On the predual of a Morrey-Lorentz space and its applications to the linear Calderón-Zygmund operators (Q6557104)

For \(0< s\), \(p \le \infty\), the norm of the Lorentz space \(L^{s,p}\) is defined by\N\[\N\| f\|_{L^{s,p} } = \begin{cases} \left( \int_0^\infty \left( t^{1/s} f^\ast (\alpha) \right)^p \frac{d \alpha}{\alpha} \right)^{1/p} & \text{if}\ p < \infty , \\\N\sup_{ \alpha >0 } \alpha ^{1/s} f^\ast (\alpha) & \text{if}\ p =\infty, \end{cases}\N\]\Nwhere \(f^\ast \) is the decreasing rearrangement of \(f\).\N\NFor \(p>1,\) \(1\le r\le \infty\) and \(\alpha \in (0,n/p]\) the Morrey-Lorentz space \(L_\alpha ^{p,r}\) is the set of all measurable functions such that\N\[\N\| f\|_{ L_\alpha ^{p,r} } =\sup_{\operatorname{Ball} \; B} |B|^{\alpha/n -1/p} \| f \chi_B \|_{L^{s,p} } <\infty.\N\]\N\NFor \( 1<q <\infty \), \(1\le r \le \infty\), and \(\beta >0\), a function \(b\) is called a \((q,r, \beta)\)-block if supp \(b \subset B\) and \( \|b\|_{L^{q,r} } \le |B|^{1/q -\beta/n}\).\N\NThe function space \(\mathcal{B} _{\beta} ^{q,r} (\mathbb{R}^n)\) is the set of all measurable functions \(f\) such that \(f\) is realized as the sum\N\[\Nf=\sum_{j\in\mathbb{N}}\lambda_{j}g_{j}\N\]\Nwith some \(\lambda=\{\lambda_{j}\}_{j\in\mathbb{N}}\in\ell^{1}(\mathbb{N})\) and \(g_{j}\) is a \((q,r, \beta)\)-block, where the convergence occurs in \(L_{\mathrm{loc}}^{1}.\) The norm of \(\mathcal{B} _{\beta} ^{q,r} (\mathbb{R}^n)\) is defined by\N\[\N\|f\|_{ \mathcal{B} _{\beta} ^{q,r} (\mathbb{R}^n)}:=\inf_{\lambda}\|\lambda\|_{\ell^{1}},\N\]\Nwhere the infimum is taken over all admissible sequences \(\lambda\).\N\NIn this paper, the authors study the block space \(\mathcal{B} _{\beta} ^{q,r} (\mathbb{R}^n)\), which is the predual of the Morrey-Lorentz space. They obtain the boundedness of the powered Hardy-Littlewood maximal function on the Morrey-Lorentz space. Using duality and the Fefferman-Stein inequality, the mapping property of some powered Hardy-Littlewood maximal function on the block space \(\mathcal{B} _{\beta} ^{q,r} (\mathbb{R}^n)\) is obtained. They also obtain a Morrey-Lorentz bound for the sharp maximal function. As corollaries, they show the boundedness of linear Calderón-Zygmund operators and their commutator with BMO functions on Morrey-Lorentz spaces and an inequality of Minkowski type on Morrey-Lorentz spaces.\N\NThe authors extend the Hardy factorization in terms of linear Calderón-Zygmund operators to Morrey-Lorentz spaces. The compactness characterization of the commutators generated by Calderón-Zygmund operators and BMO functions in Morrey-Lorentz spaces is obtained. Finally, they show that the block space \(\mathcal{B} _{\beta} ^{q,r} (\mathbb{R}^n)\) has the Fatou property.