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We prove a conjecture in fluid dynamics concerning optimal bounds for heat transportation in the infinite Prandtl number limit and for large Rayleigh number Ra,predicted in (Howard in Proceedings of the 11th International Congress of AppliedMathematics on Applied Mechanics, Munich, 1964, p. 1109, Springer, 1966) and(Malkus in Proc. R. Soc. Lond. Ser. A 225:196–212, 1954). Due to a maximum principle property for the temperature exploited by Constantin-Doering and Otto-Seis, thisamounts to showing a-priori bounds for horizontally-periodic solutions of a fourthorder equation in a strip of large width. While there have been recent nearly-optimalresults up to logarithmic divergences in Ra, we prove here sharp bounds employingFourier analysis, integral representations, and a bilinear estimate due to Coifman andMeyer which uses the Carleson measure characterization of BMO functions by Fefferman.