Abstract
In this paper, we study Sobolev type inequalities for fractional maximal functions in the half space. We also discuss Sobolev type inequalities for Green potentials in the half space.
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References
Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)
Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15, 195–208 (2008)
Bojarski, B., Hajłasz, P.: Pointwise inequalities for Sobolev functions and some applications. Studia Math. 106(1), 77–92 (1993)
Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator and fractional integrals on variable \(L^p\) spaces. Rev. Mat. Iberoamericana 23(3), 743–770 (2007)
Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. 7, 273–279 (1987)
Di Fazio, G., Ragusa, M. A.: Commutators and Morrey Spaces. Boll. Un. Mat. Ital. (7) 5-A (1991), 323–332
Gardiner, S.J.: Growth properties of \(p\)th means of potentials in the unit ball. Proc. Am. Math. Soc. 103, 861–869 (1988)
Kinnunen, J., Saksman, E.: Regularity of the fractional maximal function. Bull. London. Math. Soc. 35, 529–535 (2003)
Kruglyak, N., Kuznetsov, E.A.: Sharp integral estimates for the fractional maximal function and interpolation. Ark. Mat. 44, 309–326 (2006)
Lewis, J. L.: On very weak solutions of certain elliptic systems. Comm. Partial Differential Equations 18(9) (10) , 1515–1537, (1993)
Maeda, F.-Y., Mizuta, Y., Ohno, T., Shimomura, T.: Boundedness of maximal operators and Sobolev’s inequality on Musielak-Orlicz-Morrey spaces. Bull. Sci. Math. 137, 76–96 (2013)
Mizuta, Y.: Potential theory in Euclidean spaces. Gakk\(\overline{{\rm o}}\)tosho, Tokyo, (1996)
Mizuta, Y.: Boundary limits of Green potentials of general order. Proc. Am. Math. Soc. 101(1), 131–135 (1987)
Mizuta, Y.: Growth properties of \(p\)th hyperplane means of Green potentials in a half space. J. Math. Soc. Japan 44(1), 1–12 (1992)
Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent. Complex Vari. Elliptic Equ. 56(7–9), 671–695 (2011)
Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s inequalities for Herz-Morrey-Orlicz spaces on the half space. Math. Ineq. Appl. 21, 433–453 (2018)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)
Peetre, J.: On the theory of \({\cal{L}}_{p,\lambda }\) spaces. J. Funct. Anal. 4, 71–87 (1969)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Stoll, M.: Boundary limits of Green potentials in the unit disc. Arch. Math. (Basel) 44(5), 451–455 (1985)
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Mizuta, Y., Shimomura, T. Sobolev type inequalities for fractional maximal functions and Green potentials in half spaces. Positivity 25, 1131–1146 (2021). https://doi.org/10.1007/s11117-021-00810-z
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DOI: https://doi.org/10.1007/s11117-021-00810-z