# Maximal function theory

Below is an outline of the notes I wrote up on the basic theory Hardy—Littlewood maximal function and its variants for a seminar. The notes assume familiarity with measure theory.

Download the notes here: MaximalFunctionTheory.pdf (29 pages)

Let \(f:[a,b] \to \mathbb{R}\) be continuous. The ** fundamental theorem of calculus** states that

\[F(x) = \int_a^x f(y) \, dy\]

is differentiable in \((a,b)\) and \(F'(x) = f(x)\) for all \(x \in (a,b)\).

Could we generalize this result to a larger class of functions? Note that the above result is equivalent to

\[(1) \hspace{2em} \lim_{r \to 0} \frac{1}{2r} \int_{x-r}^{x+r} f(y) \, dy = f(x)\]

for all continuous functions \(f\). In this form, the fundamental theorem of calculus is a statement about the behavior of the integral mean value

\[(2) \hspace{2em} (\mathcal{A}_{r} f)(x) = \frac{1}{2r} \int_{x-r}^{x+r} f(y) \, dy\]

as the length of the interval \((x-r,x+r)\) centered at \(x\) decreases to 0. Since \((\mathcal{A}_rf)(x)\) is well-defined for all \(f \in L^1([x-r,x+r])\), it makes sense to try and prove (1.1) for all \(f \in L^1_{\mbox{loc}}(\mathbb{R})\), the space of measurable functions on \(\mathbb{R}\) whose integral is finite on each compact subset of \(\mathbb{R}\).

We also consider \(d\)-dimensional generalizations of (1). To this end, we must determine what we wish to take as an \(n\)-dimensional generalization of intervals. We abstract three properties of intervals: compactness, convexity, and symmetry.

Definition.A nonempty set \(B \subseteq \mathbb{R}^d\) isif, for each pair of points \(x_1\) and \(x_2\) in \(B\), theconvex\((1-\lambda) x_1 + \lambda x_2\) is in \(B\) for all \(0 \leq \lambda \leq 1\).convex combination

Definition.A nonempty set \(B \subseteq \mathbb{R}^d\) iswith respect to \(p \in B\) if \(B\) is invariant under the affine transform \(x \mapsto 2p - x\). This is equivalent to saying that \(p + h \in B\) if and only if \(p - h \in B\).centrally symmetric

We write ** centrally symmetric convex body** to refer to a subset of \(\mathbb{R}^d\) that is compact, convex, and centrally symmetric with its center \(p\) at the origin.

Since we can rewrite (2) as

\[(\mathcal{A}_r f)(x) = \frac{1}{2r} \int_{-r}^r f(x+y) \, dy,\]

it suffices to consider centrally symmetric convex bodies with respect to the origin in asking the following question:

Question.Given a centrally symmetric convex body \(B \subseteq \mathbb{R}^d\), does the integral mean value\[(\mathcal{A}_{rB}f)(x) = \frac{1}{m(rB)} \int_{rB} f(x+y) \, dy\]

of \(f \in L^1_{\mbox{loc}}(\mathbb{R}^d)\) converge pointwise to \(f(x)\) as \(r \to 0\)? Here we have used \(rB\) to denote the scaled set \(rB = \{ry : y \in B\}\).

Whenever \(f\) is continuous, the argument for the one-dimensional fundamental theorem of calculus can be applied with minor modification to answer Question 1.5 in the affirmative. If \(f\) is merely \(L^1(\mathbb{R})\), then, for each \(\varepsilon > 0\), we can find a \(g \in \mathscr{C}_c(\mathbb{R}^d)\) such that \(\| f-g \|_1 < \varepsilon\). We can then rewrite \((\mathcal{A}_{rB}f)(x) - f(x)\) as

\[\mathcal{A}_{rB}(f-g)(x) + (\mathcal{A}_{rB}g)(x) - g(x) + g(x) - f(x).\]

By continuity, we have \((\mathcal{A}_{rB}g)(x) \to g(x)\) as \(r \to 0\), whence we have the estimate

\[\begin{align*} &\limsup_{r \to 0} \vert (\mathcal{A}_{rb}f)(x) - f(x) \vert \\ \leq& \limsup_{r \to 0} \vert \mathcal{A}_{rb}(f-g)(x) \vert| \\ &+ \limsup_{r \to 0} \vert(\mathcal{A}_{rB}g)(x) - g(x) \vert \\ &+ \vert f(x) - g(x) \vert \\ =& \limsup_{r \to 0} \mathcal{A}_{rB}(\vert f - g \vert)(x) + \vert f(x) + g(x) \vert.\end{align*}\]

