Lorentz space

The Lorentz space on a measure space $(X,\; \backslash mu)$ is the space of complex-valued measurable functions $f$ on X such that the following quasinorm is finite

:$\backslash |f\backslash |\_\{L^\{p,q\}(X,\backslash mu)\}\; =\; p^\{\backslash frac\{1\}\{q\}\}\; \backslash left\; \backslash |t\backslash mu\backslash \{|f|\backslash ge\; t\backslash \}^\{\backslash frac\{1\}\{p\}\}\; \backslash right\; \backslash |\_\{L^q\; \backslash left\; (\backslash mathbf\{R\}^+,\; \backslash frac\{dt\}\{t\}\; \backslash right)\}$

where and . Thus, when ,

:$\backslash |f\backslash |\_\{L^\{p,q\}(X,\backslash mu)\}=p^\{\backslash frac\{1\}\{q\}\}\backslash left(\backslash int\_0^\backslash infty\; t^q\; \backslash mu\backslash left\backslash \{x\; :\; |f(x)|\; \backslash ge\; t\backslash right\backslash \}^\{\backslash frac\{q\}\{p\}\}\backslash ,\backslash frac\{dt\}\{t\}\backslash right)^\{\backslash frac\{1\}\{q\}\}$

= \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 $q\; =\; \backslash infty$,

:$\backslash |f\backslash |\_\{L^\{p,\backslash infty\}(X,\backslash mu)\}^p\; =\; \backslash sup\_\{t>0\}\backslash left(t^p\backslash mu\backslash left\backslash \{x\; :\; |f(x)|\; >\; t\; \backslash right\backslash \}\backslash right).$

It is also conventional to set $L^\{\backslash infty,\backslash infty\}(X,\; \backslash mu)\; =\; L^\{\backslash infty\}(X,\; \backslash mu)$.

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, $(X,\; \backslash mu)$, its decreasing rearrangement function, $f^\{\backslash ast\}:\; [0,\; \backslash infty)\; \backslash to\; [0,\; \backslash infty]$ can be defined as

:$f^\{\backslash ast\}(t)\; =\; \backslash inf\; \backslash \{\backslash alpha\; \backslash in\; \backslash mathbf\{R\}^\{+\}:\; d\_f(\backslash alpha)\; \backslash leq\; t\backslash \}$

where $d\_\{f\}$ is the so-called distribution function of $f$, given by

:$d\_f(\backslash alpha)\; =\; \backslash mu(\backslash \{x\; \backslash in\; X\; :\; |f(x)|\; >\; \backslash alpha\backslash \}).$

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

The two functions $|f|$ and $f^\{\backslash ast\}$ are equimeasurable, meaning that

:$\backslash lambda\; \backslash bigl(\; \backslash \{\; x\; \backslash in\; X\; :\; |f(x)|\; >\; \backslash alpha\backslash \}\; \backslash bigr)\; =\; \backslash lambda\; \backslash bigl(\; \backslash \{\; t\; >\; 0\; :\; f^\{\backslash ast\}(t)\; >\; \backslash alpha\backslash \}\; \backslash bigr),\; \backslash quad\; \backslash alpha\; >\; 0,$

where $\backslash lambda$ 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

:$\backslash mathbf\{R\}\; \backslash ni\; t\; \backslash mapsto\; \backslash tfrac\{1\}\{2\}\; f^\{\backslash ast\}(|t|).$

Given these definitions, for and , the Lorentz quasinorms are given by

:$\backslash |\; f\; \backslash |\_\{L^\{p,\; q\}\}\; =\; \backslash begin\{cases\}$

