Category Archives: Mathematics

Circle discrepancy for checkerboard measures

This week I am giving a talk at the Department of Mathematics and Systems Analysis of Aalto University where I will discuss results from a recent paper with Mihalis Kolountzakis. I will give a short introduction to different notions of … Continue reading

DMat0101, Notes 8: Notes Littlewood-Paley inequalities and multipliers

In this final set of notes we will study the Littlewood-Paley decomposition and the Littlewood-Paley inequalities. These consist of very basic tools in analysis which allow us to decompose a function, on the frequency side, to pieces that have almost … Continue reading

DMat0101, Notes 7: General Calderón-Zygmund Operators

After having studied the Hilbert transform in detail we now move to the study of general Calderón-Zygmund operators, that is operators given formally as for an appropriate kernel . Let us quickly review what we used in order to show … Continue reading

DMat0101, Notes 6: Introduction to singular integral operators; the Hilbert transform

This week we come to the study of singular integral operators, that is operators of the form defined initially for `nice’ functions . Here we typically want to include the case where has a singularity close to the diagonal which … Continue reading

DMat0101, Notes 5: The Hardy-Littlewood maximal function

1. Averages and maximal operators This week we will be discussing the Hardy-Littlewood maximal function and some closely related maximal type operators. In order to have something concrete let us first of all define the averages of a locally integrable … Continue reading

DMat0101, Notes 4: The Fourier transform of the Schwartz class and tempered distributions

In this section we go back to the space of Schwartz functions and we define the Fourier transform in this set up. This will turn out to be extremely useful and flexible. The reason for this is the fact that … Continue reading

DMat0101, Notes 3: The Fourier transform on L^1

1. Definition and main properties. For , the Fourier transform of is the function Here denotes the inner product of and : Observe that this inner product in is compatible with the Euclidean norm since . It is easy to … Continue reading