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I’ve just uploaded the short story “Uchiyama’s constructive proof of the Fefferman-Stein decomposition“. In 1982, Uchiyama gave a new proof of the celebrated Fefferman-Stein theorem that expressed any BMO function as the sum of a bounded function, and Riesz transforms of bounded functions. Unlike the original proof (which relied, among other things, on the Hahn-Banach theorem), Uchiyama’s proof was very explicit, constructing the decomposition by building the bounded functions one Littlewood-Paley frequency band at a time while keeping the functions taking values on or near a sphere, and then iterating away the error. Here I have written some notes on how the proof goes. The notes are a little condensed, in that a number of standard computations involving estimations of Schwartz tails, Carleson measures, etc. have been omitted, but hopefully the gist of the argument is still clear.