One of my recent main interests has been the characterization of boundedness of (integral) operators between two $L^p$ spaces equipped with two different measures. Some recent developments have indicated a need of "Banach spaces and their applications" also in this area of Classical Analysis. For instance, while the theory of two-weight $L^2$ inequalities is already rich enough to deal with a number of singular operators (like the Hilbert transform), the $L^p$ theory has been essentially restricted to positive operators so far. In fact, a counterexample of $F$. Nazarov shows that the common "Sawyer testing" or "David-Journé $T(1)$" type characterization will fail, in general, in the two-weight $L^p$ world. What comes to rescue is what we so often need to save the $L^2$ results in an Lp setting: $R$-boundedness in place of boundedness! Even in the case of positive operators, it turns out that a version of "sequential boundedness" is useful to describe the boundedness of operators from $L^p$ to $L^q$ when $q < p$. - This is about my recent joint work with T. Hänninen and K. Li, as well as the work of my student E. Vuorinen.

two-weight inequalities - boundedness - singular operators[-]