Abstract
We introduce weighted generalized Grand Morrey spaces and prove that the boundedness of linear operators from the generalized Grand Lebesgue spaces to generalized Morrey spaces may be derived from their boundedness from classical weighted Lebesgue spaces into weighted Morrey spaces. As an application we prove a theorem on mapping properties of the Riesz potential operator from weighted generalized Grand Lebesgue spaces to weighted generalized Grand Morrey spaces with Muckenhoupt–Wheeden A p,q-weights, under some natural assumptions on the way how we generalize grand spaces.
Mathematics Subject Classification (2010). Primary 46E15; Secondary 42B35.
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References
D.R. Adams, A note on Riesz potentials. Duke Math. J. 42(4) (1975), 765–778.
C. Capone and A. Fiorenza, On small Lebesgue spaces. J. Funct. Spaces Appl. 3 (2005), 73–89.
G. Di Fazio and M.A. Ragusa, Commutators and Morrey spaces. Bollettino U.M.I. 7(5-A) (1991), 323–332.
G. Di Fratta and A. Fiorenza, A direct approach to the duality of grand and small Lebesgue spaces. Nonlinear Anal. 70(7) (2009), 2582–2592,
Y. Ding and C.-C. Lin, Two-weight norm inequalities for the rough fractional integrals. Int. J. Math. Math. Sci. 25(8) (2001), 517–524.
A. Fiorenza, B. Gupta, and P. Jain, The maximal theorem in weighted grand Lebesgue spaces. Studia Math. 188(2) (2008), 123–133.
M. Giaquinta, Multiple integrals in the calculus of variations and non-linear elliptic systems. Princeton Univ. Press, 1983.
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. 2nd Edition, Springer-Verlag, Berlin, 1983.
L. Greco, T. Iwaniec, and C. Sbordone, Inverting the p-harmonic operator. Manuscripta Math. 92 (1997), 249–258.
T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119 (1992), 129–143.
V. Kokilashvili, Boundedness criterion for the Cauchy singular integral operator in weighted grand Lebesgue spaces and application to the Riemann problem. Proc. A. Razmadze Math. Inst. 151 (2009), 129–133.
V. Kokilashvili, The Riemann boundary value problem for analytic functions in the frame of grand L p) spaces. Bull. Georgian Natl. Acad. Sci. 4(1) (2010), 5–7.
V. Kokilashvili and A. Meskhi, A note on the boundedness of the Hilbert transform in weighted grand Lebesgue spaces. Georgian Math. J. 16(3) (2009), 547–551.
V. Kokilashvili, A. Meskhi, and H. Rafeiro, Riesz type potential operators in generalized grand Morrey spaces. GeorgianMath. J. 20(1) (2013), 43–64.
V. Kokilashvili and S. Samko, Boundedness of weighted singular integral operators on a Carleson curves in Grand Lebesgue spaces. In ICNAAM 2010: Intern. Conf. Numer. Anal. Appl. Math., AIP Confer. Proc., volume 1281 (2010), 490–493.
V. Kokilashvili and S. Samko, Boundedness of weighted singular integral operators in Grand Lebesgue spaces. Georgian Math. J. 18(2) (2011), 259–269.
A. Meskhi, Criteria for the boundedness of potential operators in grand Lebesgue spaces, arXiv:1007.1185.
A. Meskhi, Maximal functions and singular integrals in Morrey spaces associated with grand Lebesgue spaces. Proc. A. Razmadze Math. Inst. 151 (2009), 139–143.
A. Meskhi, Integral operators in maximal functions, potentials and singular integrals in grand Morrey spaces. Complex Var. Elliptic Equ., Doi.10.1080/ 17476933.2010.534793, 2011, 1–19.
C.B. Morrey. On the solutions of quasi-linear elliptic partial differential equations. Amer. Math. Soc. 43 (1938), 126–166.
B. Muckenhoupt and R.L. Wheeden, Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192 (1974), 261–274.
J. Peetre, On the theory of \( \mathcal{L}_{p, \lambda} \) spaces. J. Funct. Anal. 4 (1969), 71–87.
H. Rafeiro, A note on boundedness of operators in grand grand Morrey spaces. Operator Theory: Advances and Applications 229 (2013), 349-356.
N.G. Samko, Weighted Hardy and potential operators in Morrey spaces. J. Funct. Spaces Appl. 2012 (2012), Article ID 678171, doi: 10.1155/2012/678171.
N.G. Samko, Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350(1) (2009), 56–72.
S.G. Samko and S.M. Umarkhadzhiev, On Iwaniec–Sbordone spaces on sets which may have infinite measure. Azerb. J. Math. 1(1) (2011), 67–84.
Y. Sawano and H. Tanaka, Morrey spaces for non-doubling measures. Acta Math. Sin. (Engl. Ser.) 21(6) (2005), 1535–1544.
E.M. Stein and G. Weiss, Interpolation of operators with change of measures. Trans. Amer. Math. Soc. 87 (1958), 159–172.
S.M. Umarkhadzhiev, Riesz–Thorin–Stein–Weiss Interpolation Theorem in a Lebesgue–Morrey Setting. Operator Theory: Advances and Applications 229 (2013), 387–392.
S.M. Umarkhadzhiev, A generalization of the notion of Grand Lebesgue space. Russian Math. (Iz. VUZ), to appear.
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Umarkhadzhiev, S. (2014). The Boundedness of the Riesz Potential Operator from Generalized Grand Lebesgue Spaces to Generalized Grand Morrey Spaces. In: Bastos, M., Lebre, A., Samko, S., Spitkovsky, I. (eds) Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, vol 242. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0816-3_22
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DOI: https://doi.org/10.1007/978-3-0348-0816-3_22
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