Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials
Yoshihiro MIZUTA, Eiichi NAKAI, Takao OHNO, and Tetsu SHIMOMURA
Source: J. Math. Soc. Japan Volume 62, Number 3
(2010), 707-744.
Abstract
Our aim in this paper is to deal with boundedness of fractional integral operators on Morrey spaces L(1,φ)(G) and the Sobolev embeddings for generalized Riesz potentials. Target spaces are Orlicz-Morrey, Orlicz-Campanato, and generalized Hölder spaces.
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References
D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765–778.
Mathematical Reviews (MathSciNet): MR458158
Zentralblatt MATH: 0336.46038
Digital Object Identifier: doi:10.1215/S0012-7094-75-04265-9
Project Euclid: euclid.dmj/1077311348
F. Chiarenza and M. Frasca, Morrey spaces and Hardy-Littlewood maximal function, Rend. Mat. Appl., 7(7) (1987), 273–279.
Mathematical Reviews (MathSciNet): MR985999
Zentralblatt MATH: 0717.42023
A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc., 60 (1999), 187–202.
Mathematical Reviews (MathSciNet): MR1721824
Zentralblatt MATH: 0940.46015
Digital Object Identifier: doi:10.1112/S0024610799007711
D. E. Edmunds, P. Gurka and B. Opic, Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J., 44 (1995), 19–43.
Mathematical Reviews (MathSciNet): MR1336431
Zentralblatt MATH: 0826.47021
Digital Object Identifier: doi:10.1512/iumj.1995.44.1977
Eridani, On the boundedness of a generalized fractional integral on generalized Morrey spaces, Tamkang J. Math., 33 (2002), 335–340.
Mathematical Reviews (MathSciNet): MR1936313
Zentralblatt MATH: 1026.42016
Eridani, H. Gunawan and E. Nakai, On generalized fractional integral operators, Sci. Math. Jpn., 60 (2004), 539–550.
Mathematical Reviews (MathSciNet): MR2099586
H. Gunawan, A note on the generalized fractional integral operators, J. Indones. Math. Soc., 9(1) (2003), 39–43.
Mathematical Reviews (MathSciNet): MR2013135
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415–426.
Mathematical Reviews (MathSciNet): MR131498
Zentralblatt MATH: 0102.04302
Digital Object Identifier: doi:10.1002/cpa.3160140317
K. Kurata, S. Nishigaki and S. Sugano, Boundedness of integral operators on generalized Morrey spaces and its application to Schrödinger operators, Proc. Amer. Math. Soc., 128 (2000), 1125–1134.
Mathematical Reviews (MathSciNet): MR1646196
Digital Object Identifier: doi:10.1090/S0002-9939-99-05208-9
JSTOR: links.jstor.org
Y. Mizuta, E. Nakai, T. Ohno and T. Shimomura, An elementary proof of Sobolev embeddings for Riesz potentials of functions in Morrey spaces $L^{1,\nu,\beta}(G)$, Hiroshima Math. J., 38 (2008), 461–472.
Mathematical Reviews (MathSciNet): MR2477751
Zentralblatt MATH: 1175.31005
Project Euclid: euclid.hmj/1233152779
Y. Mizuta and T. Shimomura, Exponential integrability for Riesz potentials of functions in Orlicz classes, Hiroshima Math. J., 28 (1998), 355–371.
Mathematical Reviews (MathSciNet): MR1637338
Zentralblatt MATH: 0917.31004
Project Euclid: euclid.hmj/1206126767
C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126–166.
E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr., 166 (1994), 95–103.
Mathematical Reviews (MathSciNet): MR1273325
Zentralblatt MATH: 0837.42008
Digital Object Identifier: doi:10.1002/mana.19941660108
E. Nakai, On generalized fractional integrals, Taiwanese J. Math., 5 (2001), 587–602.
Mathematical Reviews (MathSciNet): MR1849780
Zentralblatt MATH: 0990.26007
E. Nakai, On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type, Sci. Math. Jpn., 54 (2001), 473–487.
Mathematical Reviews (MathSciNet): MR1874169
Zentralblatt MATH: 1007.42013
E. Nakai, On generalized fractional integrals on the weak Orlicz spaces, $\BMO_{\varphi}$, the Morrey spaces and the Campanato spaces, Function spaces, interpolation theory and related topics (Lund, 2000), de Gruyter, Berlin, 2002, 389–401.
Mathematical Reviews (MathSciNet): MR1943296
Zentralblatt MATH: 1021.42006
E. Nakai, Generalized fractional integrals on Orlicz-Morrey spaces, Banach and Function Spaces (Kitakyushu, 2003), Yokohama Publishers, Yokohama, 2004, 323–333.
Mathematical Reviews (MathSciNet): MR2146936
E. Nakai, Orlicz-Morrey spaces and the Hardy-Littlewood maximal function, Studia Math., 188 (2008), 193–221.
Mathematical Reviews (MathSciNet): MR2429821
Zentralblatt MATH: 1163.46020
Digital Object Identifier: doi:10.4064/sm188-3-1
E. Nakai and H. Sumitomo, On generalized Riesz potentials and spaces of some smooth functions, Sci. Math. Jpn., 54 (2001), 463–472.
Mathematical Reviews (MathSciNet): MR1874168
Zentralblatt MATH: 0995.43004
P. A. Olsen, Fractional integration, Morrey spaces and a Schrödinger equation, Comm. Partial Differential Equations, 20 (1995), 2005–2055.
Mathematical Reviews (MathSciNet): MR1361729
Digital Object Identifier: doi:10.1080/03605309508821161
R. O'Neil, Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc., 115 (1965), 300–328.
Mathematical Reviews (MathSciNet): MR194881
Zentralblatt MATH: 0132.09201
J. Peetre, On the theory of $\mathscr L_{p,\lambda}$ spaces, J. Funct. Anal., 4 (1969), 71–87.
J. Serrin, A remark on Morrey potential, Contemporary Math., 426 (2007), 307–315.
Mathematical Reviews (MathSciNet): MR2311532
Zentralblatt MATH: 1129.31003
S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 593–608.
Mathematical Reviews (MathSciNet): MR190729
S. Sugano and H. Tanaka, Boundedness of fractional integral operators on generalized Morrey spaces, Sci. Math. Jpn. Online, 8 (2003), 233–242.
Mathematical Reviews (MathSciNet): MR2017532
Zentralblatt MATH: 1058.42009
N. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473–483.
Mathematical Reviews (MathSciNet): MR216286
Zentralblatt MATH: 0163.36402
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