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    • Analysis on Trees with Nondoubling Flow Measures
    Authors: Matteo, Levi; Federico, Santagati; Anita, Tabacco;  Advisor: -;  Co-Author: - (2021)

    We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderón–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings.
    • Improved Sobolev regularity for linear nonlocal equations with VMO coefficients
    Authors: Simon, Nowak;  Advisor: -;  Co-Author: - (2022)

    This work is concerned with both higher integrability and differentiability for linear nonlocal equations with possibly very irregular coefficients of VMO-type or even coefficients that are merely small in BMO. In particular, such coefficients might be discontinuous. While for corresponding local elliptic equations with VMO coefficients such a gain of Sobolev regularity along the differentiability scale is unattainable, it was already observed in previous works that gaining differentiability in our nonlocal setting is possible under less restrictive assumptions than in the local setting