Hardy-Littlewood Maximal Function

In mathematics, the Hardy-Littlewood maximal operator is a significant nonlinear operator used in real analysis and harmonic analysis. It takes a locally integrable function defined on real d-dimensional space and returns a new function whose value at a point x is the supremum, over every ball centered at x, of the average value of the original function on that ball. The resulting maximal function is a standard tool for controlling the size of a function in terms of its local averages, and it underlies many other results in harmonic analysis.