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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
$\lambda_f^u (t) = \int_{ \{ x \in \mathbb R^n ;\;| f(x) | >t \} } u(x)\,dx \quad \text{and} \quad f_u^{*}(t)= \inf \{ s>0 ;\;\lambda_f^u (s) \leq t \}$
and
$\Lambda_u^p(w)(R^n) = \biggl\{ f;\;\| f \|_{\Lambda_u^p(w)} = \biggl( \int_0^{\infty} ( f_u^{*}(t))^p w(t)\,dt \biggr)^{1/p} \biggr\}.$
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