Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T05:59:30.144Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDEDNESS OF GENERALIZED RIESZ POTENTIALS ON THE VARIABLE HARDY SPACES

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  14 August 2017

PABLO ROCHA
PABLO ROCHA*
Affiliation:
Universidad Nacional del Sur, INMABB (Conicet), (8000) Bahía Blanca, Buenos Aires, Argentina email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the boundedness from $H^{p(\cdot )}(\mathbb{R}^{n})$ into $L^{q(\cdot )}(\mathbb{R}^{n})$ of certain generalized Riesz potentials and the boundedness from $H^{p(\cdot )}(\mathbb{R}^{n})$ into $H^{q(\cdot )}(\mathbb{R}^{n})$ of the Riesz potential, both results are achieved via the finite atomic decomposition developed in Cruz-Uribe and Wang [‘Variable Hardy spaces’, Indiana University Mathematics Journal63(2) (2014), 447–493].

Keywords

variable Hardy spacesfractional operators

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 2 , April 2018 , pp. 255 - 273
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Capone, C., Cruz-Uribe, D. and Fiorenza, A., ‘The fractional maximal operator and fractional integrals on variable L p spaces’, Rev. Mat. Iberoam. 23(3) (2007), 743–770.Google Scholar
Cruz-Uribe, D. and Fiorenza, A., Variable Lebesgue Spaces, Foundations and Harmonic Analysis (Birkhäuser, Basel, 2013).CrossRefGoogle Scholar
Cruz-Uribe, D., Martell, J. M. and Pérez, C., Weights, Extrapolation and the Theory of Rubio de Francia (Birkhäuser, Basel, 2011).Google Scholar
Cruz-Uribe, D. and Wang, D., ‘Variable Hardy spaces’, Indiana Univ. Math. J. 63(2) (2014), 447–493.Google Scholar
Diening, L., ‘Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces’, Bull. Sci. Math. 129(8) (2005), 657–700.Google Scholar
Diening, L., Harjulehto, P., Hästö, P. and Ruzicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017 (Springer, Berlin–Heidelberg, 2011).CrossRefGoogle Scholar
Fefferman, C. and Stein, E. M., ‘ H p spaces of several variables’, Acta Math. 129(3–4) (1972), 137–193.CrossRefGoogle Scholar
García-Cuerva, J. and Rubio de Francia, J. L., Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, New York, Oxford, 1985).Google Scholar
de Guzmán, M., Real Variable Methods in Fourier Analysis (North-Holland, Amsterdam–New York–Oxford, 1981).Google Scholar
Lerner, A. K., Ombrosi, S. and Pérez, C., ‘Sharp A1 bounds for Calderón–Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden’, Int. Math. Res. Not. IMRN 6 (2008), Article ID rnm161, 11 pp.Google Scholar
Lu, S. Z., Four Lectures on Real H p Spaces (World Scientific Publishing Co. Pte. Ltd, Singapore, 1995).CrossRefGoogle Scholar
Muckenhoupt, B., ‘Weighted norm inequalities for the Hardy maximal function’, Trans. Amer. Math. Soc. 165 (1972), 207–226.Google Scholar
Muckenhoupt, B. and Wheeden, R. L., ‘Weighted norm inequalities for fractional integrals’, Trans. Amer. Math. Soc. 192 (1974), 261–274.Google Scholar
Nakai, E., ‘Recent topics of fractional integrals’, Sugaku Expositions 20(2) (2007), 215–235.Google Scholar
Nakai, E. and Sawano, Y., ‘Hardy spaces with variable exponents and generalized Campanato spaces’, J. Funct. Anal. 262 (2012), 3665–3748.Google Scholar
Riveros, S. and Urciuolo, M., ‘Weighted inequalities for integral operators with some homogeneous kernels’, Czechoslovak Math. J. 55(2) (2005), 423–432.Google Scholar
Rocha, P. and Urciuolo, M., ‘On the H p - L q boundedness of some fractional integral operators’, Czechoslovak Math. J. 62(3) (2012), 625–635.Google Scholar
Rocha, P. and Urciuolo, M., ‘Fractional type integral operators on variable Hardy spaces’, Acta Math. Hungar. 143(2) (2014), 502–514.CrossRefGoogle Scholar
Sawano, Y., ‘Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators’, Integral Equations Operator Theory 77 (2013), 123–148.Google Scholar
Stein, E. M., Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, NJ, 1993).Google Scholar
Strömberg, J. O. and Torchinsky, A., Weighted Hardy Spaces, Lecture Notes in Mathematics, 131 (Springer, Berlin, 1989).Google Scholar
Taibleson, M. H. and Weiss, G., ‘The molecular characterization of certain Hardy spaces’, Astérisque 77 (1980), 67–149.Google Scholar