De Nápoli, Pablo L.; Drelichman, Irene; Durán, Ricardo G. On weighted inequalities for fractional integrals of radial functions. (English) Zbl 1277.26026 Ill. J. Math. 55, No. 2, 575-587 (2011). The authors prove that the weighted Hardy-Littlewood-Sobolev inequality \[ || | x|{}^{-\beta}T{}_{\gamma}v||{}_{L^{q}}{}_{(\mathbb{R}^{n})} \leq C\mid\mid \mid x\mid{}^{\alpha}v\mid\mid_{L^{p}}{}_{(\mathbb{R}^{n})} \] holds for all radially symmetric \(v\in L^{p}\left (\mathbb{R}^n,\mid x\mid^{p\alpha}dx\right)\), where \(n\geq 1\), \(0<\gamma<n\), \(1<p<\infty\), \(\alpha <\frac{n}{p}\), \(\beta <\frac{n}{q}\), \(\alpha + \beta \geq (n-1)\Big(\frac1{q}-\frac1{p}\Big)\), \(\frac1{q}=\frac1{p}+\frac{\gamma+\alpha+\beta}{n}-1\), \(p+q<\infty\) and the constant \(C\) is independent of \(v\). Furthermore, it is shown that the range of power weights appearing in the classical inequality of Stien and Weies can be improved for radially symmetric functions. Reviewer: James Adedayo Oguntuase (Abeokuta) Cited in 3 ReviewsCited in 20 Documents MSC: 26D10 Inequalities involving derivatives and differential and integral operators 47G10 Integral operators 31B10 Integral representations, integral operators, integral equations methods in higher dimensions Keywords:fractional integral operator; locally compact group; radially symmetric functions; Hardy-Littlewood-Sobolev inequality PDFBibTeX XMLCite \textit{P. L. De Nápoli} et al., Ill. J. Math. 55, No. 2, 575--587 (2011; Zbl 1277.26026) Full Text: arXiv Euclid References: [1] P. L. De Nápoli, I. Drelichman and R. G. Durán, Radial solutions for Hamiltonian elliptic systems with weights , Adv. Nonlinear Stud. 9 (2009), 579-593. · Zbl 1191.35117 [2] G. Gasper, K. Stempak and W. Trebels, Fractional integration for Laguerre expansions , Mathods Appl. Anal. 2 (1995), 67-75. · Zbl 0837.42014 [3] L. Grafakos, Classical and modern Fourier analysis , Pearson Education, Inc., Upper Saddle River, NJ, 2004. · Zbl 1148.42001 [4] K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for Wave and Schrödinger equations , Illinois J. Math. 52 (2008), 365-388. · Zbl 1183.35068 [5] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, I , Math. Z. 27 (1928), 565-606. · JFM 54.0275.05 · doi:10.1007/BF01171116 [6] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev , J. Funct. Anal. 49 (1982), 315-334. · Zbl 0501.46032 · doi:10.1016/0022-1236(82)90072-6 [7] W. Rother, Some existence theorems for the equation \(-\Delta u + K(x) u^p = 0\) . Comm. Partial Differential Equations 15 (1990), 1461-1473. · Zbl 0729.35045 · doi:10.1080/03605309908820733 [8] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces , Amer. J Math. 114 (1992), 813-874. · Zbl 0783.42011 · doi:10.2307/2374799 [9] E. M. Stein, Singular integrals and differentiability properties of functions , Princeton University Press, Princeton, 1970. · Zbl 0207.13501 [10] E. M. Stein and G. Weiss, Fractional integrals on \(n\)-dimensional Euclidean space , J. Math. Mech. 7 (1958), 503-514. · Zbl 0082.27201 [11] M. C. Vilela, Regularity solutions to the free Schrödinger equation with radial initial data , Illinois J. Math. 45 (2001), 361-370. · Zbl 1161.42306 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.