Abstract
We introduce two classes of Banach spaces \(F(q,p,\alpha )\) and \(T^{p,\;\alpha }\) (\(1\le q\le \alpha \le p\le \infty \)) which are subspaces of Herz spaces and Morrey spaces of functions and measures respectively. We study the Riesz potential operators in these spaces and obtain norm inequalities on these operators from which we deduce a necessary condition for a nonnegative Radon measure to have its Riesz potential in a given Lebesgue space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, D.R., Hedberg, L.I.: Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften, vol. 314. Springer, London, Berlin, Heidelberg, New York (1996)
Burenkov, V.I., Guliyev, V.S.: Necessary and sufficient conditions for the boundedness of the Riesz potential in Local Morrey-type spaces. Potential Anal. 30, 211–249 (2009). doi:10.1007/s11118-008-9113-5
Chinillo, S., Wheeden, R.L.: \(L^{p}\) estimates for fractionals integrals and Sobolev inequalities with applications to Schrödinger operator. Commun. Partial Differ. Equ. 10(9), 1077–1116 (1985)
Douyon, D., Fofana, I.: A Sobolev inequality for functions with locally bounded variation in \(\mathbb{R}^{d}\). Ann. Math. Afr. Sér. 1 1, 49–67 (2010)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. CRC Press, Boca Raton (1992)
Feuto, J., Fofana, I., Koua, K.: Weigthed norm inequalities for a maximal operator in some subspace of amalgams. Canad. Math. Bull. 53(2), 263–277 (2010)
Fofana, I.: Etude d’une classe d’espaces de fonctions contenant les espaces de Lorentz. Afrika Mat. (2) 1, 29–50 (1988)
Fofana, I.: Continuité de l’intégrale fractionnaire et espaces \((L^{q}, l^{p})^{\alpha }\), C.R.A.S. Paris, t.308, Série I, 525–527 (1989)
Fofana, I.: Espace \((L^{q}, l^{p})^{\alpha }(R^{d}; n)\) : espace de fonctions à moyenne fractionnaire intégrables, Thèse d’Etat es-Sciences; Université de Cocody (1995). http://greenstone.refer.bf/collect/thef/index/assoc/HASH6a39.dir/CSe02767
Fofana, I.: Espace \((L^{q}, l^{p})^{\alpha }\) et continuité de l’opérateur maximal fractionnaire de Hardy–Littlewood. Afrika Mat. Sér. 3 12, 23–37 (2001)
Fofana, I., Kpata, B.A., Douyon, D.: Some properties of Radon measures having their Riesz potential in a Lebesgue space, preprint (2011)
Grafakos, L.: Classical and modern Fourier analysis. Pearson Education, Inc., NJ, USA (2004)
Garcià-Cuerva, J., Gatto, A.E.: Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162(3), 245–261 (2004)
Giga, Y., Miyagawa, T.: Navier-Stokes flow in \(\mathbb{R}^{3}\) with measures as initial vorticity and Morrey spaces. Commun. Partial Differ. Equ. 14(5), 577–618 (1989)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis II. Springer, Berlin (1970)
Kikuchi, N., Nakai, E., Tomita, N., Yabuta, K., Yoneda, T.: Calderón-Zygmund operators on amalgam spaces and in the discrete case. J. Math. Anal. Appl. 335(1), 198–212 (2007)
Kpata, B.A., Fofana, I., Koua, K.: Necessary condition for measures which are \((L^{q}, l^{p})\) multipliers. Ann. Math. Blaise Pascal 16(2), 339–353 (2009)
Muckenhoupt, B., Wheeden, R.L.: Weighted norm inequalities for classical operators. Trans. Am. Math. Soc. 192, 261–277 (1974)
Nakai, E.: Hardy-Littlewood maximal operator, singular integral operutors and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)
Nakai, E.: Recent topics of fractional integral, Sugaku Expositions. Am. Math. Soc. 20(2), 215–235 (2007)
Phuc, N.C., Torres, M.: Characterizations of the existence and removable singularities of divergence-measure vector fields. Indiana Univ. Math. J. 57(4), 1573–1597 (2008)
Sawyer, E.T., Wheeden, R.L.: Weighted inequalities of fractional integrals on euclidean and homogeneous spaces. Am. J. Math. 114, 813–874 (1992)
Stein, E.M.: Singular integrals and Differentiability Properties of functions. Princeton University Press, Princeton, NJ (1970)
Acknowledgments
The authors would like to express their gratitude to the referees for their very valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fofana, I., Faléa, F.R. & Kpata, B.A. A class of subspaces of Morrey spaces and norm inequalities on Riesz potential operators. Afr. Mat. 26, 717–739 (2015). https://doi.org/10.1007/s13370-014-0241-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-014-0241-3