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In this final set of notes we will study the LittlewoodPaley decomposition and the LittlewoodPaley 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
After having studied the Hilbert transform in detail we now move to the study of general CalderónZygmund operators, that is operators given formally as for an appropriate kernel . Let us quickly review what we used in order to show … Continue reading
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
1. Averages and maximal operators This week we will be discussing the HardyLittlewood 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
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
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
1. Convolutions and approximations to the identity We restrict our attention to the Euclidean case . As we have seen the space is a vector space; linear combinations of functions in remain in the space. There is however a `product’ … Continue reading