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.


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.)