# Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar $L^{p}$ spaces.

The Lorentz spaces are denoted by $L^{p,q}$ . Like the $L^{p}$ spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the $L^{p}$ norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the $L^{p}$ norms, by exponentially rescaling the measure in both the range ($p$ ) and the domain ($q$ ). The Lorentz norms, like the $L^{p}$ norms, are invariant under arbitrary rearrangements of the values of a function.

## Definition

The Lorentz space on a measure space $(X,\mu )$  is the space of complex-valued measurable functions $f$  on X such that the following quasinorm is finite

$\|f\|_{L^{p,q}(X,\mu )}=p^{\frac {1}{q}}\left\|t\mu \{|f|\geq t\}^{\frac {1}{p}}\right\|_{L^{q}\left(\mathbf {R} ^{+},{\frac {dt}{t}}\right)}$

where $0  and $0 . Thus, when $q<\infty$ ,

$\|f\|_{L^{p,q}(X,\mu )}=p^{\frac {1}{q}}\left(\int _{0}^{\infty }t^{q}\mu \left\{x:|f(x)|\geq t\right\}^{\frac {q}{p}}\,{\frac {dt}{t}}\right)^{\frac {1}{q}}=\left(\int _{0}^{\infty }{\bigl (}\tau \mu \left\{x:|f(x)|^{p}\geq \tau \right\}{\bigr )}^{\frac {q}{p}}\,{\frac {d\tau }{\tau }}\right)^{\frac {1}{q}}.$

and, when $q=\infty$ ,

$\|f\|_{L^{p,\infty }(X,\mu )}^{p}=\sup _{t>0}\left(t^{p}\mu \left\{x:|f(x)|>t\right\}\right).$

It is also conventional to set $L^{\infty ,\infty }(X,\mu )=L^{\infty }(X,\mu )$ .

## Decreasing rearrangements

The quasinorm is invariant under rearranging the values of the function $f$ , essentially by definition. In particular, given a complex-valued measurable function $f$  defined on a measure space, $(X,\mu )$ , its decreasing rearrangement function, $f^{\ast }:[0,\infty )\to [0,\infty ]$  can be defined as

$f^{\ast }(t)=\inf\{\alpha \in \mathbf {R} ^{+}:d_{f}(\alpha )\leq t\}$

where $d_{f}$  is the so-called distribution function of $f$ , given by

$d_{f}(\alpha )=\mu (\{x\in X:|f(x)|>\alpha \}).$

Here, for notational convenience, $\inf \varnothing$  is defined to be $\infty$ .

The two functions $|f|$  and $f^{\ast }$  are equimeasurable, meaning that

$\mu {\bigl (}\{x\in X:|f(x)|>\alpha \}{\bigr )}=\lambda {\bigl (}\{t>0:f^{\ast }(t)>\alpha \}{\bigr )},\quad \alpha >0,$

where $\lambda$  is the Lebesgue measure on the real line. The related symmetric decreasing rearrangement function, which is also equimeasurable with $f$ , would be defined on the real line by

$\mathbf {R} \ni t\mapsto {\tfrac {1}{2}}f^{\ast }(|t|).$

Given these definitions, for $0  and $0 , the Lorentz quasinorms are given by

$\|f\|_{L^{p,q}}={\begin{cases}\left(\displaystyle \int _{0}^{\infty }\left(t^{\frac {1}{p}}f^{\ast }(t)\right)^{q}\,{\frac {dt}{t}}\right)^{\frac {1}{q}}&q\in (0,\infty ),\\\sup \limits _{t>0}\,t^{\frac {1}{p}}f^{\ast }(t)&q=\infty .\end{cases}}$

## Lorentz sequence spaces

When $(X,\mu )=(\mathbb {N} ,\#)$  (the counting measure on $\mathbb {N}$ ), the resulting Lorentz space is a sequence space. However, in this case it is convenient to use different notation.

### Definition.

for $(a_{n})_{n=1}^{\infty }\in \mathbb {R} ^{\mathbb {N} }$  (or $\mathbb {C} ^{\mathbb {N} }$  in the complex case), let $\|(a_{n})_{n=1}^{\infty }\|_{p}=\left(\sum _{n=1}^{\infty }|a_{n}|^{p}\right)^{1/p}$  denote the p-norm for $1\leq p<\infty$  and $\|(a_{n})_{n=1}^{\infty }\|_{\infty }=\sup |a_{n}|$  the ∞-norm. Denote by $\ell _{p}$  the Banach space of all sequences with finite p-norm. Let $c_{0}$  the Banach space of all sequences satisfying $\lim |a_{n}|=0$ , endowed with the ∞-norm. Denote by $c_{00}$  the normed space of all sequences with only finitely many nonzero entries. These spaces all play a role in the definition of the Lorentz sequence spaces $d(w,p)$  below.

Let $w=(w_{n})_{n=1}^{\infty }\in c_{0}\setminus \ell _{1}$  be a sequence of positive real numbers satisfying $1=w_{1}\geq w_{2}\geq w_{3}\cdots$ , and define the norm $\|(a_{n})\|_{d(w,p)}=\sup _{\sigma \in \Pi }\|(a_{\sigma (n)}w_{n}^{1/p})_{n=1}^{\infty }\|_{p}$ . The Lorentz sequence space $d(w,p)$  is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define $d(w,p)$  as the completion of $c_{00}$  under $\|\cdot \|_{d(w,p)}$ .

## Properties

The Lorentz spaces are genuinely generalisations of the $L^{p}$  spaces in the sense that, for any $p$ , $L^{p,p}=L^{p}$ , which follows from Cavalieri's principle. Further, $L^{p,\infty }$  coincides with weak $L^{p}$ . They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for $1  and $1\leq q\leq \infty$ . When $p=1$ , $L^{1,1}=L^{1}$  is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of $L^{1,\infty }$ , the weak $L^{1}$  space. As a concrete example that the triangle inequality fails in $L^{1,\infty }$ , consider

$f(x)={\tfrac {1}{x}}\chi _{(0,1)}(x)\quad {\text{and}}\quad g(x)={\tfrac {1}{1-x}}\chi _{(0,1)}(x),$

whose $L^{1,\infty }$  quasi-norm equals one, whereas the quasi-norm of their sum $f+g$  equals four.

The space $L^{p,q}$  is contained in $L^{p,r}$  whenever $q . The Lorentz spaces are real interpolation spaces between $L^{1}$  and $L^{\infty }$ .