I've been learning about ill-posed/inverse problems, and I'm having a hard time parsing/understanding the following, which seems crucial to the theory:
Say we have an operator $A:L^2(\Omega)\rightarrow Y$ such that for some $\alpha,m,M>0$ and independent of $f$ we have the estimate
$$ m\|f\|_{H_0^{-\alpha}(\Omega)}\leq \|Af\|_{Y}\leq M\|f\|_{H_0^{-\alpha}(\Omega)} $$ Here $H_0^{-\alpha}(\Omega)$ are dual Sobolev spaces. Then, I frequently see claims of the following type:
- "$A^{-1}$ exists but is unbounded as an operator from $Y$ to $L^2(\Omega)$."
- "$\alpha$ measures the degree of ill-posedness of solving the problem $Af=g$"
- "$Af$ is $\alpha$-smoother than $f$"
I roughly understand the second and third statements and how they relate to each other in terms of Fourier transforms (though I welcome second opinions). Is there a simple functional analysis argument for the first statement, showing that such an estimate guarantees that $A^{-1}$ exists but is unbounded?
Assume that $A^{-1}: Y\to L^2(\Omega)$ is bounded, i.e. $$\|A^{-1} f\|_2\leq c\|f\|_Y,\ \forall\ f\in Y\tag{1}$$
where $c>0$ is a constant. We get from $(1)$ and from your inequality that $$\|A^{-1} f\|_2\leq cM\|A^{-1}f\|_{H^{-\alpha}},\ \forall f\in Y$$
or equivalently $$\|g\|_2\leq cM\|g\|_{H^{-\alpha}},\ \forall \ g\in L^2(\Omega)$$
Do you know what is the problem with the last inequality?