Reference request on weighted $L^2$ spaces

Question

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.

Answer 1

Let me at least address the specific question you had about convergence:

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.