according to size, before it is decomposed globally into frequencies. This local-global decomposition is characteristic of many arguments in harmonic analysis.

For example, the Hilbert transform is not bounded in , but it does map into weak-type . A proof takes a function and decomposes it into a -bounded part plus a mean-bounded part over a collection of subintervals, . One then has

which bounds the norm of in terms of the norm of . The Hilbert transform is bounded on , so the good part inherits the same bound. Because

1.Previous
3.Next