Abstract
Recently, we defined the d-modified Riesz potentials \({\tilde {I}}_{\alpha ,d}\) and proved several estimates of boundedness of \({\tilde {I}}_{\alpha ,d}\) on the central Morrey spaces \(B^{p,\lambda }(\mathbb {R}^n)\), using the central Campanato spaces \(\Lambda ^{(d)}_{p,\lambda }(\mathbb {R}^n)\), the generalized σ-Lipschitz spaces \(\mathrm {Lip}^{(d)}_{\beta ,\sigma }(\mathbb {R}^n)\) and so on. In this paper, we will consider the results of the boundedness for \({\tilde {I}}_{\alpha ,d}\) on the λ-central mean oscillation spaces \(\mathrm {CMO}^{p,\lambda }(\mathbb {R}^n)\).
Dedicated to Professor Lars-Erik Persson in celebration of his 75th birthday
This work was supported by Grant-in-Aid for Scientific Research (C) (Grant No. 17K05306), Japan Society for the Promotion of Science.
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References
J. Alvarez, M. Guzmán-Partida, J. Lakey, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect. Math. 51, 1–47 (2000)
Y. Chen, K. Lau, Some new classes of Hardy spaces. J. Funct. Anal. 84, 255–278 (1989)
H. Feichtinger, An elementary approach to Wiener’s third Tauberian theorem on Euclidean n-space, in Proceedings, Conference at Cortona 1984, Symposia Mathematica, vol. 29 (Academic Press, New York, 1987), pp. 267–301
Z. Fu, Y. Lin, S. Lu, λ-central BMO estimates for commutators of singular integral operators with rough kernels. Acta Math. Sin. (Engl. Ser.) 24, 373–386 (2008)
J. García-Cuerva, Hardy spaces and Beurling algebras. J. Lond. Math. Soc. 39, 499–513 (1989)
J. García-Cuerva, M.J.L. Herrero, A theory of Hardy spaces associated to the Herz spaces. Proc. Lon. Math. Soc. 69, 605–628 (1994)
G.H. Hardy, J.E. Littlewood, Some properties of fractional integrals. I. Math. Z. 27, 565–606 (1928); II, ibid. 34, 403–439 (1932)
C. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms. J. Math. Mech. 18, 283–324 (1968)
Y. Komori-Furuya, K. Matsuoka, Some weak-type estimates for singular integral operators on CMO spaces. Hokkaido Math. J. 39, 115–126 (2010)
Y. Komori-Furuya, K. Matsuoka, Strong and weak estimates for fractional integral operators on some Herz-type function spaces, in Proceedings of the Maratea Conference FAAT 2009, Rendiconti del Circolo Mathematico di Palermo, Serie II, Supplementary, vol. 82 (2010), pp. 375–385
Y. Komori-Furuya, K. Matsuoka, E. Nakai, Y. Sawano, Integral operators on B σ-Morrey–Campanato spaces. Rev. Mat. Complut. 26, 1–32 (2013)
Y. Komori-Furuya, K. Matsuoka, E. Nakai, Y. Sawano, Applications of Littlewood–Paley theory for \(\dot {B}_{\sigma }\)-Morrey spaces to the boundedness of integral operators. J. Funct. Spaces Appl. 2013, 859402 (2013)
T. Kurokawa, Riesz potentials, higher Riesz transforms and Beppo Levi spaces. Hiroshima Math. J. 18, 541–597 (1988)
T. Kurokawa, Weighted norm inequalities for Riesz potentials. Jpn. J. Math. 14, 261–274 (1988)
S. Lu, D. Yang, The central BMO spaces and Littlewood–Paley operators. Approx. Theory Appl. (N.S.) 11, 72–94 (1995)
K. Matsuoka, B σ-Morrey–Campanato estimates and some estimates for singular integrals on central Morrey spaces and λ-CMO spaces, in Banach and Function Spaces IV (Kitakyushu 2012) (Yokohama Publishers, Yokohama, 2014), pp. 325–335
K. Matsuoka, Generalized fractional integrals on central Morrey spaces and generalized λ-CMO spaces, in Function Spaces X, Banach Center Publications, vol. 102 (Institute of Mathematics, Polish Academy of Sciences, Warsawa, 2014), pp. 181–188
K. Matsuoka, Generalized fractional integrals on central Morrey spaces and generalized σ-Lipschitz spaces, in Current Trends in Analysis and its Applications: Proceedings of the 9th ISAAC Congress, Kraków 2013. Springer Proceedings in Mathematics and Statistics (Birkhäuser, Basel, 2015), pp. 179–189
K. Matsuoka, E. Nakai, Fractional integral operators on B p, λ with Morrey–Campanato norms, in Function Spaces IX, Banach Center Publishers, vol. 92 (Institute of Mathematics, Polish Academy of Sciences, Warsawa, 2011), pp. 249–264
Y. Mizuta, On the behaviour at infinity of superharmonic functions. J. Lond. Math. Soc. 27, 97–105 (1983)
Y. Mizuta, Potential Theory in Euclidean Spaces (Gakkōtosho, Tokyo, 1996)
E. Nakai, Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)
J. Peetre, On the theory of \(\mathcal {L}_{p,\lambda }\) spaces. J. Funct. Anal. 4, 71–87 (1969)
A. Pietsch, History of Banach Spaces and Linear Operators (Birkhäuser, Boston, 2007)
S.L. Sobolev, On a theorem in functional analysis. Mat. Sbornik 4, 471–497 (1938) (in Russian)
A. Zygmund, On a theorem of Marcinkiewicz concerning interpolation of operations. J. Math. Pures Appl. 35, 223–248 (1956)
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The author would like to express his deep gratitude to the anonymous referees for their careful reading and fruitful comments.
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Matsuoka, K. (2021). d-Modified Riesz Potentials on Central Campanato Spaces. In: Bastos, M.A., Castro, L., Karlovich, A.Y. (eds) Operator Theory, Functional Analysis and Applications. Operator Theory: Advances and Applications, vol 282. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51945-2_21
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