| Title:
|
A class of pairs of weights related to the boundedness of the Fractional Integral Operator between $L^p$ and Lipschitz spaces (English) |
| Author:
|
Pradolini, Gladis |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
133-152 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In [P] we characterize the pairs of weights for which the fractional integral operator $I_{\gamma}$ of order $\gamma$ from a weighted Lebesgue space into a suitable weighted $BMO$ and Lipschitz integral space is bounded. In this paper we consider other weighted Lipschitz integral spaces that contain those defined in [P], and we obtain results on pairs of weights related to the boundedness of $I_{\gamma}$ acting from weighted Lebesgue spaces into these spaces. Also, we study the properties of those classes of weights and compare them with the classes given in [P]. Then, under additional assumptions on the weights, we obtain necessary and sufficient conditions for the boundedness of $I_{\gamma}$ between $BMO$ and Lipschitz integral spaces. For the boundedness between Lipschitz integral spaces we obtain sufficient conditions. (English) |
| Keyword:
|
two-weighted inequalities |
| Keyword:
|
fractional integral |
| Keyword:
|
weighted Lebesgue spaces |
| Keyword:
|
\newline weighted Lipschitz spaces |
| Keyword:
|
weighted BMO spaces. |
| MSC:
|
42B25 |
| MSC:
|
47B38 |
| MSC:
|
47G10 |
| idZBL:
|
Zbl 1055.42015 |
| idMR:
|
MR1825378 |
| . |
| Date available:
|
2009-01-08T19:08:50Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119229 |
| . |
| Reference:
|
[CF] Coifman R., Fefferman C.: Weighted norm inequalities for maximal functions and singular integrals.Studia Math. 51 (1974), 241-250. Zbl 0291.44007, MR 0358205 |
| Reference:
|
[HL] Hardy G., Littlewood J.: Some properties of fractional integrals.Math. Z. 27 (1928), 565-606. MR 1544927 |
| Reference:
|
[HSV] 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:
|
[MW1] Muckenhoupt B., Wheeden R.: Weighted norm inequalities for fractional integral.Trans. Amer. Math. Soc. 192 (1974), 261-274. MR 0340523 |
| Reference:
|
[MW2] Muckenhoupt B., Wheeden R.: Weighted bounded mean oscillation and Hilbert transform.Studia Math. T. LIV, pp.221-237, 1976. MR 0399741 |
| Reference:
|
[Pe] Peetre, J.: On the theory of ${\Cal L}_{p,\lambda }$ spaces.J. Funct. Anal. 4 (1969), 71-87. |
| Reference:
|
[P] Pradolini G.: Two-weighted norm inequalities for the fractional integral operator between $L^p$ and Lipschitz spaces.to appear in Comment. Math. Polish Acad. Sci. MR 1876717 |
| Reference:
|
[S] Sobolev S.L.: On a theorem in functional analysis.Math. Sb. 4 (46) (1938), 471-497; English transl.: Amer. Math. Soc. Transl. (2) 34 (1963), 39-68. |
| Reference:
|
[SWe] Stein E., Weiss G.: Fractional integrals on n-dimensional euclidean space.J. Math. Mech. 7 (1958), 503-514; MR 20#4746. Zbl 0082.27201, MR 0098285 |
| Reference:
|
[WZ] Wheeden R., Zygmund A.: Measure and Integral. An Introduction to Real Analysis.Marcel Dekker Inc, 1977. Zbl 0362.26004, MR 0492146 |
| . |
