Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,^[1][2] are generalisations of the more familiar [math]\displaystyle{ L^{p} }[/math] spaces. The Lorentz spaces are denoted by [math]\displaystyle{ L^{p,q} }[/math]. Like the [math]\displaystyle{ L^{p} }[/math] spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the [math]\displaystyle{ L^{p} }[/math] 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 [math]\displaystyle{ L^{p} }[/math] norms, by exponentially rescaling the measure in both the range ([math]\displaystyle{ p }[/math]) and the domain ([math]\displaystyle{ q }[/math]). The Lorentz norms, like the [math]\displaystyle{ L^{p} }[/math] norms, are invariant under arbitrary rearrangements of the values of a function.

Definition

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

[math]\displaystyle{ \|f\|_{L^{p,q}(X,\mu)} = p^{\frac{1}{q}} \left \|t\mu\{|f|\ge t\}^{\frac{1}{p}} \right \|_{L^q \left (\mathbf{R}^+, \frac{dt}{t} \right)} }[/math]

where [math]\displaystyle{ 0 \lt p \lt \infty }[/math] and [math]\displaystyle{ 0 \lt q \leq \infty }[/math]. Thus, when [math]\displaystyle{ q \lt \infty }[/math],

[math]\displaystyle{ \|f\|_{L^{p,q}(X,\mu)}=p^{\frac{1}{q}}\left(\int_0^\infty t^q \mu\left\{x : |f(x)| \ge t\right\}^{\frac{q}{p}}\,\frac{dt}{t}\right)^{\frac{1}{q}} = \left(\int_0^\infty \bigl(\tau \mu\left\{x : |f(x)|^p \ge \tau \right\}\bigr)^{\frac{q}{p}}\,\frac{d\tau}{\tau}\right)^{\frac{1}{q}} . }[/math]

and, when [math]\displaystyle{ q = \infty }[/math],

[math]\displaystyle{ \|f\|_{L^{p,\infty}(X,\mu)}^p = \sup_{t\gt 0}\left(t^p\mu\left\{x : |f(x)| \gt t \right\}\right). }[/math]

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

Decreasing rearrangements

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

[math]\displaystyle{ f^{\ast}(t) = \inf \{\alpha \in \mathbf{R}^{+}: d_f(\alpha) \leq t\} }[/math]

where [math]\displaystyle{ d_{f} }[/math] is the so-called distribution function of [math]\displaystyle{ f }[/math], given by

[math]\displaystyle{ d_f(\alpha) = \mu(\{x \in X : |f(x)| \gt \alpha\}). }[/math]

Here, for notational convenience, [math]\displaystyle{ \inf \varnothing }[/math] is defined to be [math]\displaystyle{ \infty }[/math].

The two functions [math]\displaystyle{ |f| }[/math] and [math]\displaystyle{ f^{\ast} }[/math] are equimeasurable, meaning that

[math]\displaystyle{ \mu \bigl( \{ x \in X : |f(x)| \gt \alpha\} \bigr) = \lambda \bigl( \{ t \gt 0 : f^{\ast}(t) \gt \alpha\} \bigr), \quad \alpha \gt 0, }[/math]

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

[math]\displaystyle{ \mathbf{R} \ni t \mapsto \tfrac{1}{2} f^{\ast}(|t|). }[/math]

Given these definitions, for [math]\displaystyle{ 0 \lt p \lt \infty }[/math] and [math]\displaystyle{ 0 \lt q \leq \infty }[/math], the Lorentz quasinorms are given by

[math]\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 \gt 0} \, t^{\frac{1}{p}} f^{\ast}(t) & q = \infty. \end{cases} }[/math]

Lorentz sequence spaces

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

Definition.

For [math]\displaystyle{ (a_n)_{n=1}^\infty\in\mathbb{R}^\mathbb{N} }[/math] (or [math]\displaystyle{ \mathbb{C}^\mathbb{N} }[/math] in the complex case), let [math]\displaystyle{ \left\|(a_n)_{n=1}^\infty\right\|_p = \left(\sum_{n=1}^\infty|a_n|^p\right)^{1/p} }[/math] denote the p-norm for [math]\displaystyle{ 1\leq p\lt \infty }[/math] and [math]\displaystyle{ \left\|(a_n)_{n=1}^\infty\right\|_\infty = \sup_{n\in\N}|a_n| }[/math] the ∞-norm. Denote by [math]\displaystyle{ \ell_p }[/math] the Banach space of all sequences with finite p-norm. Let [math]\displaystyle{ c_0 }[/math] the Banach space of all sequences satisfying [math]\displaystyle{ \lim_{n\to\infty}a_n=0 }[/math], endowed with the ∞-norm. Denote by [math]\displaystyle{ c_{00} }[/math] 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 [math]\displaystyle{ d(w,p) }[/math] below.

