Professor Cruz-Uribe’s research is in harmonic analysis. He is particularly interested in the study of the classical operators of harmonic analysis—maximal operators, the Hilbert transform and other singular integrals, Riesz potentials/fractional integrals—on weighted and variable exponent Lebesgue spaces. These kinds of spaces occur naturally in situations where there is a lack of homogeneity in the underlying problem being studied. He also works on the theory of Rubio de Francia extrapolation, which allows weighted classical norm inequalities to be extended to a variety of function space settings.
Finally, he is interested in the applications of harmonic analysis to the study of partial differential equations, particularly degenerate elliptic equations.
Variable Lebesgue Spaces and Hyperbolic Systems, Advanced Courses in Mathematics, CRM Barcelona, Birkhauser, Basel, 2014. (Joint with A. Fiorenza, M. Ruzhansky, J. Wirth.)
Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhauser, Applied and Numerical Harmonic Analysis, 2013. (Joint with A. Fiorenza.)
Weights, Extrapolation and the Theory of Rubio de Francia, Operator Theory: Advances and Applications, Birkhauser, Basel, 2011. (Joint with J.M. Martell and C. Pérez.)
Fourier Analysis, translation and revision of the Spanish edition by Javier Duoandikoetxea, Amer. Math. Soc., Providence, December, 2000.