## Lorentz spaces of weak-type.(English)Zbl 0943.42010

The weak-type Lorentz space $$\Lambda^{p,\infty}(w)$$, $$0<p<\infty$$, is defined as the space of measurable functions $$f$$ satisfying $$\|f\|_{\Lambda^{p,\infty}(w)}= \sup_{t>0} \{(\int_0^tw(s) ds)^{1/p} f^*(t)\}<\infty$$, where $$w$$ is a weight in $$(0,\infty)$$ and $$f^*$$ denotes the non-increasing rearrangement of $$f$$. The interest on introducing and studying these spaces is that they generalize and include classical Lorentz spaces as $$L^{p,\infty}$$ and $$L^{q,p}$$.
The problem, taken into consideration and solved by the author, is about the characterization of weights $$w$$ for which $$\Lambda^{p,\infty}(w)$$ becomes a Banach space. The main result, in Theorem 3.1, asserts the equivalence between the following facts:
1) $$\Lambda^{p,\infty}(w)$$ is a Banach space;
2) The quasi-norm $$\|\cdot\|_{\Lambda^{p,\infty}(w)}$$ is equivalent to the norm defined by $\|f\|_{\Lambda^{p,\infty}(w)}^*= \sup_{t>0}\left\{\Bigl(\int_0^tw(s) ds\Bigr)^{1/p}\Bigl(t^{-1}\int_0^t f^*(s) ds\Bigr)\right\};$
3) The Hardy-Littlewood maximal operator is bounded on $$\Lambda^{p,\infty}(w)$$;
4) The weight $$w$$ satisfies the growth condition $\int_0^R \Biggl(\int_0^rw(s) ds \Biggr)^{-1/p} dr\leq cR \Biggl(\int_0^Rw(s) ds \Biggr)^{-1/p} \quad\text{for all $$R>0$$;}$
5) The weight $$w$$ satisfies the $$B_p$$ condition $\int_\infty^R w(s)s^{-p} ds dr\leq cR^{-p}\int_0^R w(s) ds \quad\text{for all $$R>0$$.}$ In 4) and 5), $$c$$ is a fixed nonnegative constant.
Among the keys used in the proofs is the reduction of the problem to the boundedness of the Hardy operator $$(Hf)(t)=t^{-1}\int_0^tf(s)ds$$ from $$L^{p,\infty}_{\text{dec}}(w)$$ to $$L^{p,\infty}(w)$$, where $$L^{p,\infty}_{\text{dec}}(w)$$ denotes the class of non-increasing functions in $$L^{p,\infty}(w)$$. Actually, a characterization of the weights $$u$$ and $$v$$ for which $$H$$ is bounded from $$L^{p,\infty}_{\text{dec}}(v)$$ to $$L^{q,r}(u)$$ is obtained in Theorem 4.1.

### MSC:

 42B30 $$H^p$$-spaces 42B25 Maximal functions, Littlewood-Paley theory 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)