Radial oscillation of harmonic functions in the Korenblum class

We study radial behavior of harmonic functions in the unit disk belonging to the Korenblum class. We prove that functions which admit two-sided Korenblum estimate either oscillate or have slow growth along almost all radii

Lecture notes on quantitative unique continuation for solutions of second order elliptic equations

In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline a proof of a recent result on propagation of smallness. The ideas are also useful in the study of the zero sets of eigenfunctions of Laplace-Beltrami operator and we discuss the connection. Some basic facts about second order elliptic PDEs in divergent form are collected in the Appendix at the end of the notes.Comment: Lecture notes for Graduate Summer School in Park City (PCMI), July 201

Two types of Rubio de Francia operators on Triebel--Lizorkin and Besov spaces

We discuss generalizations of Rubio de Francia's inequality for Triebel--Lizorkin and Besov spaces, continuing the research from [5]. Two versions of Rubio de Francia's operator are discussed: it is shown that a rotation factor is needed for the boundedness of the operator in some smooth spaces while it is not essential in other spaces. We study the operators on some "end" spaces of the Triebel--Lizorkin scale and then use usual interpolation methods.Comment: 12 pages, 3 figure

Radial growth of harmonic functions in the unit ball

We study harmonic functions which admit a certain majorant in the unit ball in $\R^m$. We prove that when the majorant fulfills a doubling condition, the extremal growth or decay may occur only along small sets of radii, and we give precise estimates of these exceptional sets