Non-tangential Maximal Functions
The non-tangential maximal function takes a function defined on the upper-half plane and produces a function defined on via the expression
Observe that for a fixed, the set is a cone in with vertex at and axis perpendicular to the boundary of . Thus, the non-tangential maximal operator simply takes the supremum of the function over a cone with vertex at the boundary of .
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“Empirical science is apt to cloud the sight, and, by the very knowledge of functions and processes, to bereave the student of the manly contemplation of the whole.”
—Ralph Waldo Emerson (1803–1882)