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 where and. Thus, when q < \infty, and, when q = \infty, It is also conventional to set.

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, can be defined as where d_{f} is the so-called distribution function of f, given by Here, for notational convenience, is defined to be \infty. The two functions |f| and f^{\ast} are equimeasurable, meaning that 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 Given these definitions, for and, the Lorentz quasinorms are given by

Lorentz sequence spaces

When (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 (or in the complex case), let denote the p-norm for and 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, 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 be a sequence of positive real numbers satisfying, and define the norm. 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.

Properties

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

where, , , and.

Dual space

If (X,\mu) is a nonatomic σ-finite measure space, then (i) for 0<p<1, or ; (ii) for, or 0<q\le p=1; (iii) for. 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. (i). (ii) where f_n has disjoint support, with measure \le2^n, on which almost everywhere, and. (iii) almost everywhere, where and. (iv) where f_n has disjoint support E_n, with nonzero measure, on which almost everywhere, B_0,B_1 are positive constants, and. (v) almost everywhere, where.