Lorentz Space
The purpose of this note is to give a few equivalent definitions, \eqref{def1}, \eqref{def2}, \eqref{def3} and \eqref{def4} for Lorentz norm L ^{p, q}.
Distribution function and Decreasing Rearrangement
Similar as Lebesgue space, Lorentz space only cares about the height of a function and the space it occupies at each height. Let f \in \mathcal L ^0 \pth{X, \d \mu} be any complex-valued measurable function. Any definition below that works for f \ge 0 can be generalized for complex-valued (and in general Banach space-valued) f by replacing f by \abs{f}.
Definition. The distribution function of f is defined as the measure of the superlevel set, that is
\begin{align} d _f (\alpha) := \mu \pth{\set{x \in X: |f (x)| > \alpha}}, \qquad \alpha \ge 0. \end{align}
This definition is the opposite to the definition of cdf (cumulative distribution function) for a random variable that we are used to in probability. So in contrast to cdf, d _f is left/lower semicontinuous, and is a decreasing function.
Definition. The decreasing rearrangement of f is defined on [0, \infty) by
\begin{align} f ^* (\lambda) := \inf \set{t > 0: d _f (t) \le \lambda} \end{align}
This is almost like the “inverse function” of the distribution function. It is also left/lower semicontinuous and decreasing. Notice that the superlevel set of f ^* has the same measure as f.
One can take the following example to understand the construction of the distribution function and decreasing rearrangement. Assume the following is the graph of f = \ind{S _1} + 4 \ind{S _2} + 2 \ind{S _3} where \mu (S _1) = \mu (S _2) = \mu (S _3) = 1.
To find its distribution function, rotate by 90 degrees and then find the measure of superlevel set at each level \alpha.
Finally, rotate back and flip the distribution function to obtain decreasing rearrangement.
Compare f and f ^*, it looks like just push all the level sets to the left onto the \alpha axis, and end up with something decreasing.
Layer Cake Representation and L ^p Norm
The layer cake representation states the following. For f \ge 0,
\begin{align} \int _X f \d \mu = \int _0 ^\infty d _f (\alpha) \d \alpha. \end{align}
This comes from thinking integration over the hypograph,
\begin{align} \Omega _f = \set{(x, \alpha) \in X \times \R _+: f (x) > \alpha}. \end{align}
Therefore
\begin{align} \int _X f \d \mu &= \int _X \int _0 ^{f(x)} \d \alpha \d \mu \notag\\\ &= \int _{\Omega _f} \d \alpha \d \mu \notag\\\ &= \int _0 ^\infty \int _X \ind{\set{f(x) > \alpha}} \d \mu \d \alpha \notag \\\ &= \int _0 ^\infty \mu (f > \alpha) \d \alpha \notag \\\ &= \int _0 ^\infty d _f (\alpha) \d \alpha. \notag \end{align}Using decreasing rearrangement, we can also think
\begin{align} \int _X f \d \mu = \int _0 ^\infty f ^* \d \lambda = \int _{\Omega _{f ^*}} \d \alpha \d \lambda. \notag \end{align}
Here \Omega _{f ^*} is the hypograph of f ^* (and also the hypograph of d _f)
\begin{align} \Omega _{f ^*} = \set{(\lambda, \alpha) \in \R _+ ^2: f ^* (\lambda) > \alpha} = \set{(\lambda, \alpha) \in \R _+ ^2: d _f (\alpha) > \lambda}. \notag \end{align}
Similarly, for L ^p norm,
\begin{align} \int _X f ^p \d \mu &= \int _{\Omega _{f ^*}} \d (\alpha ^p) \d \lambda = \int _0 ^\infty d _f (\alpha) \d (\alpha ^p) = \int _0 ^\infty p \alpha ^{p - 1} d _f (\alpha) \d \alpha. \notag \end{align}
So the integral of f ^p can be just thought as the \d(\alpha ^p) \tensor \d \lambda measure of the hypograph.
