Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, ^{[1] [2]} are generalisations of the more familiar ${\displaystyle L^{p}}$ spaces.

The Lorentz spaces are denoted by ${\displaystyle L^{p,q}}$. Like the ${\displaystyle L^{p}}$ spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the ${\displaystyle L^{p}}$ 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 ${\displaystyle L^{p}}$ norms, by exponentially rescaling the measure in both the range (${\displaystyle p}$) and the domain (${\displaystyle q}$). The Lorentz norms, like the ${\displaystyle L^{p}}$ norms, are invariant under arbitrary rearrangements of the values of a function.

Definition

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

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

where ${\displaystyle 0<p<\infty }$ and ${\displaystyle 0<q\leq \infty }$. Thus, when ${\displaystyle q<\infty }$,

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

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

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

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

Decreasing rearrangements

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

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

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

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

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

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

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

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

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

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

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

Lorentz sequence spaces

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

Definition.

For ${\displaystyle (a_{n})_{n=1}^{\infty }\in \mathbb {R} ^{\mathbb {N} }}$ (or ${\displaystyle \mathbb {C} ^{\mathbb {N} }}$ 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 ${\displaystyle 1\leq p<\infty }$ and ${\textstyle \left\|(a_{n})_{n=1}^{\infty }\right\|_{\infty }=\sup _{n\in \mathbb {N} }|a_{n}|}$ the ∞-norm. Denote by ${\displaystyle \ell _{p}}$ the Banach space of all sequences with finite p-norm. Let ${\displaystyle c_{0}}$ the Banach space of all sequences satisfying ${\displaystyle \lim _{n\to \infty }a_{n}=0}$, endowed with the ∞-norm. Denote by ${\displaystyle c_{00}}$ 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 ${\displaystyle d(w,p)}$ below.

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

Properties

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

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

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

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

Hölder's inequality

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

Dual space

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

Atomic decomposition

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

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 .

Notes

1. G. Lorentz, "Some new function spaces", Annals of Mathematics51 (1950), pp. 37-55.
2. G. Lorentz, "On the theory of spaces Λ", Pacific Journal of Mathematics1 (1951), pp. 411-429.