Lorentz space

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' 1 (1951), pp. 411-429. are generalisations of the more familiar $L^$ spaces.

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

:$\, f\, _ = p^ \left \, t\mu\^ \right \, _$

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

:$\, f\, _=p^\left(\int_0^\infty t^q \mu\left\^\,\frac\right)^ = \left(\int_0^\infty \bigl(\tau \mu\left\\bigr)^\,\frac\right)^ .$

and, when $q = \infty$,

:$\, f\, _^p = \sup_\left(t^p\mu\left\\right).$

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

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^: , \infty) \to [0, \infty/math> can be defined as

:$f^(t) = \inf \$

where $d_$ is the so-called distribution function of $f$, given by

:$d_f(\alpha) = \mu(\).$

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

The two functions $, f,$ and $f^$ are equimeasurable, meaning that

:$\mu \bigl( \ \bigr) = \lambda \bigl( \ \bigr), \quad \alpha > 0,$

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

:$\mathbf \ni t \mapsto \tfrac f^(, t, ).$

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

:$\, f \, _ = \begin \left( \displaystyle \int_0^ \left (t^ f^(t) \right )^q \, \frac \right)^ & q \in (0, \infty), \\ \sup\limits_ \, t^ f^(t) & q = \infty. \end$

Lorentz sequence spaces

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

Definition.

For $(a_n)_^\infty\in\mathbb^\mathbb$ (or $\mathbb^\mathbb$ in the complex case), let $\left\, (a_n)_^\infty\right\, _p = \left(\sum_^\infty, a_n, ^p\right)^$ denote the p-norm for $1\leq p<\infty$ and $\left\, (a_n)_^\infty\right\, _\infty = \sup_, a_n,$ 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 $\lim_a_n=0$, endowed with the ∞-norm. Denote by $c_$ 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 $w=(w_n)_^\infty\in c_0\setminus\ell_1$ be a sequence of positive real numbers satisfying $1 = w_1 \geq w_2 \geq w_3 \geq \cdots$, and define the norm $\left\, (a_n)_^\infty\right\, _ = \sup_\left\, (a_w_n^)_^\infty\right\, _p$. 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 $c_$ under $\, \cdot\, _$.

Properties

The Lorentz spaces are genuinely generalisations of the $L^$ spaces in the sense that, for any $p$, $L^ = L^$, which follows from Cavalieri's principle. Further, $L^$ coincides with weak $L^$. 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 $p = 1$, $L^ = L^$ is equipped with a norm, but it is not possible to define a norm equivalent to the quasinorm of $L^$, the weak $L^$ space. As a concrete example that the triangle inequality fails in $L^$, consider

:$f(x) = \tfrac \chi_(x)\quad \text \quad g(x) = \tfrac \chi_(x),$

whose $L^$ quasi-norm equals one, whereas the quasi-norm of their sum $f + g$ equals four.

The space $L^$ is contained in $L^$ whenever $q < r$. The Lorentz spaces are real interpolation spaces between $L^$ and $L^$.

Hölder's inequality

$\, fg\, _\le A_\, f\, _\, g\, _$ where 0, 0, $1/p=1/p_1+1/p_2$, and $1/q=1/q_1+1/q_2$.

Dual space

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

Atomic decomposition

The following are equivalent for 0.

(i) $\, f\, _\le A_C$.

(ii) $f=\textstyle\sum_f_n$ where $f_n$ has disjoint support, with measure $\le2^n$, on which 0 almost everywhere, and $\, H_n2^\, _\le A_C$.

(iii) $, f, \le\textstyle\sum_H_n\chi_$ almost everywhere, where $\mu(E_n)\le A_'2^n$ and $\, H_n2^\, _\le A_C$.

(iv) $f=\textstyle\sum_f_n$ 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 $\, 2^n\mu(E_n)^\, _\le A_C$.

(v) $, f, \le\textstyle\sum_2^n\chi_$ almost everywhere, where $\, 2^n\mu(E_n)^\, _\le A_C$.

See also

* Interpolation space
* Hardy–Littlewood inequality

References

*.

Notes

{{Functional analysis

Banach spaces