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Weighted norm inequalities for the Hardy-Littlewood maximal function
| Content Provider | Semantic Scholar |
|---|---|
| Author | Young, Wo Sang |
| Copyright Year | 1982 |
| Abstract | A characterization is obtained for weight functions V for which the Hardy-Littlewood maximal operator is bounded from l1I'(R", wdttx) to 1I)(Rfl, vd.'V) for sonme nontrivial wv. In this note we obtain a necessary and sufficient condition on weight functions v > 0 such that the Hardy-Littlewood maximal operator is bounded from LP(RW, wdx) to LP(R"', vdx) for some w 0 for which there are nontrivial v's was solved independently by J. L. Rubio de Francia [4] and L. Carleson and P. W. Jones [1]. Let M be the Hardy-Littlewood maximal operator defined by Mf(x) =sup I If Idy, r>O I B(x, r) B(x, r) where B(x, r) is the ball of radius r centered at x and I B(x, r) I is its Lebesgue measure. Our result is as follows. THEOREM. Given v > 0 and 1 < p < xo, the following conditions are equivalent: (a) There is w < ox a.e. such that J MfIlPvdx , Ci flfPwdx R" R~~~~~~~n for allf E LP(RW, wdx). (b) X (b)~ ~ v(x) pdx |
| Starting Page | 24 |
| Ending Page | 26 |
| Page Count | 3 |
| File Format | PDF HTM / HTML |
| Volume Number | 85 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1982-085-01/S0002-9939-1982-0647890-4/S0002-9939-1982-0647890-4.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1982-0647890-4 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
