Preface for the special issue in honor of Robert Fefferman

This special issue of the de Gruyter journal Advanced Nonlinear Studies is dedicated to Professor Robert Fefferman from the University of Chicago to celebrate his many remarkable achievements and his distinguished career. Robert Fefferman is an outstanding mathematician widely known for his many important contributions to harmonic analysis and partial differential equations. He is regarded as one of the founding fathers of the research areas of multiparameter harmonic analysis and has made many pioneering contributions in this direction.

Robert Fefferman has worked to extend classical harmonic analysis on Euclidean spaces to the multiparameter setting. In particular, he explored the extension of the theory of maximal functions and the differentiation of the integral, the theory of Calderon–Zygmund singular integrals, of Hardy spaces, BMO and weighted inequalities to the multiparameter setting. For example, differentiation of the integral via rectangles parallel to the coordinate axes needn’t hold almost everywhere for functions in L 1 ( R 2 ) . Rather, according to a ground-breaking result of Jessen, Marcinkiewicz and Zygmund, this fundamental property of differentiation applies to functions in LlogL locally in the plane, and they obtained the sharp result for R n , n > 1. They proved sharp weak type results for the corresponding maximal function, the “strong maximal function”, by iterating sharp results for the one dimensional Hardy–Littlewood maximal function. Together with A. Cordoba, R. Fefferman proved the sharp covering lemma for the corresponding rectangles in R n which provided an alternate proof of the results of Jessen, Marcinkiewicz and Zygmund. This approach had the advantages of providing a new path to obtaining estimates for maximal functions corresponding to other collections of multiparameter families of sets for which no results were known. A basic example was provided by J. Stromberg, who proved bounds for maximal functions corresponding to averages over rectangles in the plane oriented in a lacunary sequence of directions. This also leads to results on partial sum operators for multiple Fourier series of L ^p functions on the torus carried out by Cordoba and R. Fefferman. In addition to the covering lemma leading to these results, an understanding of the geometry of rectangles and their covering properties has played an important role in the theory of multiparameter singular integrals and multipliers. In order to obtain this, R. Fefferman worked with S.Y. Alice Chang in order to create a theory of multiparameter BMO which would be dual to the corresponding Hardy space H ¹. An obstacle to a simple extension of the fundamental results of Charles Fefferman and E.M. Stein, R. Coifman and R. Latter on H ^p spaces and the atomic decomposition in the one parameter setting was illustrated by a very important result of L. Carleson who showed that “rectangle atoms” i.e. the simplest extension to the multiparameter setting of the one parameter ones did not span Hardy space so that they could not be used in the most obvious way to give simple proofs of properties of singular integrals in this new setting. J.L. Journé then proved that singular integrals in the product setting were bounded from L ^∞ to BMO using a covering type argument for rectangles, and R. Fefferman extended this to show that in many settings while Carleson’s counterexample applied, one could still simply check the action of the operator on a rectangle atom, as though the atoms did span in order to prove boundedness on H ^p . This is a remarkable and surprising progress in studying the boundedness of multiparameter singular integrals on multiparameter Hardy spaces in view of L. Carleson’s counterexample. This boundedness principle has been widely regarded as “the R. Fefferman boundedness criterion” by mathematicians working in harmonic analysis. R. Fefferman also extended the theory of weighted norm inequalities to the product setting as well. What all of this illustrated was that the multiparameter theory was not as simple as invoking basic iteration arguments from the classical cases, and rather involved quite a number of surprises.

In addition to the work on multiparameter harmonic analysis, R. Fefferman worked on questions related to the L ^p Dirichlet problem for elliptic operators whose coefficients are non-smooth. The main question he worked on involved what happens to the solvability of the L ^p Dirichlet problem when one perturbs a given elliptic operator whose L ^p Dirichlet problem is known to be solvable. More precisely, suppose one starts with an elliptic operator with coefficients A(x), with a solvable L ^p Dirichlet problem and looks at a perturbation with coefficients B(x) (say A and B are defined in the unit ball of R ⁿ ). Then, in a remarkable piece of work, B. Dahlberg showed that if the difference A(x) − B(x) satisfies a Carleson measure condition with vanishing trace, then one can conclude that perturbed operator also has its L ^p Dirichlet problem solvable for the same value of p. The question that remained to be answered was: What happens if the Carleson measure condition on the difference of the coefficients does not have vanishing trace? Together with C. Kenig and J. Pipher, R. Fefferman proved that the L ^p Dirichlet problem associated with the perturbation is still solvable for some range of p, but not necessarily the original value of p of the operator one started with, and this is sharp. The proof of this involved applications of harmonic analysis and in particular the theory of weights that arise in the theory of singular integrals and maximal functions, and in the process Kenig, Pipher and R. Fefferman obtained a new characterization of A _∞ weights. This was yet another example of the extensive interaction of harmonic analysis with the theory of partial differential equations.

Thus, R. Fefferman repeatedly produced deep, important contributions to harmonic analysis, bringing to light highly original ideas.

Robert Fefferman was born in 1951 in Washington, D.C. and attended the University of Maryland for his college education. Following graduation from the University of Maryland, from 1972 to 1975 he attended graduate school at Princeton University, working under the guidance of his advisor, Elias Stein. He then became an L.E. Dickson Instructor for one year at the University of Chicago, and then joined the regular faculty of the University’s Math Department where he has worked since then. During his career at U Chicago, R. Fefferman was awarded a Sloan Fellowship and an excellence in teaching award, and was invited to give several named lectures at various mathematics departments. He also served as an administrator: R. Fefferman was the Chair of the Department and later served as the Dean of Physical Sciences at the University of Chicago. He was extraordinarily effective in both roles. He has performed service by participating in a number of visiting committees at various universities and has served on several AMS committees over the years. R. Fefferman is currently the holder of a chaired professorship, and is a Fellow of the AMS as well as of the American Academy of Arts and Sciences.

In this special issue, we invited articles from many well-established mathematicians. Many of them are leading researchers in the areas of harmonic analysis and partial differential equations, including some of the world’s most prominent mathematicians. The wide range of topics covered in this special issue are also closely related to and reflect the broad scope of Bob’s research interests. We are grateful to all of the authors and reviewers for making this special issue a success.