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IV.12. Partial Differential Equations 455 tance of the underlying geometry (in this case, the combinatorics of metric balls) in harmonic analysis. Further Reading Stein, E. M. 1970. Singular Integrals and Differentiability Properties of Functions. Princeton, NJ: Princeton Univer- sity Press. . 1993. Harmonic Analysis. Princeton, NJ: Princeton University Press. Wolff, T. H. 2003. Lectures on Harmonic Analysis, edited by I. Laba and C. Shubin. University Lecture Series, vol- ume 29. Providence, RI: American Mathematical Society. Many arguments in harmonic analysis will, at some point, involve a combinatorial statement about certain types of geometric objects such as cubes, balls, or boxes. For instance, one useful such statement is the Vitali covering lemma, which asserts that, given any col- lection B 1 , . . . , B k of balls in Euclidean space R n , there will be a subcollection B i 1 , . . . , B i m of balls that are dis- joint, but that nevertheless contain a significant frac- tion of the volume covered by the original balls. To be precise, one can choose the disjoint balls so that m vol j=1 B i j 5 -n vol k B j . j=1 (The constant 5 -n can be improved, but this will not concern us here.) This result is obtained by a "greedy algorithm": one picks balls one by one, at each stage choosing the largest ball among the B j that is disjoint IV.12 Partial Differential Equations Sergiu Klainerman Introduction