you some information on how $\omega$ and $\nu$ have to behave relatively to each other.

The above calculation already motivates which aspects of a weight function $\omega$ might be relevant in the definition the corresponding weighted $L^2$-space:

• Poles/unboundedness of $\omega$
• Zeros of $\omega$
• Asymptotic behavior of $\omega(x)$ as "$x\to \partial D$", i.e. when approaching the boundary of your domain.

When comparing weighted $L^2$-spaces, these are the aspects you should pay attention to.