The classical Marcel Riesz inequality was announced in 1924 in Comptes Rendus de L’Academie des Sciences. It states that the Hilbert transform H—one of the most important operators in harmonic analysis—continuously maps an Lp-space into itself for p greater than one.
A complete proof appeared only in 1927 in the celebrated paper “Sur les Fonctions Conjugees,” Math Zeitschrift, vol. 27. Riesz proves his inequality first for even integers, then for all p that are not odd integers, and at the end he discusses the exceptional cases.
Attempts to understand this result eventually led to the creation of interpolation theory of function spaces. Riesz remarked in his paper that it would be of interest to see how the best constant C(p) in his inequality depends on p. Challenging problems of this type are known to be very difficult, usually requiring new ideas and the sharpest available tools, along with good luck.
In 1968, Israel Gohberg and Nahum Krupnik gave an exact calculation for C(p) for p a power of 2 and showed that a better estimate cannot hold. They also found a lower estimate for the best constant B(p) for the analytic projection P, another important operator, which maps Lp onto the Hardy space Hp of functions whose Fourier coefficients with negative indices are zeros.
Gohberg and Krupnik also conjectured the value of B(p). Note that the analytic projection, together with its twin brother co-analytic projection, is used in hundreds of papers on harmonic analysis, operator theory, control theory, etc. As a consequence, one gets sharp criteria for the solvability of systems of Wiener-Hopf equations, which are widely used in astronomy, physics and prediction theory.
In 1972, Stylianos Pichorides, in his dissertation at the University of Chicago, proved the C(p) conjecture and also found best constants in other related inequalities. Pichorides received the Salem Prize for this work.
The proof given by Pichorides was based on inequalities with sharp constants for certain subharmonic functions, which further developed an earlier idea of Alberto Calderon. New proofs are now available, including probabilistic ones, which make use of deep methods developed by Donald Burkholder.
The B(p) conjecture for the Riesz projection turned out to be more difficult. It was stated as an open problem by Alexander Pelczynski in 1985 and was mentioned in several other publications. Some partial results in that direction, which strengthened the estimate of Pichorides, were obtained in 1980 by MU’s Igor Verbitsky, and independently in 1984 by Matt Essen.
A breakthrough came in fall 1998, when Verbitsky and his PhD student Brian Hollenbeck discovered, using Mathematica graphing tools, that a certain function of two complex variables had a plurisubharmonic minorant.
The hunt for that function had taken almost a year and many hours of computer-aided analysis. After that, in two weeks an analytical proof resulted, confirming the conjecture made 30 years before. More general results with sharp constants, for half-space Fourier multipliers, spaces with weights, etc., were established as well.