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|\ge 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)| \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}} .$$

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 $\left\|(a_n)_{n=1}^\infty\right\|_p = \left(\sum_{n=1}^\infty|a_n|^p\right)^{1/p}$ denote the p-norm for $$1\leq p<\infty$$ and $\left\|(a_n)_{n=1}^\infty\right\|_\infty = \sup_{n\in\N}|a_n|$  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 $\left\|(a_n)_{n=1}^\infty\right\|_{d(w,p)} = \sup_{\sigma\in\Pi}\left\|(a_{\sigma(n)}w_n^{1/p})_{n=1}^\infty\right\|_p$. 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
$$\|fg\|_{L^{p,q}}\le A_{p_1,p_2,q_1,q_2}\|f\|_{L^{p_1,q_1}}\|g\|_{L^{p_2,q_2}}$$ where $$0<p,p_1,p_2<\infty$$, $$0<q,q_1,q_2\le\infty$$, $$1/p=1/p_1+1/p_2$$, 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 $$1<p<\infty,0<q\le\infty$$, or $$0<q\le p=1$$; (iii) $$(L^{p,\infty})^*\ne\{0\}$$ for $$1\le p\le\infty$$. Here $$p'=p/(p-1)$$ for $$1<p<\infty$$, $$p'=\infty$$ for $$0<p\le1$$, and $$\infty'=1$$.

Atomic decomposition
The following are equivalent for $$0<p\le\infty, 1\le q\le\infty$$.

(i) $$\|f\|_{L^{p,q}}\le A_{p,q}C$$.

(ii) $$f=\textstyle\sum_{n\in\mathbb{Z}}f_n$$ where $$f_n$$ has disjoint support, with measure $$\le2^n$$, on which $$0<H_{n+1}\le|f_n|\le H_n$$ almost everywhere, and $$\|H_n2^{n/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C$$.

(iii) $$|f|\le\textstyle\sum_{n\in\mathbb{Z}}H_n\chi_{E_n}$$ almost everywhere, where $$\mu(E_n)\le A_{p,q}'2^n$$ and $$\|H_n2^{n/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C$$.

(iv) $$f=\textstyle\sum_{n\in\mathbb{Z}}f_n$$ where $$f_n$$ has disjoint support $$E_n$$, with nonzero measure, on which $$B_02^n\le|f_n|\le B_12^n$$ almost everywhere, $$B_0,B_1$$ are positive constants, and $$\|2^n\mu(E_n)^{1/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C$$.

(v) $$|f|\le\textstyle\sum_{n\in\mathbb{Z}}2^n\chi_{E_n}$$ almost everywhere, where $$\|2^n\mu(E_n)^{1/p}\|_{\ell^q(\mathbb{Z})}\le A_{p,q}C$$.