\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}

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

For $(a\_n)\_\{n=1\}^\backslash infty\backslash in\backslash mathbb\{R\}^\backslash mathbb\{N\}$ (or $\backslash mathbb\{C\}^\backslash mathbb\{N\}$ in the complex case), let $\backslash left\backslash |(a\_n)\_\{n=1\}^\backslash infty\backslash right\backslash |\_p\; =\; \backslash left(\backslash sum\_\{n=1\}^\backslash infty|a\_n|^p\backslash right)^\{1/p\}$ denote the p-norm for and $\backslash left\backslash |(a\_n)\_\{n=1\}^\backslash infty\backslash right\backslash |\_\backslash infty\; =\; \backslash sup\_\{n\backslash in\backslash N\}|a\_n|$ the ∞-norm. Denote by $\backslash ell\_p$ the Banach space of all sequences with finite p-norm. Let $c\_0$ the Banach space of all sequences satisfying $\backslash lim\_\{n\backslash to\backslash infty\}a\_n=0$, 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 $w=(w\_n)\_\{n=1\}^\backslash infty\backslash in\; c\_0\backslash setminus\backslash ell\_1$ be a sequence of positive real numbers satisfying $1\; =\; w\_1\; \backslash geq\; w\_2\; \backslash geq\; w\_3\; \backslash geq\; \backslash cdots$, and define the norm $\backslash left\backslash |(a\_n)\_\{n=1\}^\backslash infty\backslash right\backslash |\_\{d(w,p)\}\; =\; \backslash sup\_\{\backslash sigma\backslash in\backslash Pi\}\backslash left\backslash |(a\_\{\backslash sigma(n)\}w\_n^\{1/p\})\_\{n=1\}^\backslash infty\backslash right\backslash |\_p$. 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 $\backslash |\backslash cdot\backslash |\_\{d(w,p)\}$.

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, $L^\{p,\; \backslash infty\}$ coincides with weak $L^\{p\}$. They are quasi-Banach spaces (that is, quasi-normed spaces which are also complete) and are normable for and $1\; \backslash leq\; q\; \backslash leq\; \backslash infty$. 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 $L^\{1,\backslash infty\}$, the weak $L^\{1\}$ space. As a concrete example that the triangle inequality fails in $L^\{1,\backslash infty\}$, consider

:$f(x)\; =\; \backslash tfrac\{1\}\{x\}\; \backslash chi\_\{(0,1)\}(x)\backslash quad\; \backslash text\{and\}\; \backslash quad\; g(x)\; =\; \backslash tfrac\{1\}\{1-x\}\; \backslash chi\_\{(0,1)\}(x),$

whose $L^\{1,\backslash infty\}$ 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 . The Lorentz spaces are real interpolation spaces between $L^\{1\}$ and $L^\{\backslash infty\}$.

$\backslash |fg\backslash |\_\{L^\{p,q\}\}\backslash le\; A\_\{p\_1,p\_2,q\_1,q\_2\}\backslash |f\backslash |\_\{L^\{p\_1,q\_1\}\}\backslash |g\backslash |\_\{L^\{p\_2,q\_2\}\}$ where

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

The following are equivalent for

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

(ii) $f=\backslash textstyle\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}f\_n$ where $f\_n$ has disjoint support, with measure $\backslash le2^n$, on which

(iii) $|f|\backslash le\backslash textstyle\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}H\_n\backslash chi\_\{E\_n\}$ almost everywhere, where $\backslash mu(E\_n)\backslash le\; A\_\{p,q\}\text{'}2^n$ and $\backslash |H\_n2^\{n/p\}\backslash |\_\{\backslash ell^q(\backslash mathbb\{Z\})\}\backslash le\; A\_\{p,q\}C$.

(iv) $f=\backslash textstyle\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}f\_n$ where $f\_n$ has disjoint support $E\_n$, with nonzero measure, on which $B\_02^n\backslash le|f\_n|\backslash le\; B\_12^n$ almost everywhere, $B\_0,B\_1$ are positive constants, and $\backslash |2^n\backslash mu(E\_n)^\{1/p\}\backslash |\_\{\backslash ell^q(\backslash mathbb\{Z\})\}\backslash le\; A\_\{p,q\}C$.

(v) $|f|\backslash le\backslash textstyle\backslash sum\_\{n\backslash in\backslash mathbb\{Z\}\}2^n\backslash chi\_\{E\_n\}$ almost everywhere, where $\backslash |2^n\backslash mu(E\_n)^\{1/p\}\backslash |\_\{\backslash ell^q(\backslash mathbb\{Z\})\}\backslash le\; A\_\{p,q\}C$.

• Interpolation space
• Hardy–Littlewood inequality

• {{Citation|last1=Grafakos|first1=Loukas|title=Classical Fourier analysis | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-09431-1 | doi= 10.1007/978-0-387-09432-8 |mr=2445437 | year=2008 | volume=249}}.

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Category:Banach spaces

Category:Lp spaces