×

Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces. (English) Zbl 0942.42011

Let \(M_{p,q}f(x)= \sup_{x\in Q}\|f\chi_Q\|_{p,q}/\|\chi_Q\|_{p,q}\), \(1\leq p<\infty\), \(1\leq q\leq\infty\), be the Hardy-Littlewood maximal functions associated with Lorentz spaces \(L(p,q)\), where \(\|.\|_{p,q}\) denote the \(L(p,q)\) norms. Because \(L(p,p)= L^p\), one has that \(M_{p,p}f\cong M_pf= (M|f|^p)^{1/p}\), where \(M_1f= Mf\) is the standard Hardy-Littlewood maximal function. Let \(f^*\) denote the non-increasing rearrangement of a function \(f\). The \(K\)-functional for a pair of Banach spaces \((X,Y)\) is defined by \(K(t,f; X,Y)= \inf\{\|u\|_X+ t\|v\|_Y\}\), \(t>0\), where the inf runs over all possible decompositions \(f= u+v\) with \(u\in X\) and \(v\in Y\). It is known that there are constants \(C\) and \(c\) such that \[ (M_pf)^*(t)\leq Ct^{-1/p} K(t^{1/p}, f; L^p,L^\infty),\tag{1} \]
\[ c(M_p f)^*(t)\geq t^{-1/p} K(t^{1/p},f; L^p,L^\infty).\tag{2} \] Thus a natural question is if similar relationships (1) and (2) exist between \((M_{p,q}f)^*(t)\) and the corresponding \(t^{-1/p}K(t^{1/p},f; L(p,q),L^\infty)\). In this reviewed paper, the authors obtain both positive and negative answers depending on \(p>q\) and \(p<q\).

MSC:

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.)
PDFBibTeX XMLCite
Full Text: DOI