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 $0<p<\infty$ and $0<q\leq \infty$. 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

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

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 $0<p<\infty$ and $0<q\leq \infty$, 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 ${\textstyle \left\|(a_{n})_{n=1}^{\infty }\right\|_{p}=\left(\sum _{n=1}^{\infty }|a_{n}|^{p}\right)^{1/p}}$ denote the p-norm for $1\leq p<\infty$ and ${\textstyle \left\|(a_{n})_{n=1}^{\infty }\right\|_{\infty }=\sup _{n\in \mathbb {N} }|a_{n}|}$ 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 $\lim _{n\to \infty }a_{n}=0$, 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 $1=w_{1}\geq w_{2}\geq w_{3}\geq \cdots$, and define the norm ${\textstyle \left\|(a_{n})_{n=1}^{\infty }\right\|_{d(w,p)}=\sup _{\sigma \in \Pi }\left\|(a_{\sigma (n)}w_{n}^{1/p})_{n=1}^{\infty }\right\|_{p}}$. 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 $1<p<\infty$ 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 $q<r$. The Lorentz spaces are real interpolation spaces between $L^{1}$ and $L^{\infty }$.

Hölder's inequality

$\|fg\|_{L^{p,q}}\leq A_{p_{1},p_{2},q_{1},q_{2}}\|f\|_{L^{p_{1},q_{1}}}\|g\|_{L^{p_{2},q_{2}}}$ where $0<p,p_{1},p_{2}<\infty$, $0<q,q_{1},q_{2}\leq \infty$, $1/p=1/p_{1}+1/p_{2}$, and $1/q=1/q_{1}+1/q_{2}$.

Dual space

If $(X,\mu )$ is a nonatomic σ-finite measure space, then
(i) $(L^{p,q})^{*}=\{0\}$ for $0<p<1$, or $1=p<q<\infty$;
(ii) $(L^{p,q})^{*}=L^{p',q'}$ for $1<p<\infty ,0<q\leq \infty$, or $0<q\leq p=1$;
(iii) $(L^{p,\infty })^{*}\neq \{0\}$ for $1\leq p\leq \infty$.
Here $p'=p/(p-1)$ for $1<p<\infty$, $p'=\infty$ for $0<p\leq 1$, and $\infty '=1$.

Atomic decomposition

The following are equivalent for $0<p\leq \infty ,1\leq q\leq \infty$.
(i) $\|f\|_{L^{p,q}}\leq A_{p,q}C$.
(ii) $f=\textstyle \sum _{n\in \mathbb {Z} }f_{n}$ where $f_{n}$ has disjoint support, with measure $\leq 2^{n}$, on which $0<H_{n+1}\leq |f_{n}|\leq H_{n}$ almost everywhere, and $\|H_{n}2^{n/p}\|_{\ell ^{q}(\mathbb {Z} )}\leq A_{p,q}C$.
(iii) $|f|\leq \textstyle \sum _{n\in \mathbb {Z} }H_{n}\chi _{E_{n}}$ almost everywhere, where $\mu (E_{n})\leq A_{p,q}'2^{n}$ and $\|H_{n}2^{n/p}\|_{\ell ^{q}(\mathbb {Z} )}\leq A_{p,q}C$.
(iv) $f=\textstyle \sum _{n\in \mathbb {Z} }f_{n}$ where $f_{n}$ has disjoint support $E_{n}$, with nonzero measure, on which $B_{0}2^{n}\leq |f_{n}|\leq B_{1}2^{n}$ almost everywhere, $B_{0},B_{1}$ are positive constants, and $\|2^{n}\mu (E_{n})^{1/p}\|_{\ell ^{q}(\mathbb {Z} )}\leq A_{p,q}C$.
(v) $|f|\leq \textstyle \sum _{n\in \mathbb {Z} }2^{n}\chi _{E_{n}}$ almost everywhere, where $\|2^{n}\mu (E_{n})^{1/p}\|_{\ell ^{q}(\mathbb {Z} )}\leq A_{p,q}C$.

See also

• Interpolation space
• Hardy–Littlewood inequality

References

• Grafakos, Loukas (2008), Classical Fourier analysis, Graduate Texts in Mathematics, vol. 249 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09432-8, ISBN 978-0-387-09431-1, MR 2445437.