Weighted weak-type inequalities for some fractional integral operators. (English) Zbl 1318.26015

Summary: For \(0<\alpha<1\), let \(W_{\alpha}\) and \(R_{\alpha}\) denote Weyl fractional integral operator and Riemann-Liouville fractional integral operator, respectively. We establish sharp versions of Muckenhoupt-Wheeden conjecture for these operators. Specifically, we prove that for any weight \(w\) on \([0,\infty)\), we have \[ \|{W}_{\alpha} f\|_{L^{1/(1-\alpha),\infty}(w)}\leq \alpha^{-1}\|{f}\|_{L^{1}((M_{-}w)^{1-{\alpha}})} \] and \[ \|{R}_{\alpha} f\|_{L^{1/(1-{\alpha}),\infty}(w)}\leq \alpha^{-1}\|{f}\|_{L^{1}((M_{+}w)^{1-{\alpha}})}. \]
Here \(M_{-}\), \(M_{+}\) denote the one-sided Hardy-Littlewood maximal operators on \([0,\infty)\). In each of the estimates, the constant \(\alpha^{-1}\) is the best possible.


26A33 Fractional derivatives and integrals
26D10 Inequalities involving derivatives and differential and integral operators
Full Text: DOI Euclid


[1] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), no. 2, 547-558. · Zbl 0664.26002 · doi:10.2307/2001091
[2] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253-272. · Zbl 0795.42011 · doi:10.2307/2154555
[3] S. Chanillo and R. L. Wheeden, Some weighted norm inequalities for the area integral, Indiana Univ. Math. J. 36 (1987), no. 2, 277-294. · Zbl 0598.34019 · doi:10.1512/iumj.1987.36.36016
[4] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. · Zbl 0222.26019 · doi:10.2307/2373450
[5] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities , reprint of the 1952 ed., Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 1988.
[6] A. K. Lerner, S. Ombrosi and C. Pérez, Sharp \(A_{1}\) bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, Int. Math. Res. Not. IMRN 2008 , no. 6, Art. ID rnm161, 11 pp.
[7] A. K. Lerner, S. Ombrosi and C. Pérez, Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden, J. Fourier Anal. Appl. 15 (2009), no. 3, 394-403. · Zbl 1175.42005 · doi:10.1007/s00041-008-9032-2
[8] A. K. Lerner, S. Ombrosi and C. Pérez, \(A_{1}\) bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Math. Res. Lett. 16 (2009), no. 1, 149-156. · Zbl 1169.42006 · doi:10.4310/MRL.2009.v16.n1.a14
[9] F. Nazarov, A. Reznikov, V. Vasyunin and A. Volberg, \(A_{1}\) conjecture: Weak norm estimates of weighted singular operators and Bellman functions. (Preprint).
[10] C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. (2) 49 (1994), no. 2, 296-308. · Zbl 0797.42010 · doi:10.1112/jlms/49.2.296
[11] M. C. Reguera, On Muckenhoupt-Wheeden conjecture, Adv. Math. 227 (2011), no. 4, 1436-1450. · Zbl 1221.42026 · doi:10.1016/j.aim.2011.03.009
[12] M. C. Reguera and C. Thiele, The Hilbert transform does not map \(L^{1}(Mw)\) to \(L^{1,\infty}(w)\), Math. Res. Lett. 19 (2012), no. 1, 1-7. · Zbl 1271.42025 · doi:10.4310/MRL.2012.v19.n1.a1
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.