Two-weight inequalities and local Tb theorems: quadratic testing and big piece methods
Emil Vuorinen · Työväentutkimus Vuosikirja · 2017
This dissertation belongs to the field of harmonic analysis, which is a subfield of mathematical analysis. It consists of an introductory part and four research articles. Two of the articles study testing characterizations of two-weight inequalities and the other two study local Tb theorems. An operator, such as a singular integral or a dyadic integral operator, satisfies a two-weight inequality if it is bounded between two weighted function spaces Testing characterizations show that to have the two-weight inequality it suffices to test the operator with a very restricted class of functions. Local Tb theorems provide criteria for boundedness of a given operator, usually of a singular integral or a square function. The basic idea is similar to testing of two-weight inequalities: to have boundedness it is enough to test the operator with suitable test functions. Nazarov, Treil and Volberg characterized two-weight inequalities of well localized operators between two L^2-spaces in terms of so-called Sawyer type testing conditions. The first article in the thesis introduces a quadratic testing condition and shows that it can be used in place of the Sawyer type testing condition to obtain an L^p-version of the mentioned L^2-result. The second article introduces a quadratic version of the Muckenhoupt type two-weight A_p condition on the weights. It is shown that if the weights satisfy the quadratic A_p condition, and a dyadic shift operator satisfies the quadratic testing conditions in L^p, then a certain quantitative two-weight estimate holds. This is an L^p-version of the L^2-result by Hytönen, Pérez, Treil and Volberg. The third article introduces a new method to prove local Tb theorems. It relies on ``big piece thinking'', which appears in connection with other problems in mathematical analysis, and on the non-homogeneous good lambda method of Tolsa. By using the new method, the article improves known results concerning local Tb theorems for vertical square functions in the upper half space. The final article studies conical square functions which are defined with respect to a general closed subset of the Euclidean space. The method from the third article is applied to prove a corresponding local Tb theorem in this set-up.