Therefore, the study of the integral mean values of \(f\) at \(x \in \mathbb{R}^d\) depends crucially on the quantity

\[\limsup_{r \to 0} \mathcal{A}_{rB}(\vert f - g \vert)(x),\]

which is bounded from above by the *maximal function*

\[\sup_{r > 0} \mathcal{A}_{rB}(\vert f - g \vert)(x).\]

This motivates us to introduce our main object of study:

Definition.Theof \(h \in L^1_{\mbox{loc}}(\mathbb{R}^d)\) over a centrally symmetric convex body \(B \subseteq \mathbb{R}^d\) isHardy—Littlewood maximal function\[\begin{align*}(\mathcal{M}_Bf)(x) &= \sup_{r > 0}(\mathcal{A}_{rB}\vert f \vert)(x) \\ &= \sup_{r > 0} \frac{1}{(m(rB)} \int_{rB} \vert f(x+y) \vert \, dy.\end{align*}\]

What we have discussed so far assures us that if \(f \in L^1(\mathbb{R}^d)\), then, for each \(\varepsilon > 0\), there exists a \(g \in L^1(\mathbb{R}^d)\) that yields the estimate

\[(3) \hspace{3em} \limsup_{r \to 0} \vert (\mathcal{A}_{rB}f)(x) - f(x) \vert \leq \mathcal{M}_B(f-g)(x) + \vert f(x) - g(x)\vert.\]

Ideally, we would have liked to show that

\[\mathcal{M}_B(f-g)(x) + \vert f(x) - g(x) \vert C \varepsilon\]

for some constant \(C\) independent of \(\varepsilon\), thus proving that \(\mathcal{A}_{rB} f \to f\) pointwise *everywhere*. But this is too much to hope for, as

\[(\mathcal{A}_{(-r,r)}\chi_{(0,1)}(0) = \frac{1}{2r} \int_0^r \, dy = \frac{1}{2}\]

for all \(0 < r < 1\), which does not converge to \(\chi_{(0,1)}(0) = 0\) as \(r \to 0\).

So then, if we hope to obtain an affirmative answer to the above Question, then we must settle for an almost-everywhere statement. This equivalent to the statement that the set

\[\left\{x \in \mathbb{R}^d : \limsup_{r \to 0} \vert (\mathcal{A}_{rB}f)(x) - f(x) \vert > 0 \right\}\]

is of Lebesgue measure zero. Since the above set is the intersection of the sets

\[(4) \hspace{3em} E_k = \left\{x \in \mathbb{R}^d : \limsup_{r \to 0} \vert (\mathcal{A}_{rB}f)(x) - f(x)\vert > \frac{1}{k}\right\},\]

it suffices to show that \(m(E_k) = 0\) for all \(k \in \mathbb{N}\).

Now, Estimate (3) implies that

\[(5) \hspace{3em} \begin{align*} m(E_k) \leq& m \left( \left\{ x : \mathcal{M}_B(f-g)(x) > \frac{1}{2k}\right\}\right) \\ &+ m \left( \left\{ x : \vert f(x) - g(x) \vert > \frac{1}{2k} \right\}\right) \end{align*}\]

Observe that

\[\begin{align*} m\left(\left\{ x : \vert f(x) - g(x) \vert > \frac{1}{2k} \right\}\right) &= \int_{\{x : \vert f(x) - g(x) \vert > \frac{1}{2k}\}} 1 \, dy \\ &\leq \int_{\{x : \vert f(x) - g(x) \vert > \frac{1}{2k}\}} \frac{\vert f(y) - g(y)\vert}{1/2k} \, dy \\ &\leq \int_{\mathbb{R}^d} \frac{\vert f(y) - g(y)\vert}{1/2k} \, dy \\ &= 2k \|f-g\|_1. \end{align*}\]

It is therefore natural to hope for a bound of the form

\[(6) \hspace{3em} m \left(\{x : \mathcal{M}_B(f-g)(x) > \frac{1}{2k}\}\right) \leq 2kA \|f-g\|_1\]

for some constant \(A\) independent of \((f-g)\), so that (5) can be written as

\[m(E_k) \leq 2k(A+1) \|f-g\|_1 < 2k(A+1)\varepsilon.\]

Since \(\varepsilon\) was arbitrary, we can then conclude that \(m(E_k) = 0\).

(6) is established by the following foundational result in maximal function theory:

Theorem(Weak-type \((1,1)\)-bound on the maximal function). If \(B \subseteq \mathbb{R}^d\) is a centrally symmetric convex body, then there exists a constant \(A_{d,1,B}\) such that\[m(\{x \in \mathbb{R}^d : \mathcal{M}_Bf(x) > \alpha\}) \leq \frac{A_{d,1,B}}{\alpha}\|f\|_1\]

for each \(\alpha > 0\) and every \(f \in L^1(\mathbb{R}^d)\). The constant \(A_{d,1,B}\) depends only on the dimension \(d\) and the body \(B\).

