| Title:
|
Relations between weighted Orlicz and $BMO_\phi$ spaces through fractional integrals (English) |
| Author:
|
Harboure, E. |
| Author:
|
Salinas, O. |
| Author:
|
Viviani, B. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
1 |
| Year:
|
1999 |
| Pages:
|
53-69 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We characterize the class of weights, invariant under dilations, for which a modified fractional integral operator $I_\alpha $ maps weak weighted Orlicz$-\phi $ spaces into appropriate weighted versions of the spaces $BMO_\psi $, where $\psi (t)=t^{\alpha /n}\phi ^{-1}(1/t)$. This generalizes known results about boundedness of $I_\alpha $ from weak $L^p$ into Lipschitz spaces for $p>n/\alpha $ and from weak $L^{n/\alpha }$ into $BMO$. It turns out that the class of weights corresponding to $I_\alpha $ acting on weak$-L_\phi $ for $\phi $ of lower type equal or greater than $n/\alpha $, is the same as the one solving the problem for weak$-L^p$ with $p$ the lower index of Orlicz-Maligranda of $\phi $, namely $\omega ^{p'}$ belongs to the $A_1$ class of Muckenhoupt. (English) |
| Keyword:
|
theory of weights |
| Keyword:
|
Orlicz spaces |
| Keyword:
|
$BMO$ spaces |
| Keyword:
|
fractional integrals |
| MSC:
|
26A33 |
| MSC:
|
42B25 |
| MSC:
|
46E30 |
| MSC:
|
47G10 |
| idZBL:
|
Zbl 1060.46509 |
| idMR:
|
MR1715202 |
| . |
| Date available:
|
2009-01-08T18:49:41Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119063 |
| . |
| Reference:
|
[B] Boyd D.W.: Indices of function spaces and their relationship to interpolation.Canadian J. Math. 21 (1969), 1245-1254. Zbl 0184.34802, MR 0412788 |
| Reference:
|
[GP] Gustavson J., Peetre J.: Interpolation of Orlicz spaces.Studia Math. 60 (1997), 33-59. MR 0438102 |
| Reference:
|
[HSV1] Harboure E., Salinas O., Viviani B.: Acotación de la integral Fraccionaria en espacios de Orlicz y de Oscilación media $\phi $-Acotada''.Actas del 2$^{do}$ Congreso Dr. A. Monteiro, Bahía Blanca, 1993, pp.41-50. MR 1253076 |
| Reference:
|
[HSV2] Harboure E., Salinas O., Viviani B.: Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces.Trans. Amer. Math. Soc. 349 (1997), 235-255. Zbl 0865.42017, MR 1357395 |
| Reference:
|
[KK] Kokilashvili V., Krbec M.: Weighted inequalities in Lorentz and Orlicz spaces.World Scientific (1991). Zbl 0751.46021, MR 1156767 |
| Reference:
|
[MW] Muckenhoupt B., Wheeden R.: Weighted norm inequalities for fractional integrals.Trans. Amer. Math. Soc. 192 (1974), 261-274. Zbl 0289.26010, MR 0340523 |
| Reference:
|
[RR] Rao M.M., Ren Z.D.: Theory of Orlicz Spaces.M. Dekker, Inc., New York, 1991. Zbl 0724.46032, MR 1113700 |
| . |
