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mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, 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 Norm, the Norm or NORM may refer to: In academic disciplines * Normativity, phenomenon of designating things as good or bad * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative e ...
(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 A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(X, \mu) is the space of complex-valued
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s 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 In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
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 : \lambda \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 In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
. 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 In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
. 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 space In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
s 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 In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one ...
*
Hardy–Littlewood inequality In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean spa ...


References

*.


Notes

{{Functional analysis Banach spaces Lp spaces