×
Sawano, Yoshihiro; Shimomura, Tetsu

Sobolev embeddings for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces over non-doubling measure spaces. (English) Zbl 1301.31007

Integral Transforms Spec. Funct. 25, No. 12, 976-991 (2014).
Summary: In this paper we are concerned with Sobolev embeddings for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces over non-doubling measure spaces. We deal with the boundedness of the generalized maximal operators, Sobolev’s inequality, Trudinger’s inequality and continuity for Riesz potentials of variable order. We prove the results in full generality: our function spaces will cover Morrey spaces, variable Lebesgue spaces as well as Lebesgue spaces with a Radon measure \(\mu\). Also, we can readily handle function spaces on metric measure spaces. Our results will cover the ones in our earlier paper [Collect. Math. 64, No. 3, 313–350 (2013; Zbl 1280.31001)]. We shall show that the Trudinger exponential integrability is available in our generalized setting, whose counterpart on Lebesgue spaces was not considered in [Y. Mizuta et al., Rev. Mat. Complut. 25, No. 2, 413–434 (2012; Zbl 1273.31005)].

Cited in 10 Documents

MSC:

31B15 Potentials and capacities, extremal length and related notions in higher dimensions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

Riesz potentials; maximal functions; Sobolev inequality; Trudinger inequality; Musielak-Orlicz-Morrey spaces; non-doubling measure

Citations:

Zbl 1280.31001; Zbl 1273.31005
Cite Review PDF
Full Text: DOI

References:

[1] DOI: 10.1155/S1073792897000469 · Zbl 0889.42013 · doi:10.1155/S1073792897000469
[2] DOI: 10.1155/S1073792898000312 · Zbl 0918.42009 · doi:10.1155/S1073792898000312
[3] DOI: 10.14492/hokmj/1285766231 · Zbl 1088.42010 · doi:10.14492/hokmj/1285766231
[4] Terasawa Y., J Inequal Appl (2006)
[5] DOI: 10.1007/s13348-013-0082-7 · Zbl 1280.31001 · doi:10.1007/s13348-013-0082-7
[6] Almeida A, Georgian Math J 15 pp 195– (2008)
[7] DOI: 10.2969/jmsj/06020583 · Zbl 1161.46305 · doi:10.2969/jmsj/06020583
[8] Guliyev VS, Math Scand 107 pp 285– (2010) · Zbl 1213.42077 · doi:10.7146/math.scand.a-15156
[9] DOI: 10.1007/s10958-010-0095-7 · Zbl 1307.46018 · doi:10.1007/s10958-010-0095-7
[10] DOI: 10.1007/s13163-011-0074-7 · Zbl 1273.31005 · doi:10.1007/s13163-011-0074-7
[11] Mizuta Y, Osaka J Math 46 pp 255– (2009)
[12] DOI: 10.1080/17476930903276068 · Zbl 1205.26014 · doi:10.1080/17476930903276068
[13] DOI: 10.1080/17476933.2010.504837 · Zbl 1228.31004 · doi:10.1080/17476933.2010.504837
[14] DOI: 10.1007/s10114-005-0660-z · Zbl 1129.42403 · doi:10.1007/s10114-005-0660-z
[15] DOI: 10.5486/PMD.2013.5508 · doi:10.5486/PMD.2013.5508
[16] DOI: 10.1017/S0004972709000343 · Zbl 1176.42012 · doi:10.1017/S0004972709000343
[17] Sawano Y, J Inequal Appl (2006)
[18] Sawano Y, J Inequal Appl 12 (2013)
[19] DOI: 10.1090/S0002-9939-1972-0312232-4 · doi:10.1090/S0002-9939-1972-0312232-4
[20] Trudinger N., J Math Mech 17 pp 473– (1967)
[21] DOI: 10.1007/978-3-662-03282-4 · doi:10.1007/978-3-662-03282-4
[22] Futamura T, Math Inequal Appl 8 (4) pp 619– (2005)
[23] Futamura T, Ann Acad Sci Fenn Math 31 pp 495– (2006)
[24] DOI: 10.1016/j.jmaa.2010.01.053 · Zbl 1193.46016 · doi:10.1016/j.jmaa.2010.01.053
[25] Mizuta Y, Potential theory in Euclidean spaces (1996)
[26] Mizuta Y, Hiroshima Math J. 38 pp 425– (2008)
[27] DOI: 10.2969/jmsj/06230707 · Zbl 1200.26007 · doi:10.2969/jmsj/06230707
[28] Nakai E. Generalized fractional integrals on Orlicz–Morrey spaces, Banach and function spaces. Yokohama: Yokohama Publ. 2004. p. 323–333. · Zbl 1118.42005
[29] DOI: 10.1007/978-1-4612-1015-3 · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3
[30] Harjulehto P, Rev Mat Complut 17 pp 129– (2004) · Zbl 1072.46021 · doi:10.5209/rev_REMA.2004.v17.n1.16790
[31] Mizuta Y, Math Inequal Appl 13 pp 99– (2010)
[32] Diening L, Math Inequal Appl 7 (2) pp 245– (2004)
[33] Musielak J, Lecture notes in mathematics 1034 (1983)
[34] DOI: 10.1002/mana.19941660108 · Zbl 0837.42008 · doi:10.1002/mana.19941660108
[35] DOI: 10.1007/s00030-008-6032-5 · Zbl 1173.42317 · doi:10.1007/s00030-008-6032-5
[36] DOI: 10.4171/RMI/398 · Zbl 1099.42021 · doi:10.4171/RMI/398
[37] DOI: 10.1155/2007/914143 · Zbl 1135.42010 · doi:10.1155/2007/914143
[38] Harjulehto P, Real Anal Exchange pp 87– (2004)
[39] Kokilashvili V, Georgian Math J 15 (4) pp 683– (2008)
[40] DOI: 10.1080/10652460412331320322 · Zbl 1069.47056 · doi:10.1080/10652460412331320322
[41] DOI: 10.1215/S0012-7094-75-04265-9 · Zbl 0336.46038 · doi:10.1215/S0012-7094-75-04265-9
[42] Chiarenza F, Rend Mat Appl 7 (7) pp 273– (1987)
[43] DOI: 10.1016/j.jfa.2012.01.004 · Zbl 1244.42012 · doi:10.1016/j.jfa.2012.01.004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.