Light PhD Seminar: The Hardy-Littlewood maximal function

Abstract: 
One of the most used operators in Harmonic Analysis is the Hardy-Littlewood maximal operator. The maximal operator is deeply connected to some important problems in classical analysis, such as the differentiation of the integral. The maximal operator and some similar operators play a key role in many aspects of Mathematical Analysis, and in particular in Harmonic Analysis.

In this talk, we will give the definition of the maximal function as given by Hardy and Littlewood in 1930. Their framework was the theory of analytic functions on the disk, a classical topic in Complex Analysis. We will present the maximal function as we know it today and compare the modern operator with the original one.

Finally, we will give an application of the maximal operator in Harmonic Analysis. We will state a domination result using the maximal function. More precisely, we are going to describe a weighted norm inequality between Calder�n-Zygmund operators and the Hardy-Littlewood maximal operator.

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