ON THE BOUNDEDNESS OF THE MAXIMAL OPERATOR AND RIESZ POTENTIAL IN THE MODIFIED MORREY SPACES

Year 2014, Volume: 63 Issue: 2, 1 - 11, 01.08.2014

Canay Aykol M. Esra Yıldırım

https://doi.org/10.1501/Commua1_0000000707

Keywords

Morrey space, modified Morrey space, maximal operator, fractionalmaximal operator, Riesz potential

References

• [1] D.R. Adams, A note on Riesz potentials, Duke Math. 42 (1975), 765-778.
• [2] D.R. Adams, Choquet integrals in potential theory, Publ. Mat. 42 (1998), 3-66.
• [3] V.I. Burenkov and H.V. Guliyev, Necessary and su¢ cient conditions for boundedness of the maximal operator in the local Morrey-type spaces, Studia Mathematica 163, 2 (2004), 157-176.
• [4] V.I. Burenkov and V.S. Guliyev, Necessary and su¢ cient conditions for the boundedness of the Riesz operator in local Morrey-type spaces, Potential Analysis 30, 3 (2009), 211-249.
• [5] L. Ca§arelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1990), 253-284.
• [6] F. Chiarenza and M. Frasca Morrey spaces and Hardy- Littlewood maximal function, Rend. Math. 7 (1987), 273-279.
• [7] G. Di Fazio and M. A. Ragusa Commutators and Morrey spaces, Bollettino U.M.I., 7 (5- A)(1991), 323-332.
• [8] G. Di Fazio, D.K. Palagachev and M.A. Ragusa, Glocal Morrey regularity of strong solutions to the Dirichlet problemfor elliptic equations with discontinuous coe¢ cients, J. Funct. Anal. 166 (1999), 179-196.
• [9] V.S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009, Art ID 503948, 20 pp.
• [10] V.S. Guliyev, On maximal function and fractional integral, associated with the Bessl differentai operator, Mathematical Inequalities and Applications 6, 2(2003), 317-330.
• [11] V.S. Guliyev and J. Hasanov, Necessary and su¢ cient conditions for the boundedness of BRiesz potential in the B-Morrey spaces, Journal of Mathematical Analysis and Applications 347, 1 (2008), 113-122.
• [12] V.S. Guliyev, J. Hasanov and Y. Zeren, Necessary and su¢ cient conditions for the boundedness of B-Riesz potential in modiÖed Morrey spaces, Journal of Mathematical Inequalities 5, 4 (2011), 491-506.
• [13] V.S. Guliyev and Y.Y. Mammadov, Riesz potential on the Heisenberg group and modiÖed Morrey spaces, An. St. Univ. Ovidius Constanta 20(1) (2012), 189-212.
• [14] V.S. Guliyev and K. Rahimova, Parabolic fractional maximal operator and modiÖed parabolic Morrey spaces, Journal of Function Spaces and Applications, 2012, Article ID 543475, 20 pages, 2012.
• [15] V.S. Guliyev and K. Rahimova Parabolic fractional integral operator in modiÖed parabolic Morrey spaces, Proc. Razmadze Mathematical Institute, 163 (2013), 85-106.
• [16] V. Kokilashvili, A. Meskhi and H. Rafeiro, Riesz type potential operators in generalized grand Morrey spaces, Georgian Math. J. 20 (2013), no. 1, 43-64.
• [17] A. Kufner, O. John and S. Fucik, Function spaces, Noordho§, Leyden and Academia, Prague,1977.
• [18] A.L. Mazzucato, Besov-Morrey spaces: function theory and applications to non-linear PDE. Trans. Amer. Math. Soc. 355 (2003), 1297-1364.
• [19] C.B. Morrey, On the solutions of quasi-linear elliptic partial di§ erential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166.
• [20] B. Muckenhoupt and R. Whedeen, Weighted norm inequalities for fractional integrals, Trans. Amer.Math. Soc. 192 (1974), 261-274.
• [21] A. Ruiz and L. Vega, Unique continuation for Schrˆdinger operators with potential in Morrey spaces, Publ. Mat. 35 (1991), 291-298.
• [22] A. Ruiz and L. Vega, On local regularity of Schrˆdinger equations, Int. Math. Res. Notices 1993, 1 (1993), 13-27.
• [23] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton Univ. Press (1971).
• [24] E.M. Stein, Singular integrals and di§ erentiability properties of functions. Princeton Math. Ser. 30. Princeton University Press, Princeton(1971).
• [25] M.E. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evolution equations, Comm. Partial Di§erential Equations 17(1992), 1407-1456.