Computation and understanding some wording around the Hardy Littlewood Maximal function

Question

I have just started to read Stein's Singular Integrals and Differentiability properties of functions.

The Hardy-Littlewood maximal function has just been introduced i.e. $$M(f)(x):= \sup_{r > 0} \frac{1}{m(B(x,r))}\int_{B(x,r)}|f(y)|dy$$

where $m(B(x,r))$ denotes the measure of the Ball

Stein then states "We shall now be interested in giving a concise expression for the relative size of a function". Let $g(x)$ be defined on $\mathbb{R}^{n}$ and for each $\alpha$ consider the following set $\{x:|g(x)| > \alpha\}$. Then the function $\lambda(\alpha)$ defined to be the measure of this set is the distribution function of $|g|$.

Questions:

(1): Stein states, "The decrease of $\lambda(\alpha)$ as $\alpha$ grows describes the relative largeness of the function" why is this describing the largeness (i'd have thought it would be saying how small the function is, and relative to what, other functions?)

(2): If $g \subset L^{p}$ then one has $\int_{\mathbb{R}^{n}}|g(y)|^{p}dy = - \int_{0}^{\infty}\alpha^{p}d \lambda(\alpha)$. How does one get the RHS of this equality?

Answer 1

To determine in which $L^p$ spaces a function $g$ belongs, the behavior of $\lambda(\alpha)$ for large $\alpha is pertinent.

If $0=\alpha_0<\alpha_1<\alpha_2<\ldots$ is a partition of $[0,\infty)$, then $$\int_{\mathbb{R}^{n}}|g(y)|^{p}dy = \sum_{k=1}^\infty \int_{y: \alpha_ {k-1} <|g(y)| \le \alpha_k} |g(y)|^{p}dy \,,$$ so $$\sum_{k=1}^\infty \alpha_{k-1}^p \Bigl(\lambda(\alpha_ {k-1})- \lambda(\alpha_k) \Bigr) \le \int_{\mathbb{R}^{n}}|g(y)|^{p}dy \le \sum_{k=1}^\infty \alpha_{k}^p \Bigl(\lambda(\alpha_ {k-1})- \lambda(\alpha_k)\Bigr) \,,$$

Passing to the limit as the partition is refined gives the desired relation.

Answer 2

For additional intuition about the properties of the distribution function, assume for simplicity that the domain of $g$ is two-dimensional and consider the related function $y= h(x_1, x_2)= |g(x_1, x_2)| $. Its graph defines a surface in three-space, which we can imagine is a volcanic island that lies above sea level. The contours (level sets) of $h$ are defined by $h=\alpha$. Now regard $\alpha >0$ as the height of a rising tide. One sees that the distribution function defined as

$\lambda(\alpha) = \lambda_h(\alpha) $ = area of the measurable set in the plane defined by the inequality $h>\alpha$

represents the amount of acreage on the island that is above the tide line. Some important monotonicity properties of this function are that

(i) as $\alpha$ increases the function $\lambda(\alpha) $ decreases; dry land becomes scarcer;

(ii) if for fixed $\alpha$ the function $h$ is increased, then the quantity $\lambda _h(\alpha)$ increases: the quantity of dry land expands as the mountain rises.

Since the function $\lambda(\alpha)$ is monotone and non-negative, it is locally Riemann integrable; thus the improper Riemann integral is well-defined, (possibly equal to $\infty$). That is, there is a well-defined notion of the area under the graph of the non-negative function $\lambda(\alpha)$. The function has at most countably many jump discontinuities.

Lebesgue integration is constructed precisely so that the Lebesgue integral of $h$ equals the Riemann integral of $\lambda$. That is, the Lebesgue method for approximating $\int h \ d\lambda$ in which one partitions the range of $h$ into fine subintervals and bounds $h$ from above and below on each such zone is essentially identical to the Riemann sum approximation process for evaluating the integral of the distribution function. Thus as Yuval demonstrates in detail in his excellent answer, with some work one can show that $ \int_{R^2} h (x_1, x_2) \ d\lambda = \int _0^{\infty} \lambda(\alpha) \ d\alpha$. This generalizes to the higher-order moments (different powers of $h$) using a change of variables argument, replacing $\alpha$ by $\alpha^p$. .