Lorentz space explained

In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,^{[1] [2]} 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

(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 ^{(1)/(q)} \left \|t\mu\{|f|\ge t\} ^{(1)/(p)} \right \| _{L ^{q} \left ( R ^{+,} (dt)/(t) \right)}

where

0 < p < inf ty and 0 < q \leq inf ty . Thus, when q < inf ty ,\|f\| _{L ^{p,q} (X,\mu)} = p ^{(1)/(q)} \left(\int ^{inf ty} _{0} t ^{q} \mu\left\{x : |f(x)| \ge t\right\} ^{(q)/(p)} (dt)/(t) \right) ^{(1)/(q)} = \left(\int ^{inf ty} _{0} l (\tau \mu\left\{x : |f(x)| ^{p} \ge \tau \right\} r ) ^{(q)/(p)} (d\tau)/(\tau) \right) ^{(1)/(q)} .

and, when

q = inf ty ,\|f\| ^{p} _{L ^{p,infty} (X,\mu)} = \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 , inf ty) \to [ 0 , inf ty] can be defined asf ^{\ast} (t) = inf \{\alpha \in R ^{+} : d _{f(\alpha)} \leq t\}

where

d _{f} is the so-called distribution function of f , given byd _{f(\alpha)} = \mu(\{x \in X : |f(x)| > \alpha\}).

Here, for notational convenience,

inf \varnothing is defined to be inf ty .

The two functions

|f| and f ^{\ast} are equimeasurable, meaning thatλ l ( \{ x \in X : |f(x)| > \alpha\} r ) = λ l ( \{ t > 0 : f ^{\ast} (t) > \alpha\} r ), \alpha > 0 ,

where

λ 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 byR \ni t \mapsto \tfrac{ 1 }{ 2 } f ^{\ast} (|t|).

Given these definitions, for

0 < p < inf ty and 0 < q \leq inf ty , the Lorentz quasinorms are given by\| f \| _{L ^{p,}} = \begin{cases} \left( \displaystyle \int ^{inf ty} _{0} \left (t ^{(1)/(p)} f ^{\ast} (t) \right ) ^{q} (dt)/(t) \right) ^{(1)/(q)} & q \in ( 0 , inf ty), \\ \sup\limits _{t} t ^{(1)/(p)} f ^{\ast} (t) & q = inf ty. \end{cases}

Lorentz sequence spaces

When

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

Definition.

For

(a _{n)} ^{inf ty\in R} ^{n =1} ^{N} (or C ^{N} in the complex case), let $\backslash left\backslash |(a\_n)\_^\backslash infty\backslash right\backslash |\_p\; =\; \backslash left(\backslash sum\_^\backslash infty|a\_n|^p\backslash right)^$ denote the p-norm for 1 \leq p< inf ty and $\backslash left\backslash |(a\_n)\_^\backslash infty\backslash right\backslash |\_\backslash infty\; =\; \backslash sup\_|a\_n|$ 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 _{n\toinfty} 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)} ^{inf ty\in} ^{n =1} c _{0\setminus\ell} _{1} be a sequence of positive real numbers satisfying 1 = w _{1} \geq w _{2} \geq w _{3} \geq … , and define the norm $\backslash left\backslash |(a\_n)\_^\backslash infty\backslash right\backslash |\_\; =\; \backslash sup\_\backslash left\backslash |(a\_w\_n^)\_^\backslash infty\backslash right\backslash |\_p$. 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 \| ⋅ \| _{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,} 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 < inf ty and 1 \leq q \leq inf ty . When p = 1 , L ^{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) and 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,} 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\leinfty} , 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< inf ty ;
(ii) (L ^{p,q} ) ^{*=L} ^{p',q'} for 1 <p< inf ty, 0 <q\le inf ty , or 0 <q\le p =1 ;
(iii) (L ^{p,infty} ) ^{*\ne\{0\}} for 1 \le p\le inf ty . Here p' = p/(p -1 ) for 1 <p< inf ty , p' =inf ty for 0 <p\le 1 , and inf ty' =1 .

Atomic decomposition

The following are equivalent for

0 <p\le inf ty, 1 \le q\le inf ty .
(i) \|f\| _{L ^{p,q}} \le A _{p,q} C .
(ii) f = style\sum _{n\inZ} f _{n} where f _{n} has disjoint support, with measure \le 2 ^{n} , on which 0 <H _{n+1} \le|f _{n|\le} H _{n} almost everywhere, and \|H ^{n/p} _{n 2} \| _{\ell ^{q(Z)}} \le A _{p,q} C .
(iii) |f|\le style\sum _{n\inZ} H _{n\chi} ^{E _{n}} almost everywhere, where \mu(E _{n)\le} A _{p,q} ' 2 ^{n} and \|H ^{n/p} _{n 2} \| _{\ell ^{q(Z)}} \le A _{p,q} C .
(iv) f = style\sum _{n\inZ} f _{n} where f _{n} has disjoint support E _{n} , with nonzero measure, on which B ^{n\le|f} _{n|\le} B ^{n} _{12} almost everywhere, B _{0,B} _{1} are positive constants, and \| 2 ^{1 /p} _{n)} \| _{\ell ^{q(Z)}} \le A _{p,q} C .
(v) |f|\le style\sum _{n\inZ} 2 ^{n\chi} _{E _{n}} almost everywhere, where \| 2 ^{1 /p} _{n)} \| _{\ell ^{q(Z)}} \le A _{p,q} C .

See also

• Interpolation space
• Hardy–Littlewood inequality

References

• .

Notes and References

1. G. Lorentz, "Some new function spaces", Annals of Mathematics 51 (1950), pp. 37-55.
2. G. Lorentz, "On the theory of spaces Λ", Pacific Journal of Mathematics 1 (1951), pp. 411-429.