Maximal Function

In mathematics, the maximal function is the branch of geometry applied in the forms of harmonic analysis. Hardy–Little wood maximal function is the most important type in the maximal function. The singular integrals, differentiability properties of functions, and partial differential equations are mainly used in maximal function for easy understanding. Comparing to other methods, these are usual method to provide a simplified approach for easy understanding of problems.

Types of maximal function:

The Hardy–Little wood maximal function.

Non-tangential maximal functions.

The sharp maximal function.

The Hardy–Little wood maximal function:

The maximal function was first introduced by G.H.Hardy. It is based on cricket score. According to him, f is a function on Rn, the Hardy–Little wood maximal function M (f) is given as,

M (f) (x) = sup 1/ (│B│) ∫B│f│

Where, x € Rn

Here, the | B | is the measure of B. The centered maximal function is taken from over balls B with centre x.

Properties of Hardy–Little wood maximal function:

(a) When f € Lp (Rn) (1≤p≤∞), M (f) is almost finite.

(b) Whether f € L1 (Rn), for all α > 0,

│ {x│M(f)(x)>α}│≤ (c/ α) ∫Rn │f│.

From the above properties the second property is known as weak-type bound. According to Morkov inequality for integrable function, M (f) is not a integrable function.
Applications:

The Hardy–Littlewood maximal function is function mainly used to prove the Lebesgue differentiation theorem and Fatou's theorem in the singular integral theory.

Non-tangential maximal function:

According to non-tangential maximal function,

Rn+1 = {(x, t) x € Rn, t> 0} and the F*(x) is given as,

F*(x) = SUP │F(y, t) │.

In the non-tangential maximal function takes the function F above a cone with vertex at the boundary of Rn.

Identity Property of Non-tangential maximal functions:

Identity property is the most important function used in the Non-tangential maximal functions. It is given as,

∫ Rn Φ = 1

Φt(x) = 1/ tn Φ (x/t)

for t > 0.

F(x, t) = f* Φt(x) = ∫ Rn f(x - t) Φt(y) dy

The sharp maximal function:

The sharp maximal function is maximal function (f # ) is defined as

f # (x) = SUP (1/│B│)∫ B │f(y) – fB │dy

Where, x € Rn