Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-04T21:42:05.015Z Has data issue: false hasContentIssue false
  • English
  • Français

SOBOLEV INEQUALITIES FOR RIESZ POTENTIALS OF FUNCTIONS IN $L^{p(\cdot )}$ OVER NONDOUBLING MEASURE SPACES

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  12 November 2015

TAKAO OHNO  and
TETSU SHIMOMURA
TAKAO OHNO*
Affiliation:
Faculty of Education and Welfare Science, Oita University, Dannoharu Oita-city 870-1192, Japan email [email protected]
TETSU SHIMOMURA
Affiliation:
Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.

Keywords

Riesz potentialSobolev inequalityvariable exponentmetric measure spacenondoubling measure

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 128 - 136
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Björn, A. and Björn, J., Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics, 17 (European Mathematical Society (EMS), Zurich, 2011).CrossRefGoogle Scholar
Cruz-Uribe, D. and Fiorenza, A., Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, Heidelberg, 2013).CrossRefGoogle Scholar
Diening, L., ‘Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p (⋅) and W k, p (⋅)’, Math. Nachr. 268 (2004), 31–43.CrossRefGoogle Scholar
Diening, L., Harjulehto, P., Hästö, P. and Růžička, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017 (Springer, Heidelberg, 2011).CrossRefGoogle Scholar
Edmunds, D. E. and Rákosník, J., ‘Sobolev embeddings with variable exponent, II’, Math. Nachr. 246–247 (2002), 53–67.3.0.CO;2-T>CrossRefGoogle Scholar
Futamura, T. and Mizuta, Y., ‘Continuity properties of Riesz potentials for functions in L p (⋅) of variable exponent’, Math. Inequal. Appl. 8(4) (2005), 619–631.Google Scholar
Hajłasz, P. and Koskela, P., ‘Sobolev met Poincaré’, Mem. Amer. Math. Soc. 145 (2000), number 688.Google Scholar
Hedberg, L. I., ‘On certain convolution inequalities’, Proc. Amer. Math. Soc. 36 (1972), 505–510.CrossRefGoogle Scholar
Mizuta, Y., Potential Theory in Euclidean Spaces (Gakkōtosho, Tokyo, 1996).Google Scholar
Mizuta, Y., Ohno, T. and Shimomura, T., ‘Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev’s exponent’, Bull. Sci. Math. 134 (2010), 12–36.CrossRefGoogle Scholar
Mizuta, Y., Shimomura, T. and Sobukawa, T., ‘Sobolev’s inequality for Riesz potentials of functions in non-doubling Morrey spaces’, Osaka J. Math. 46 (2009), 255–271.Google Scholar
Sawano, Y., ‘Sharp estimates of the modified Hardy–Littlewood maximal operator on the nonhomogeneous space via covering lemmas’, Hokkaido Math. J. 34 (2005), 435–458.CrossRefGoogle Scholar
Sawano, Y. and Shimomura, T., ‘Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents’, Collect. Math. 64 (2013), 313–350.CrossRefGoogle Scholar
Sawano, Y. and Shimomura, T., ‘Sobolev’s inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces’, J. Inequal. Appl. 2013(12) (2013), 19 pages.CrossRefGoogle Scholar
Strömberg, J.-O., ‘Weak type L 1 estimates for maximal functions on noncompact symmetric spaces’, Ann. of Math. (2) 114(1) (1981), 115–126.CrossRefGoogle Scholar