One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like. In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach. A key step is the solution of an inverse problem with the following flavor. Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$. What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?
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