In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis.
Definition The operator takes a locally integrable function f : R d → C {\displaystyle f:\mathbb {R} ^{d}\to \mathbb {C} } and returns another function M f : R d → [ 0 , ∞ ] {\displaystyle Mf:\mathbb {R} ^{d}\to [0,\infty ]} , where M f ( x ) {\displaystyle Mf(x)} is the supremum of the average of f {\displaystyle f} among all possible balls centered on x {\displaystyle x} . Formally,
M f ( x ) = sup r > 0 1 | B ( x , r ) | ∫ B ( x , r ) | f ( y ) | d y {\displaystyle Mf(x)=\sup _{r>0}{\frac {1}{|B(x,r)|}}\int _{B(x,r)}|f(y)|\,dy} , where |E| denotes the d-dimensional Lebesgue measure of a subset E ⊂ Rd, and B ( x , r ) {\displaystyle B(x,\,r)} is the ball of radius, r > 0 {\displaystyle r>0} , centered at the point x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} . Since f {\displaystyle f} is locally integrable, the averages are jointly continuous in x and r, so the maximal function Mf, being the supremum over r > 0, is measurable. A nontrivial corollary of the Hardy–Littlewood maximal inequality states that M f {\displaystyle Mf} is finite almost everywhere for functions in L 1 {\displaystyle L^{1}} .
Hardy–Littlewood maximal inequality This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from Lp(Rd) to itself for p > 1. That is, if f ∈ Lp(Rd) then the maximal function Mf is weak L1-bounded and Mf ∈ Lp(Rd). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}. Now we have:
Theorem (Weak Type Estimate). For d ≥ 1, there is a constant Cd > 0 such that for all λ > 0 and f ∈ L1(Rd), we have:
| { M f > λ } | < C d λ ‖ f ‖ L 1 ( R d ) . {\displaystyle \left|\{Mf>\lambda \}\right|<{\frac {C_{d}}{\lambda }}\Vert f\Vert _{L^{1}(\mathbf {R} ^{d})}.}
With the Hardy–Littlewood maximal inequality in hand, the following strong-type estimate is an immediate consequence of the Marcinkiewicz interpolation theorem:
Theorem (Strong Type Estimate). For d ≥ 1, 1 < p ≤ ∞, and f ∈ Lp(Rd), there is a constant Cp,d > 0 such that
‖ M f ‖ L p ( R d ) ≤ C p , d ‖ f ‖ L p ( R d ) . {\displaystyle \Vert Mf\Vert _{L^{p}(\mathbf {R} ^{d})}\leq C_{p,d}\Vert f\Vert _{L^{p}(\mathbf {R} ^{d})}.}
In the strong type estimate the best bounds for Cp,d are unknown. However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following:
Theorem (Dimension Independence). For 1 < p ≤ ∞ one can pick Cp,d = Cp independent of d.
Proof While there are several proofs of this theorem, a common one is given below, that uses the following version of the Vitali covering lemma to prove the weak-type estimate. (See the article for the proof of the lemma.)
Note that the constant C = 5 d {\displaystyle C=5^{d}} in the proof can be improved to 3 d {\displaystyle 3^{d}} by using the inner regularity of the Lebesgue measure, and the finite version of the Vitali covering lemma. See the Discussion section below for more about optimizing the constant.
Applications Some applications of the Hardy–Littlewood Maximal Inequality include proving the following results:
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