×
Sawano, Yoshihiro; Sugano, Satoko; Tanaka, Hitoshi

Orlicz-Morrey spaces and fractional operators. (English) Zbl 1242.42017

Potential Anal. 36, No. 4, 517-556 (2012).
The purpose of this paper is to study the boundedness properties of the generalized fractional integral operators on Orlicz-Morrey spaces.
Some Orlicz-Morrey norm inequalities (the trace inequality and the Olsen inequality) for the generalized fractional integral operators are established and the local boundedness property of the Orlicz maximal operators is also verified. Furthermore, some Morrey norm inequalities with small parameters for the generalized fractional integral operators are investigated.
Reviewer: Koichi Saka (Akita)

Cited in 1 Review
Cited in 97 Documents

MSC:

42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Orlicz-Morrey space; generalized fractional integral operator; generalized fractional maximal operator; Orlicz maximal operator; trace inequality, Olsen’s inequality; Wiener-Stein equivalence
Cite Review PDF
Full Text: DOI

References:

[1] Adams, D.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975) · Zbl 0336.46038 · doi:10.1215/S0012-7094-75-04265-9
[2] Adams, D., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53, 1629–1663 (2004) · Zbl 1100.31009
[3] Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted norm inequalities and related topics. In: Mathematics Student, vol. 116. North-Holland, Amsterdam (1985) · Zbl 0578.46046
[4] Gilbarg, D., Trudinger, S.N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983) · Zbl 0562.35001
[5] Kerman, R., Sawyer, E.: The trace inequality and eigenvalue estimates for Schrödinger operators. Ann. Inst. Fourier (Grenoble) 36, 207–228 (1986) · Zbl 0591.47037 · doi:10.5802/aif.1074
[6] Kita, H.: On maximal functions in Orlicz spaces. Proc. Am. Math. Soc. 124, 3019–3025 (1996) · Zbl 0862.42014 · doi:10.1090/S0002-9939-96-03807-5
[7] Kita, H.: On Hardy–Littlewood maximal functions in Orlicz spaces. Math. Nachr. 183, 135–155 (1997) · Zbl 0864.42007 · doi:10.1002/mana.19971830109
[8] Nakai, E.: On generalized fractional integrals. Taiwan. J. Math. 5, 587–602 (2001) · Zbl 0990.26007
[9] Nakai, E.: Generalized fractional integrals on Orlicz–Morrey spaces. In: Banach and Function Spaces, pp. 323–333. Yokohama Publishers, Yokohama (2004) · Zbl 1118.42005
[10] Nakai, E.: Orlicz–Morrey spaces and the Hardy–Littlewood maximal function. Stud. Math. 188, 193–221 (2008) · Zbl 1163.46020 · doi:10.4064/sm188-3-1
[11] Olsen, P.: Fractional integration, Morrey spaces and Schrödinger equation. Commun. Partial Differ. Equ. 20, 2005–2055 (1995) · Zbl 0838.35017 · doi:10.1080/03605309508821161
[12] Pérez, C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43, 663–683 (1994) · Zbl 0809.42007 · doi:10.1512/iumj.1994.43.43028
[13] Pérez, C.: Sharp L p -weighted Sobolev inequalities. Ann. Inst. Fourier (Grenoble) 45, 809–824 (1995) · Zbl 0820.42008 · doi:10.5802/aif.1475
[14] Rao, M.M., Ren, D.Z.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991) · Zbl 0724.46032
[15] Sawano, Y., Sobukawa, T., Tanaka, H.: Limiting case of the boundedness of fractional integral operators on nonhomogeneous space. J. Inequal. Appl. 16 pp. (2006). doi: 10.1155/JIA/2006/92470 · Zbl 1193.42087
[16] Sawano, Y., Sugano, S., Tanaka, H.: A note on generalized fractional integral operators on generalized Morrey spaces. Boundary Value Problems 2009, 18 pp. (2009). doi: 10.1155/2009/835865 · Zbl 1202.47056
[17] Sawano, Y., Sugano, S., Tanaka, H.: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. Trans. Am. Math. Soc. (2011, to appear) · Zbl 1229.42024
[18] Stein, M.E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) · Zbl 0207.13501
[19] Tanaka, H.: Morrey spaces and fractional operators. J. Aust. Math. Soc. 88, 247–259 (2010) · Zbl 1193.42095 · doi:10.1017/S1446788709000457
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.