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Recent developments in the theory of Lorentz spaces and weighted inequalities. (English) Zbl 1126.42005
Mem. Am. Math. Soc. 877, 128 p. (2007).
Let \(M\) be the Hardy-Littlewood maximal function. Then
\[ M \;\text{is bounded on} \;L^p(u)(\mathbb R^n), 1<p< \infty, \;\text{if and only if} \;u \in A_p(\mathbb R^n). \] Here \(L^p(u)\) is weighted \(L^p\) space and \(A_p\) is the Muckenhoupt weight class. Let \(\Lambda^p(w)(\mathbb R^n)\) be the generalized Lorentz space with weight \(w\); \[ \| f \|_{\Lambda^p(w)} = \biggl( \int_0^{\infty} ( f^{*}(t))^p w(t)\,dt \biggr)^{1/p}, \]
where \(f^{*}\) is the nonincreasing rearrangement; \( f^{*}(t)= \inf \{ s>0 ; \lambda_f (s) \leq t \} \) and \( \lambda_f(t) =\) \({| \{ x \in \mathbb R^n;}\) \(| f(x) | >t \} | \). Note that if \(w(t) = t^{p/q -1}\) then \(\Lambda^p(w) = L^{q,p}\), the classical Lorentz space. M. A. Ariño and B. Muckenhoupt [Trans. Am. Math. Soc. 320, No. 2, 727–735 (1990; Zbl 0716.42016)] proved the following.
\[ M \;\text{is bounded on} \;\Lambda^p(w)(\mathbb R^n), 1<p< \infty, \;\text{if and only if} \;w \in B_p(\mathbb R^{+}), \] where
\[ B_p = \biggl\{w;\;\int_r^{\infty} \biggl( \frac{r}{t} \biggr)^p w(t)\,dt \leq C \int_0^r w(t) \,dt \quad \text{for all} \quad r>0 \biggr\}. \]
The authors’ aim to give a unified and generalized version of these two theories. The following proposition is important for this purpose. Let \(A\) be the Hardy operator
\[ Af(t) = \frac{1}{t}\int_0^t f(s)\,ds, \quad t>0. \]
Then \( (Mf)^{*}(t) \approx A f^{*}(t). \) Therefore
\[ \int_0^{\infty} (Af(t))^p w(t) \,dt \leq C \int_0^{\infty} f(t)^p w(t) \,dt \quad \text{for any positive nonincreasing function} \quad f \]
if and only if \(w \in B_p\).
They define the following function space. Let
\[ \lambda_f^u (t) = \int_{ \{ x \in \mathbb R^n ;\;| f(x) | >t \} } u(x)\,dx \quad \text{and} \quad f_u^{*}(t)= \inf \{ s>0 ;\;\lambda_f^u (s) \leq t \} \]
\[ \Lambda_u^p(w)(R^n) = \biggl\{ f;\;\| f \|_{\Lambda_u^p(w)} = \biggl( \int_0^{\infty} ( f_u^{*}(t))^p w(t)\,dt \biggr)^{1/p} \biggr\}. \]
In Chapter 2, they study analytic properties of this function space, for example, normability, duality, interpolation theorem. One of main results of this article is the following.
If \(0<p< \infty\), the following results are equivalent:
(i) \(M\) is bounded on \(\Lambda_u^p(w)\).
(ii) \(\| M \chi_E \|_{\Lambda_u^p(w)} \leq C \| \chi_E \|_{\Lambda_u^p(w)}\), for any \(E \subset \mathbb R^n. \)
(iii) \(((M \chi_E)_u^{*}(t))^q \leq C \frac{W(u(E))}{W(t)}\) for any \(E \subset \mathbb R^n\), with \(0<q<p\),
where \(u(E) = \int_E u(x)dx\) and \(W(r) = \int_0^r w(t)\,dt\).
They also consider weak type estimates and the relation between \(A_p\) and \(B_p\).

42B25 Maximal functions, Littlewood-Paley theory
26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)
47G10 Integral operators
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
Zbl 0716.42016
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