Cianchi, Andrea; Stroffolini, Bianca An extension of Hedberg’s convolution inequality and applications. (English) Zbl 0914.42014 J. Math. Anal. Appl. 227, No. 1, 166-186 (1998). In his celebrated paper [Proc. Am. Math. Soc. 36, 505-510 (1993; Zbl 0283.26003)], L. I. Hedberg presented a new proof of the Sobolev inequality for potentials, which asserts, roughly speaking, that the Riesz potential of order \(\alpha\in(0,n)\) is bounded from \(L^p(\mathbb R^n)\) into \(L^q(\mathbb R^n)\) when \(p\in(1,n/\alpha)\) and \(q=np/(n-\alpha p)\). Hedberg’s method is based on a pointwise inequality linking together the Riesz potential and the Hardy-Littlewood maximal operator. The paper under review presents an Orlicz-space version of Hedberg’s pointwise inequality and its very interesting applications. In particular, sharp embeddings of Orlicz-Sobolev spaces into Orlicz spaces are obtained. Examples including the limiting case \(p=n/\alpha\) are discussed. Reviewer: L.Pick (Praha) Cited in 1 ReviewCited in 10 Documents MSC: 42B25 Maximal functions, Littlewood-Paley theory 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:Riesz potential; maximal operator; Hedberg’s convolution inequality; Orlicz spaces Citations:Zbl 0283.26003 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Adams, D. R.; Hedberg, L. I., Function Spaces and potential Theory (1996), Springer-Verlag: Springer-Verlag Berlin [2] Aissaoui, N.; Benkirane, A., Capacités dans les espaces d’Orlicz, Ann. Sci. Math. Québec, 18, 1-23 (1994) · Zbl 0822.31006 [3] Bennett, C.; Sharpley, R., Interpolation of Operators (1988), Academic Press: Academic Press Boston · Zbl 0647.46057 [4] Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1987), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0617.26001 [5] Boyd, D. W., Indices for the Orlicz spaces, Pacific J. Math., 38, 315-323 (1971) · Zbl 0227.46039 [6] Cianchi, A., A sharp embedding theorem for Orlicz-Sobolev spaces, Indiana Univ. Math. J., 45, 36-65 (1996) · Zbl 0860.46022 [7] A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc.; A. Cianchi, Strong and weak type inequalities for some classical operators in Orlicz spaces, J. London Math. Soc. · Zbl 0940.46015 [8] Donaldson, T. K.; Trudinger, N. S., Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal., 8, 52-75 (1971) · Zbl 0216.15702 [9] Edmunds, D. E.; Gurka, P.; Opic, B., Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces, Indiana Univ. Math. J., 44, 19-43 (1995) · Zbl 0826.47021 [10] Fusco, N.; Lions, P. L.; Sbordone, C., Some remarks on Sobolev embeddings in borderline cases, Proc. Amer. Math. Soc., 70, 561-565 (1996) · Zbl 0841.46023 [11] Hedberg, L. I., On certain convolution inequalities, Proc. Amer. Math. Soc., 36, 505-510 (1972) · Zbl 0283.26003 [12] Kokilashvili, V.; Krbec, M., Weighted inequalities in Lorentz and Orlicz spaces (1991), World Scientific: World Scientific Singapore · Zbl 0751.46021 [13] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces II (1979), Springer: Springer Berlin · Zbl 0403.46022 [14] Moscariello, G., A pointwise inequality for Riesz potentials in Orlicz spaces, Rend. Accad. Sci. Fis. Mat., 53, 41-47 (1986) · Zbl 1145.46301 [15] O’Neil, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc., 115, 300-328 (1965) · Zbl 0132.09201 [16] Strichartz, R. S., A note on Trudinger’s extension of Sobolev’s inequalities, Indiana Univ. Math. J., 21, 841-842 (1972) · Zbl 0241.46028 [17] Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17, 473-483 (1967) · Zbl 0163.36402 [18] Ziemer, W. P., Weakly Differentiable Functions (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0177.08006 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.