Exercise 1.6.19 (Hardy-Littlewood maximal inequality II)

We prove the Hardy-Littlewood inequality, which we will do with the constant $C_d:=2^d$. It suffices to verify the claim with strict inequality,

as the non-strict case then follows by perturbing $\lambda$ slightly and then taking limits. Thus, fix $f$ and $\lambda$. By inner regularity (Exercise 1.2.15), it suffices to show that

and then takes supremum over $K$. Hence, let $K$ be arbitrary. By definition of $K$, for each $x\in K$ there exists a $r>0$ such that

By the axiom of choice we construct a collection $B{(x,r_x)}_{x\in K}$ of such balls with $K\subseteq {\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{r>0}B(x,r_x)$. In other words, $B{(x,r_x)}_{x\in K}$ is an open cover of $K$. By compactness, we then must be able to find a finite subcover $B_1,\dots ,B_n$ of $B{(x,r_x)}_{x\in K}$. We now introduce an algorithm (similar to the one provided in Vitali lemma to construct a disjoint subcollection $B_1^{\prime },\dots ,B_m^{\prime }$ of $B_1,\dots ,B_n$) such that its doubled version covers $K$. The trick is to introduce a tolerance of $𝜖\in (0,1)$.

(i)

Around each $x\in K$ consider the ball $B{(x,𝜖r_x)}_{x\in K}$ (notice the epsilon! - we have made the balls smaller than originally thought). Then obviously ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{x\in K}B(x,𝜖r_x)$ is an open cover of $K$. Thus, by compactness of $K$ there exists a finite subcover $B{(x_i,𝜖r_i)}_{1\le i\le n}$ which still covers all of the $K$.

(ii)

If $B{(x_i,𝜖r_i)}_{1\le i\le n}$ covers $K$, then so does $B{(x_i,r_i)}_{1\le i\le n}$ as it is simply a larger cover.

(iii)

Using the same construction steps as we used for the Vitali covering lemma (Exercise 1.6.18) we construct a disjoint subcollection $B{(x_{(j)},r_{(j)})}_{1\le j\le m}$ of $B{(x_i,r_i)}_{1\le i\le n}$ such that $r_{(1)}\ge \dots \ge r_{(m)}$ and

$$K\subseteq \bigcup _{j=1}^{m}B(x_{(j)},3\cdot r_{(j)})$$

We now argue that the constant $3$ can be improved to $2$ in the above statement by noticing that in the construction used to prove the Vitali covering lemma (Exercise 1.6.18), the centres of the balls $B_i,1\le i\le n$ were contained in ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^{m}2B_j^{\prime }$ and not just in ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^{m}3B_j^{\prime }$.

(iv)

At this stage we employ the main trick. Instead of showing that

$$K\subseteq \bigcup _{i=1}^{n}B(x_i,r_i)\subseteq \bigcup _{j=1}^{m}B(x_{(j)},2r_{(j)})$$

(which is not necessarily true in the middle), we give ourselves an epsilon of room and show instead that

$$K\subseteq \bigcup _{i=1}^{n}B(x_i,𝜖r_i)\subseteq \bigcup _{j=1}^{m}B(x_{(j)},(2+𝜖)r_{(j)}).$$

Thus, pick an arbitrary $B(x_i,𝜖r_i)$ from the original collection $B{(x_i,𝜖r_i)}_{1\le i\le n}$. By Vitali construction in the previous step, there exists a $B(x_{(j)},r_{(j)})$ of minimal $j\le m$ such that $B(x_i,r_i)\cap B(x_{(j)},r_{(j)})\ne \varnothing$. Also $r_{(j)}\ge r_i$ by construction. By triangle inequality, this implies that the centre $x_i$ of the ball $B(x_i,r_i)$ is contained inside $B(x_{(j)},2\cdot r_{(j)})$:

$$\text{for }z\in B_i\cap B_{(j)}:d(x_i,x_{(j)})\le d(x_i,z)+d(z,x_{(j)})\le r_i+r_{(j)}\le 2r_{(j)}$$

From here we see that the smaller ball $B(x_i,𝜖r_i)$ has a non-empty intersection with $B(x_{(j)},2\cdot r_{(j)})$ as well. Furthermore, we see that it paid off to take an epsilon of room as we obtain $B(x_i,𝜖r_i)\subseteq B(x_{(j)},(2+𝜖)r_{(j)})$ by triangle inequality:

$$\begin{array}{llll}\hfill \forall <!--\; FUNCTION\; APPLICATION\; -->p\in B(x_i,𝜖r_i):d(p,x_{(j)})& \le d(p,x_i)+d(x_i,x_{(j)})& \hfill & \\ \hfill & \le 𝜖r_i+2r_{(j)}& \hfill & \\ \hfill & \le r_{(j)}(2+𝜖)& \hfill & \end{array}$$

We now integrate this into the proof of the Hardy-Littlewood theorem. We have shown

to be true.

We cannot take infimum as $𝜖\to 0$ as $n$ and $m$ may explode; instead, we rewrite the above using an upper bound for measure without involving $m$:

Now we can safely take $𝜖$ to $0$ and obtain the desired result.