The weak-type bound now implies the below pointwise (almost everywhere) convergence result for the \(L^1\) case. The general case of the theorem follows easily from the \(L^1\) case by considering the *compact cutoff function* \(f\chi_{B(0;k)}\) with respect to closed balls \(B(0;k)\) of radius \(k \in \mathbb{N}\) centered at the origin.

Theorem(Lebesgue differentiation theorem). If \(f \in L^1_{\textrm{loc}}(\mathbb{R}^d)\), and if \(B \subseteq \mathbb{R}^d\) is a centrally symmetric convex body, then\[\lim_{r \to 0}(\mathcal{A}_{rB}f)(x) = f(x)\]

for almost every \(x \in \mathbb{R}^d\).

The method of establishing a weak-type bound to prove a pointwise convergence result turns out to be extremely powerful. In fact, we cannot do better:

Theorem(Stein's \(L^1\) maximal principle). Let \(G\) be a compact, Hausdorff, abelian topological group equipped with the Haar measure \(\mu\). If \((\varphi_n)_{n=1}^\infty\) is a sequence of operators in \(L^\infty(G)\) such that, for each \(f \in L^1(G)\), we have the "pointwise convergence criterion"\[\limsup_{n \to \infty} \vert (f \ast \varphi_n)(x) \vert < \infty\]

on a set \(E_f\) of positive measure, then the maximal operator

\[Mf(x) = \sup_{n \in \mathbb{N}} \vert (f \ast \varphi_n)(x) \vert\]

satisfies the weak-type \((1,1)\)-bound.

The weak-type bound of a general maximal operator is typically established through a judicious use of the Hardy–Littlewood maximal operator. It is thus of interest to tighten the bound

\[m(\{x \in \mathbb{R}^d : \mathcal{M}_Bf(x) > \alpha\}) \leq \frac{A_{d,1,B}}{\alpha}\|f\|_1\]

by reducing the size of the constant \(A_{d, 1, B}\) as much as possible.

The classical proof yields \(A_{d,1,B} = O(5^d)\) when \(B\) is the Euclidean ball. This is a consequence of the following lemma:

Lemma(Infinitary Vitali covering lemma). If \(\{B(x_\beta,r_\beta)\}_{\beta}\) is a collection of Euclidean balls in \(\mathbb{R}^d\) whose radii are uniformly bounded, then there exists a pairwise-disjoint countable subcollection \(\{B(x_n,r_n)\}_n\) such that\[\bigcup_{\beta} B(x_\beta,r_\beta) \subseteq \bigcup_n B(x_n,5r_n).\]

An 1988 result of Stein and StrÃ¶mberg reduces the constant to \(O(d)\) for the Euclidean ball and \(O(d \log d)\) for an arbitrary centrally symmetric convex body. Naor and Tao showed in 2011 that \(d \log d\) is, in fact, optimal for a broad class of metric measure spaces.

The question of finding a tight bound for the Euclidean-space case remains open. In 2003, Melas showed that \(\frac{11+\sqrt{61}}{21}\) is the optimal constant for the weak-type bound of the Hardy–Littlewood maximal function on the one-dimensional Euclidean ball. This remains the only known tight bound.

Stein and StrÃ¶mberg conjectured that the bound may, in fact, be \(O(1)\). While the conjecture has not been settled for the Euclidean ball case, Aldaz showed in 2011 that the constant grows without bound on the \(l^\infty\) ball as the dimension increase.

Much work has been done on establishing dimension-independent bound for \(L^p\) when \(p > 1\). A classical proof using real interpolation methods gives us the bound

\[\|\mathcal{M}_Bf\|_p \leq A_{d, p, B} \|f\|_p\]

for all \(p > 1\) with constant

\[A_{d, p, B} = 2^{\frac{p-1}{p}} A^{\frac{1}{p}}_{d, 1, B} \left( \frac{p}{p-1} \right)^{1/p}.\]

Stein showed in 1982 that, given a fixed centrally symmetric convex body \(B\), the constant can be made \(O(1)\) with respect to the dimension \(d\). Moreover, Bourgain's far-reaching generalization in 1986 shows that, for each \(p > 3/2\),

\[\sup_{d, p, B} A_{d, p, B} < \infty\]

where the supremum is taken over all \(d \geq 1\) and centrally symmetric convex bodies \(B \subseteq \mathbb{R}^d\). In 2014, Bourgain brought \(p\) down to \(p > 1\) for the \(l^\infty\) ball. The question of establishing the above uniform bound for a large class of centrally symmetric convex bodies in the range \(p > 1\) remains open.