On certain convolution inequalities
Author:
Lars Inge Hedberg
Journal:
Proc. Amer. Math. Soc. 36 (1972), 505510
MSC:
Primary 46E30; Secondary 46E35
MathSciNet review:
0312232
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Abstract: It is proved that certain convolution inequalities are easy consequences of the HardyLittlewoodWiener maximal theorem. These inequalities include the HardyLittlewoodSobolev inequality for fractional integrals, its extension by Trudinger, and an interpolation inequality by Adams and Meyers. We also improve a recent extension of Trudinger's inequality due to Strichartz.
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 D. R. Adams and N. G. Meyers, Bessel potentials. Inclusion relations among classes of exceptional sets, Bull. Amer. Math. Soc. 77 (1971), 968970. MR 0284607 (44:1831)
 [1a]
 , Bessel potentials. Inclusion relations among classes of exceptional sets (to appear).
 [2]
 E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat. 8 (1959), 2451. MR 22 #181. MR 0109295 (22:181)
 [3]
 G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), 565606. MR 1544927
 [4]
 J. A. Hempel, G. R. Morris and N. S. Trudinger, On the sharpness of a limiting case of the Sobolev imbedding theorem, Bull. Austral. Math. Soc. 3 (1970), 369373. MR 0280998 (43:6717)
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 I. I. Hirschman, Jr., A convexity theorem for certain groups of transformations, J. Analyse Math. 2 (1953), 209218. MR 15, 295; 1139. MR 0057936 (15:295b)
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 J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 10771092. MR 0301504 (46:662)
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 L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115162. MR 22 #823. MR 0109940 (22:823)
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 S. L. Sobolev, On a theorem of functional analysis, Mat. Sb. 4 (46) (1938), 471497; English transl., Amer. Math. Soc. Transl. (2) 34 (1963), 3968.
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 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton, Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
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 R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841842. MR 0293389 (45:2466)
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 N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473483. MR 35 #7121. MR 0216286 (35:7121)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197203122324
PII:
S 00029939(1972)03122324
Article copyright:
© Copyright 1972 American Mathematical Society
