# Relations between weighted Orlicz and $BM{O}_{\phi}$ spaces through fractional integrals

Eleonor Ofelia Harboure; Oscar Salinas; Beatriz E. Viviani

Commentationes Mathematicae Universitatis Carolinae (1999)

- Volume: 40, Issue: 1, page 53-69
- ISSN: 0010-2628

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topHarboure, Eleonor Ofelia, Salinas, Oscar, and Viviani, Beatriz E.. "Relations between weighted Orlicz and $BMO_\phi $ spaces through fractional integrals." Commentationes Mathematicae Universitatis Carolinae 40.1 (1999): 53-69. <http://eudml.org/doc/248439>.

@article{Harboure1999,

abstract = {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 $\omega ^\{p^\{\prime \}\}$ belongs to the $A_1$ class of Muckenhoupt.},

author = {Harboure, Eleonor Ofelia, Salinas, Oscar, Viviani, Beatriz E.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {theory of weights; Orlicz spaces; $BMO$ spaces; fractional integrals; theory of weights; Orlicz spaces; BMO spaces; fractional integrals},

language = {eng},

number = {1},

pages = {53-69},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Relations between weighted Orlicz and $BMO_\phi $ spaces through fractional integrals},

url = {http://eudml.org/doc/248439},

volume = {40},

year = {1999},

}

TY - JOUR

AU - Harboure, Eleonor Ofelia

AU - Salinas, Oscar

AU - Viviani, Beatriz E.

TI - Relations between weighted Orlicz and $BMO_\phi $ spaces through fractional integrals

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 1999

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 40

IS - 1

SP - 53

EP - 69

AB - 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 $\omega ^{p^{\prime }}$ belongs to the $A_1$ class of Muckenhoupt.

LA - eng

KW - theory of weights; Orlicz spaces; $BMO$ spaces; fractional integrals; theory of weights; Orlicz spaces; BMO spaces; fractional integrals

UR - http://eudml.org/doc/248439

ER -

## References

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- 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. (1997) Zbl0865.42017MR1357395
- Kokilashvili V., Krbec M., Weighted inequalities in Lorentz and Orlicz spaces, World Scientific (1991). Zbl0751.46021MR1156767
- Muckenhoupt B., Wheeden R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. (1974) Zbl0289.26010MR0340523
- Rao M.M., Ren Z.D., Theory of Orlicz Spaces, M. Dekker, Inc., New York, 1991. Zbl0724.46032MR1113700

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