Just to add a bit more on what Terry and Denis said (it shouldn't be surprising the people who jumped at this question all work in PDEs): a simple example of the power of the Hardy Inequality can be illustrated by the following "qualitative" description of the inequality:

The Hardy Inequality allows you to control low derivative norms by "weighted" high derivative norms subject to boundary conditions.

This is particularly useful in the study of hyperbolic partial differential equations. Consider the linear wave equation:

$$ \Box u = 0 $$

Ignoring for now Fourier analytic methods, by just integration by parts in physical space, you get the "energy estimate"

$$ \frac{d}{ds}E(s) = 0, \quad E(s) = \int_{t = s} (\partial_tu(t,x))^2 + (\nabla u(t,x))^2 dx$$

which provides you with global-in-time control of the $L^2$ norm of one derivative of a solution $u$.

In general, you can get more complicated energy estimates by integrating the equation against different "weighted derivatives" of the solution $u$. This is often called the "ABC-method" of Morawetz or the "vector-field method" (depending on whom you talk to). The method can often give you a conserved, almost conserved, or monotonically decreasing scalar quantity that dominates a weighted $L^2$ norm of some positive number of derivatives of the function $u$. The key here is that we have a somewhat systematic way of constructing these energy estimates for hyperbolic PDEs.

But sometimes you may need to estimate $L^2$ of the function itself, without derivatives, in the course of the argument. The construction of energy described above does not generally extend to work on the case of zero derivatives. This is where Hardy inequalities becomes useful. For hyperbolic equations, if the prescribed initial data has compact support, the "finite speed of propagation" property (often available for these systems) implies that for any future time, the solution will also have compact spatial support: therefore the boundary requirement of Hardy's inequality is satisfied. With this we can convert a weighted energy estimate on derivatives of the function $u$ to a weighted (with different weight) $L^2$ control for the function itself.

To give a limited sample of how this is used in the context of wave equations

• A Hardy-type inequality is used in Rodnianski and Sterbenz's proof of stable blow-up for the co-rotational wave-map in 1+2 dimensions for a similar reason to what I described above: they need to extend a certain energy control available for higher order derivatives to lower order derivatives, in order to control the lower-order terms in the evolution.
• Hardy-type inequalities is also used quite often in the work of Dafermos and Rodnianski studying the decay of linear waves on black hole backgrounds. See, e.g. this recent paper.