Parabolic Hausdorff measure

The parabolic Hausdorff measure is Hausdorff measure for the parabolic metric. A common equivalent convention covers $E\subset\mathbb R^n\times\mathbb R$ by cylinders of spatial radius $r_j$ and time length $r_j^2$ , and sets

\mathcal P^s(E)=\lim_{\delta\downarrow0}\inf\left\{\sum_jr_j^s: E\subseteq\bigcup_jQ_{r_j},\ 0<r_j<\delta\right\}.

The cylinder and metric-diameter versions are comparable up to constants and have the same null sets. State the convention if numerical normalization matters.

Meaning of a null-set conclusion

The condition $\mathcal P^1(E)=0$ says that arbitrarily fine parabolic covers can have arbitrarily small sum of radii. It does not say that $E$ is empty. This distinction matters in partial-regularity statements for fluid equations.

References

Lenya Ryzhik, Math 256B lecture notes (2024), §9, parabolic Hausdorff measure.

Fluid regularity

The Caffarelli–Kohn–Nirenberg theorem controls the interior singular set of suitable three-dimensional Navier–Stokes solutions using this measure.