Andersen, K. F.; Sawyer, E. T. Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators. (English) Zbl 0664.26002 Trans. Am. Math. Soc. 308, No. 2, 547-558 (1988). This paper studies weighted norm inequalities \[ [\int^{\infty}_{0}| (Tf)(x)u(x)|^ q dx]^{1/q}\leq C[\int^{\infty}_{0}| f(x)v(x)|^ p dx]^{1/p}, \] where u and v are nonnegative weight functions, \(\alpha >0,\quad 1<p<1/\alpha,\quad 1/q=1/p-\alpha,\) and C is a constant depending on p, q, \(\alpha\), u, v but independent of f and T is one of Riemann-Liouville or Weyl fractional integral from which inequalities for other fractional integral operators T, such as Erdélyi-Kober, can be deduced. Some questions raised by B. Muckenhoupt [Proc. Symp. Pure Math. 35, No.1, 69-83 (1979; Zbl 0428.26009)] have been answered through Theorems 2 and 3. Reviewer: R.N.Kalia Cited in 4 ReviewsCited in 56 Documents MSC: 26A33 Fractional derivatives and integrals 26D10 Inequalities involving derivatives and differential and integral operators 42B25 Maximal functions, Littlewood-Paley theory Keywords:Riemann-Liouville fractional integrals; weighted norm inequalities; Weyl fractional integral Citations:Zbl 0428.26009 PDFBibTeX XMLCite \textit{K. F. Andersen} and \textit{E. T. Sawyer}, Trans. Am. Math. Soc. 308, No. 2, 547--558 (1988; Zbl 0664.26002) Full Text: DOI References: [1] K. F. Andersen, Weighted inequalities for fractional integrals, Fractional calculus (Glasgow, 1984) Res. Notes in Math., vol. 138, Pitman, Boston, MA, 1985, pp. 12 – 25. · Zbl 0645.42016 [2] K. F. Andersen and H. P. Heinig, Weighted norm inequalities for certain integral operators, SIAM J. Math. Anal. 14 (1983), no. 4, 834 – 844. · Zbl 0527.26010 · doi:10.1137/0514064 [3] Kenneth F. Andersen and Benjamin Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9 – 26. · Zbl 0501.47011 [4] J. Scott Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), no. 4, 405 – 408. · Zbl 0402.26006 · doi:10.4153/CMB-1978-071-7 [5] L. Carleson and P. Jones, Weighted norm inequalities and a theorem of Koosis, Mittag-Leffler Inst. Rep. 2 (1981). [6] Angel E. Gatto and Cristian E. Gutiérrez, On weighted norm inequalities for the maximal function, Studia Math. 76 (1983), no. 1, 59 – 62. · Zbl 0536.42021 [7] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press, 1967. [8] Benjamin Muckenhoupt, Weighted norm inequalities for classical operators, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 69 – 83. [9] Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261 – 274. · Zbl 0289.26010 [10] Eric T. Sawyer, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), no. 2, 533 – 545. · Zbl 0665.42023 [11] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), no. 1, 53 – 61. · Zbl 0627.42009 [12] Eric T. Sawyer, Weighted norm inequalities for fractional maximal operators, 1980 Seminar on Harmonic Analysis (Montreal, Que., 1980) CMS Conf. Proc., vol. 1, Amer. Math. Soc., Providence, R.I., 1981, pp. 283 – 309. Eric T. Sawyer, Two weight norm inequalities for certain maximal and integral operators, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 102 – 127. · Zbl 0508.42024 [13] Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1 – 11. · Zbl 0508.42023 [14] Elias M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482 – 492. · Zbl 0072.32402 [15] G. V. Welland, Weighted norm inequalities for fractional integrals, Proc. Amer. Math. Soc. 51 (1975), 143 – 148. · Zbl 0306.26007 [16] Wo Sang Young, Weighted norm inequalities for the Hardy-Littlewood maximal function, Proc. Amer. Math. Soc. 85 (1982), no. 1, 24 – 26. · Zbl 0489.42019 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.