Lorenze Space
Definition. For p, q > 0, we define the Lorenze norm of f \ge 0 by
\begin{align} \label{def1} \nor{f} _{L ^{p, q}} ^q = \int _{\Omega _{f ^*}} \d (\alpha ^q) \d \pth{\frac pq \lambda ^{\frac qp}} = \int _{\Omega _{f ^*}} \lambda ^{\frac qp - 1} \d (\alpha ^q) \d \lambda. \end{align}
That is, we change the measure we put on domain, and put L ^q-typed measure on range. If we integrate \lambda first, we end up with
\begin{align} \nor{f} _{L ^{p, q}} ^q &= \int _0 ^\infty \frac pq \bkt{d _f (\alpha)} ^{\frac qp} q \alpha ^{q - 1} \d \alpha \notag \\\ &= p \int _0 ^\infty \bkt{d _f (\alpha)} ^{\frac qp} \alpha ^{q} \frac{\d \alpha}\alpha \label{split} ,\\\ \nor f _{L ^{p,q}} &= p ^\frac1q \nor{\alpha [d _f (\alpha)] ^\frac1p} _{L ^q \pth{\R _+, \frac{\d \alpha}\alpha}} . \label{def2} \end{align}If we integrate \alpha first, we end up with
\begin{align} \nor f _{L ^{p, q}} ^q &= \int _0 ^\infty [f ^* (\lambda)] ^q \lambda ^{\frac qp - 1} \d \lambda \notag \\ &= \int _0 ^\infty \lambda ^{\frac qp} [f ^* (\lambda)] ^q\frac{\d \lambda}\lambda \notag ,\\ \nor f _{L ^{p, q}} &= \nor{\lambda ^{\frac1p} f ^* (\lambda)} _{L ^q \pth{\R _+, \frac{\d \lambda}\lambda}}. \label{def3} \end{align}Dyadic Decomposition in Range
We decompose f \ge 0 by range as the following. Let E _k = \set{f > 2 ^k} denote a sequence of nested dyadic level sets, and define
\begin{align*} f _k &= 2 ^{k-1} \ind{E _{k + 1}} + (f - 2^{k - 1})(\ind{E _k} - \ind{E _{k + 1}}) \\\ &= f(\ind{E _k} - \ind{E _{k + 1}}) + 2 ^k \ind{E _{k + 1}} - 2^{k - 1} \ind{E _k} . \end{align*}Then f = \sum _{k \in \mathbb Z} f _k, f _k is supported in E _k, and f _k is comparible to 2 ^k \ind{E _k}:
\begin{align*} \frac12 2 ^k \ind{E _k} \le f _k \le \frac32 2 ^k \ind{E _k}. \end{align*}
Now we also split the integral \eqref{split} dyadically,
\begin{align*} \frac1p \nor{f} _{L ^{p, q}} ^q &= \int _0 ^\infty \bkt{d _f (\alpha)} ^{\frac qp} \alpha ^{q} \frac{\d \alpha}\alpha \notag \\\ &= \sum _{k \in \Z} \int _{2 ^k} ^{2 ^{k + 1}} \bkt{d _f (\alpha)} ^{\frac qp} \alpha ^{q} \frac{\d \alpha}\alpha .\notag \\\ \end{align*}Since d _f is decreasing,
\begin{align*} 2 ^{kq} \bkt{d _f \pth{2 ^{k + 1}}} ^{\frac qp} \lesssim \int _{2 ^k} ^{2 ^{k + 1}} \bkt{d _f (\alpha)} ^{\frac qp} \alpha ^{q} \frac{\d \alpha}\alpha \lesssim 2 ^{kq} \bkt{d _f \pth{2 ^{k}}} ^{\frac qp}. \notag \end{align*}By definition of E _k,
\begin{align*} 2 ^{kq} \mu\pth{E _{k + 1}} ^{\frac qp} \lesssim \int _{2 ^k} ^{2 ^{k + 1}} \bkt{d _f (\alpha)} ^{\frac qp} \alpha ^{q} \frac{\d \alpha}\alpha \lesssim 2 ^{kq} \mu\pth{E _{k}} ^{\frac qp}. \notag \end{align*}Since \nor{f _k} _{L ^p} \sim \nor{2 ^k \ind {E _k}} _{L ^p} = 2 ^k \mu \pth{E _k} ^{\frac 1p},
\begin{align*} \nor{f _{k + 1}} _{L ^p} ^q \lesssim \int _{2 ^k} ^{2 ^{k + 1}} \bkt{d _f (\alpha)} ^{\frac qp} \alpha ^{q} \frac{\d \alpha}\alpha \lesssim \nor{f _k} _{L ^p} ^q. \notag \end{align*}By taking the summation over k, we obtain an equivalent way of defining L ^{p, q} norm,
\begin{align} \label{def4} \nor f _{L ^{p, q}} \sim _{p,q} \nor{ \set{\nor{f _k} _{L ^p}} _{k \in \Z}} _{\ell ^q}. \end{align}