Let [math]\displaystyle{ w=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1 }[/math] be a sequence of positive real numbers satisfying [math]\displaystyle{ 1 = w_1 \geq w_2 \geq w_3 \geq \cdots }[/math], and define the norm [math]\displaystyle{ \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 }[/math]. The Lorentz sequence space [math]\displaystyle{ d(w,p) }[/math] is defined as the Banach space of all sequences where this norm is finite. Equivalently, we can define [math]\displaystyle{ d(w,p) }[/math] as the completion of [math]\displaystyle{ c_{00} }[/math] under [math]\displaystyle{ \|\cdot\|_{d(w,p)} }[/math].

Properties

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

[math]\displaystyle{ f(x) = \tfrac{1}{x} \chi_{(0,1)}(x)\quad \text{and} \quad g(x) = \tfrac{1}{1-x} \chi_{(0,1)}(x), }[/math]

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

The space [math]\displaystyle{ L^{p,q} }[/math] is contained in [math]\displaystyle{ L^{p, r} }[/math] whenever [math]\displaystyle{ q \lt r }[/math]. The Lorentz spaces are real interpolation spaces between [math]\displaystyle{ L^{1} }[/math] and [math]\displaystyle{ L^{\infty} }[/math].

Hölder's inequality

[math]\displaystyle{ \|fg\|_{L^{p,q}}\le A_{p_1,p_2,q_1,q_2}\|f\|_{L^{p_1,q_1}}\|g\|_{L^{p_2,q_2}} }[/math] where [math]\displaystyle{ 0\lt p,p_1,p_2\lt \infty }[/math], [math]\displaystyle{ 0\lt q,q_1,q_2\le\infty }[/math], [math]\displaystyle{ 1/p=1/p_1+1/p_2 }[/math], and [math]\displaystyle{ 1/q=1/q_1+1/q_2 }[/math].

Dual space

If [math]\displaystyle{ (X,\mu) }[/math] is a nonatomic σ-finite measure space, then
(i) [math]\displaystyle{ (L^{p,q})^*=\{0\} }[/math] for [math]\displaystyle{ 0\lt p\lt 1 }[/math], or [math]\displaystyle{ 1=p\lt q\lt \infty }[/math];
(ii) [math]\displaystyle{ (L^{p,q})^*=L^{p',q'} }[/math] for [math]\displaystyle{ 1\lt p\lt \infty,0\lt q\le\infty }[/math], or [math]\displaystyle{ 0\lt q\le p=1 }[/math];
(iii) [math]\displaystyle{ (L^{p,\infty})^*\ne\{0\} }[/math] for [math]\displaystyle{ 1\le p\le\infty }[/math]. Here [math]\displaystyle{ p'=p/(p-1) }[/math] for [math]\displaystyle{ 1\lt p\lt \infty }[/math], [math]\displaystyle{ p'=\infty }[/math] for [math]\displaystyle{ 0\lt p\le1 }[/math], and [math]\displaystyle{ \infty'=1 }[/math].

Atomic decomposition

The following are equivalent for [math]\displaystyle{ 0\lt p\le\infty, 1\le q\le\infty }[/math].
(i) [math]\displaystyle{ \|f\|_{L^{p,q}}\le A_{p,q}C }[/math].
(ii) [math]\displaystyle{ f=\textstyle\sum_{n\in\mathbb{Z}}f_n }[/math] where [math]\displaystyle{ f_n }[/math] has disjoint support, with measure [math]\displaystyle{ \le2^n }[/math], on which [math]\displaystyle{ 0\lt H_{n+1}\le|f_n|\le H_n }[/math] almost everywhere, and [math]\displaystyle{ \|H_n2^{n/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C }[/math].
(iii) [math]\displaystyle{ |f|\le\textstyle\sum_{n\in\mathbb{Z}}H_n\chi_{E_n} }[/math] almost everywhere, where [math]\displaystyle{ \mu(E_n)\le A_{p,q}'2^n }[/math] and [math]\displaystyle{ \|H_n2^{n/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C }[/math].
(iv) [math]\displaystyle{ f=\textstyle\sum_{n\in\mathbb{Z}}f_n }[/math] where [math]\displaystyle{ f_n }[/math] has disjoint support [math]\displaystyle{ E_n }[/math], with nonzero measure, on which [math]\displaystyle{ B_02^n\le|f_n|\le B_12^n }[/math] almost everywhere, [math]\displaystyle{ B_0,B_1 }[/math] are positive constants, and [math]\displaystyle{ \|2^n\mu(E_n)^{1/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C }[/math].
(v) [math]\displaystyle{ |f|\le\textstyle\sum_{n\in\mathbb{Z}}2^n\chi_{E_n} }[/math] almost everywhere, where [math]\displaystyle{ \|2^n\mu(E_n)^{1/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C }[/math].

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 .

Notes

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