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

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

where $0<p<\mathrm{\infty }$ and $0<q\le \mathrm{\infty }$. Thus, when $q<\mathrm{\infty }$,

$\Vert f{\Vert }_{L^{p,q}(X,\mu )}=p^{\frac{1}{q}}{\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}{t}^q\mu {\{x:|f(x)|\ge t\}}^{\frac{q}{p}}\frac{dt}{t}\right)}^{\frac{1}{q}}={\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}\right(\tau \mu \{x:|f(x){|}^p\ge \tau \}{)}^{\frac{q}{p}}\frac{d\tau }{\tau })}^{\frac{1}{q}}.$

and, when $q=\mathrm{\infty }$,

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

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

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, $f^{\ast }:[0,\mathrm{\infty })\to [0,\mathrm{\infty }]$ can be defined as

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

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 $\mathrm{\infty }$.

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

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

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

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

Given these definitions, for $0<p<\mathrm{\infty }$ and $0<q\le \mathrm{\infty }$, the Lorentz quasinorms are given by

$\Vert f{\Vert }_{L^{p,q}}=\{\begin{array}{ll}{\left(\underset{0}{\overset{\mathrm{\infty }}{\int }}{(t^{\frac{1}{p}}{f}^{\ast }(t))}^q\frac{dt}{t}\right)}^{\frac{1}{q}}& q\in (0,\mathrm{\infty }),\\ \sup {}_{t>0}{t}^{\frac{1}{p}}{f}^{\ast }(t)& q=\mathrm{\infty }.\end{array}$

Lorentz sequence spaces

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

Definition.

For $(a_n{)}_{n=1}^{\mathrm{\infty }}\in {\mathbb{ℝ}}^{\mathbb{\mathbb{N}}}$ (or ${\mathbb{ℂ}}^{\mathbb{\mathbb{N}}}$ in the complex case), let ${\Vert (a_n{)}_{n=1}^{\mathrm{\infty }}\Vert }_p={({\sum }_{n=1}^{\mathrm{\infty }}|a_n{|}^p)}^{1/p}$ denote the p-norm for $1\le p<\mathrm{\infty }$ and ${\Vert (a_n{)}_{n=1}^{\mathrm{\infty }}\Vert }_{\mathrm{\infty }}=\underset{n\in \mathbb{\mathbb{N}}}{\sup }|a_n|$ 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 $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=0$, 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 $w=(w_n{)}_{n=1}^{\mathrm{\infty }}\in c_0\setminus {\ell }_1$ be a sequence of positive real numbers satisfying $1=w_1\ge w_2\ge w_3\ge \cdots$, and define the norm ${\Vert (a_n{)}_{n=1}^{\mathrm{\infty }}\Vert }_{d(w,p)}=\underset{\sigma \in \mathrm{\Pi }}{\sup }{\Vert (a_{\sigma (n)}{w}_n^{1/p}{)}_{n=1}^{\mathrm{\infty }}\Vert }_p$. 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 $\Vert \cdot {\Vert }_{d(w,p)}$.

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, $L^{p,\mathrm{\infty }}$ coincides with weak $L^p$. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for $1<p<\mathrm{\infty }$ and $1\le q\le \mathrm{\infty }$. 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 $L^{1,\mathrm{\infty }}$, the weak $L^1$ space. As a concrete example that the triangle inequality fails in $L^{1,\mathrm{\infty }}$, consider

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

whose $L^{1,\mathrm{\infty }}$ 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 $L^{\mathrm{\infty }}$.

Hölder's inequality

$\Vert fg{\Vert }_{L^{p,q}}\le A_{p_1,p_2,q_1,q_2}\Vert f{\Vert }_{L^{p_1,q_1}}\Vert g{\Vert }_{L^{p_2,q_2}}$ where $0<p,p_1,p_2<\mathrm{\infty }$, $0<q,q_1,q_2\le \mathrm{\infty }$, $1/p=1/p_1+1/p_2$, 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 $1=p<q<\mathrm{\infty }$;
(ii) $(L^{p,q}{)}^{*}=L^{p^{\prime },q^{\prime }}$ for $1<p<\mathrm{\infty },0<q\le \mathrm{\infty }$, or $0<q\le p=1$;
(iii) $(L^{p,\mathrm{\infty }}{)}^{*}\ne \{0\}$ for $1\le p\le \mathrm{\infty }$. Here $p^{\prime }=p/(p-1)$ for $1<p<\mathrm{\infty }$, $p^{\prime }=\mathrm{\infty }$ for $0<p\le 1$, and ${\mathrm{\infty }}^{\prime }=1$.

Atomic decomposition

The following are equivalent for $0<p\le \mathrm{\infty },1\le q\le \mathrm{\infty }$.
(i) $\Vert f{\Vert }_{L^{p,q}}\le A_{p,q}C$.
(ii) $f=\sum _{n\in \mathbb{\mathbb{Z}}}{f}_n$ where $f_n$ has disjoint support, with measure $\le 2^n$, on which $0<H_{n+1}\le |f_n|\le H_n$ almost everywhere, and $\Vert H_n{2}^{n/p}{\Vert }_{{\ell }^q(\mathbb{\mathbb{Z}})}\le A_{p,q}C$.
(iii) $|f|\le \sum _{n\in \mathbb{\mathbb{Z}}}{H}_n{\chi }_{E_n}$ almost everywhere, where $\mu (E_n)\le A_{p^{\prime },q}{2}^n$ and $\Vert H_n{2}^{n/p}{\Vert }_{{\ell }^q(\mathbb{\mathbb{Z}})}\le A_{p,q}C$.
(iv) $f=\sum _{n\in \mathbb{\mathbb{Z}}}{f}_n$ where $f_n$ has disjoint support $E_n$, with nonzero measure, on which $B_0{2}^n\le |f_n|\le B_1{2}^n$ almost everywhere, $B_0,B_1$ are positive constants, and $\Vert 2^n\mu (E_n{)}^{1/p}{\Vert }_{{\ell }^q(\mathbb{\mathbb{Z}})}\le A_{p,q}C$.
(v) $|f|\le \sum _{n\in \mathbb{\mathbb{Z}}}{2}^n{\chi }_{E_n}$ almost everywhere, where $\Vert 2^n\mu (E_n{)}^{1/p}{\Vert }_{{\ell }^q(\mathbb{\mathbb{Z}})}\le A_{p,q}C$.

See also

• Interpolation space
• Hardy–Littlewood inequality

References

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