Category Archives: Dmat0101 – Harmonic Analysis

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

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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

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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

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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

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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

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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

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DMat0101, Notes 2: Convolution, Dense subspaces and interpolation of operators

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

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