×

Integral operators on weighted amalgams. (English) Zbl 0824.42015

The authors investigate boundedness conditions for the operator \[ Pf(x)= \int^ x_{- \infty} f \] considered as a map of weighted amalgams \(l^{q_ 1}(L^{p_ 1}_ v)\) into \(l^ q(L^ p_ u)\). The weighted amalgam spaces are defined by \[ l^ q(L^ p_ u)= \Biggl\{f: \| f\|_{p, u, q}= \Biggl( \sum_{n\in \mathbb{Z}} \Biggl( \int^{n+ 1}_ n | f(x)|^ p u(x) dx\Biggr)^{q/p}\Biggr)^{1/q}< +\infty\Biggr\}. \] The conditions on the weights are amalgam type analogues of the corresponding conditions for spaces with an absolutely continuous measure. Recall that for the Hardy operator to map \(L^{p_ 1}_ v\) into \(L^ p_ u\) with \(p_ 1\leq p\) the condition is \[ \sup_{y\in \mathbb{R}} \Biggl( \int^ \infty_ y u(x) dx\Biggr)^{1/p} \Biggl( \int^ y_{-\infty} v(x)^{1- p_ 1'} dx\Biggr)^{1/p_ 1'}= B_ 1< \infty. \] There is a more complicated condition which must be used when \(p_ 1> p\). The necessary and sufficient condition for the estimate \[ \| Pf\|_{p, u, q}\leq B\| f\|_{p_ 1, v, q_ 1} \] requires two conditions. The first condition can be described as follows: take the above condition on \(u\) and calculate instead the \(l^{q/p}\) norm of a sum of integrals over intervals of length one from \(m\) to \(\infty\) and proceed similarly for the condition on \(v^{1- p_ 1'}\) taking the \(l^{q_ 1'/p_ 1'}\) norm of a sum of integrals over intervals of length one from \(-\infty\) to \(m- 1\). Here \(p'\) denotes the conjugate index, \(p'= p/(p- 1)\). One condition is that the sup of appropriate powers of these sums over all choices of \(m\) be finite. A second condition requires that the integeral of \(u\) over fractional subintervals \([n+ \alpha, n+ 1]\) and of \(v^{1- p_ 1'}\) over \([n, n+ \alpha]\) to appropriate powers be bounded for all \(n\) and \(0< \alpha< 1\). The results are applied to the Hardy-Littlewood maximal operator. Here \[ Mf(x)= \sup_{h> 0} 1/2h \int^{x+ h}_{x- h} | f(x)| dx. \] They prove that if \(w\in A_ p\cap A_{2- (p/q)}\) and if \(w\in RH_{q/p}\), then \(M: l^ q(L^ p_ w)\to l^ q(L^ p_ w)\) for \(1< p< q< +\infty\). They prove that the condition \(w\in RH_{q/p}\) is optimal in the sense that there is a weight \(w\in A_ 1\) with \(w\in RH_{q/p- \varepsilon}\) and an \(f\in l^ q(L^ p_ w)\) for which \(Mf\equiv \infty\). The \(A_ p\) condition is sufficient for the above result when \(M\) is replaced by \(M_ 1\), the local maximal operator. In \(M_ 1\), the sup is taken over intervals of length one, \[ M_ 1 f(x)= \sup_{x\in I, | I|\leq 1} 1/| I| \int_ I | f(t)| dt. \] They prove that if \(w\in A_ p\), then \(M_ 1: l^ q(L^ p_ w)\to l^ q(L^ p_ w)\) for \(1< p< q< +\infty\).

MSC:

42B25 Maximal functions, Littlewood-Paley theory
26D10 Inequalities involving derivatives and differential and integral operators
46B45 Banach sequence spaces
46E40 Spaces of vector- and operator-valued functions
PDFBibTeX XMLCite
Full Text: DOI EuDML