Saker, Samir H.; Krnić, Mario; Pečarić, Josip Higher summability theorems from the weighted reverse discrete inequalities. (English) Zbl 1524.40003 Appl. Anal. Discrete Math. 13, No. 2, 423-439 (2019). Summary: Motivated by higher integrability theorems due to Muckenhoupt and Gehring, in this paper we establish some related higher summability results for nonincreasing sequences, verifying the weighted reverse discrete inequalities. Our main result will be proved by employing the weighted Hardy-type inequality designed and proved for this purpose. Cited in 9 Documents MSC: 40A05 Convergence and divergence of series and sequences 26D15 Inequalities for sums, series and integrals 40C99 General summability methods Keywords:Hardy-type inequality; higher summability; reverse inequality PDFBibTeX XMLCite \textit{S. H. Saker} et al., Appl. Anal. Discrete Math. 13, No. 2, 423--439 (2019; Zbl 1524.40003) Full Text: DOI References: [1] H. Alzer: An improvment of the constants in a reverse integral inequality, J. Math. Anal. Appl. 190 (1995), 774-779. · Zbl 0823.26013 [2] J. Bober, E. Carneiro, K. Hughes, L.B. Pierce: On a discrete version of Tanaka’s theorem for maximal functions, Proc. Amer. Math. Soc. 140 (2012), 1669-1680. · Zbl 1245.42017 [3] B. Bojarski, C. Sbordone, I. Wik: The Muckenhoupt class A1(R), Studia Math. 101 (1992), 155-163. · Zbl 0808.42010 [4] S.I. Butt, J. Pečarić, I. Perić: Refinement of integral inequalities for monotone functions, J. Ineq. Appl. (2012), 2012:301 · Zbl 1281.26011 [5] R.R. Coifman, C. Fefferman: Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51, 1974, 241-250. · Zbl 0291.44007 [6] D.J.H. Garling: Inequalities: A Journey into Linear Analysis, Cambridge University Press, UK (2007). · Zbl 1135.26014 [7] F.W. Gehring: The L p -integrability of the partial derivatives of a quasi-conformal mapping, Acta Math. 130 (1973), 265-277. · Zbl 0258.30021 [8] L. Grafakos: Classical Fourier Analysis, Springer, New York (2014). · Zbl 1304.42001 [9] L. Grafakos, J. Kinnunen: Sharp inequalities for maximal functions associated with general measures, Proc. Royal Soc. Edinb. Section A: Mathematics 128 (1998), 717-723. · Zbl 0918.42010 [10] G.H. Hardy, J.E. Littlewood, G. Polya: Inequalities, 2nd ED. Cambridge Univ. Press, 1934. · Zbl 0010.10703 [11] T. Iwaniec: On L p -integrability in p.d.e and quasiconformal mapping for large expo-nents, Ann. Acad. Sci. Fen. Math. Ser. A 7 (1982), 301-322. · Zbl 0505.30011 [12] J. Kinnunen, J.L. Lewis: Higher integrability for parabolic systems of p-Laplacian type, Duke Math. J. 102 (2000), 253-272. · Zbl 0994.35036 [13] J. Kinnunen, P. Lindqvist: The derivative of the maximal function, J. Reine Angew. Math. 503 (1998), 161-167. · Zbl 0904.42015 [14] J. Kinnunen, E. Saksman: Regularity of the fractional maximal function, Bull. London Math. Soc. 35 (2003), 529-535. · Zbl 1021.42009 [15] F. Liu: Endpoint regularity of discrete multisublinear fractional maximal operators associated with l 1 -balls, J. Ineq. Appl. (2018), 2018:33. · Zbl 1383.42019 [16] J. Madrid: Sharp inequalities for the variation of the discrete maximal function, Bull. Austr. Math. Soc. 95 (2017), 94-107. · Zbl 1364.42022 [17] A. Magyar, E.M. Stein, S. Wainger: Discrete analogues in harmonic analysis: Spherical averages, Ann. Math. 155 (2002), 189-208. · Zbl 1036.42018 [18] M. Milman: A note on Gehring’s lemma, Ann. Acad. Sci. Fenn. Math. 21 (1996), 389-398. · Zbl 0903.42008 [19] M. Muckenhoupt: Weighted norm inequalities for the Hardy maximal function, Tran. Amer. Math. Soc. 165 (1972), 207-226. · Zbl 0236.26016 [20] L. Nania: On some reverse integral inequalities, J. Aust. Math. Soc. (Series A) 49 (1990), 319-326. · Zbl 0715.26010 [21] J. Pečarić, I. Perić, L.E. Persson: Integral inequalities for monotone functions, J. Math. Anal. Appl. 215 (1997), 235-251. · Zbl 0937.26012 [22] D.T. Shum: On integral inequalities related to Hardy’s, Canad. Math. Bull. 14 (1971), 225-230. · Zbl 0213.34503 [23] E.M. Stein: Harmonic analysis, Princeton, New Jersey, Princeton University Press, 1993. · Zbl 0821.42001 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.