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On weighted inequalities for fractional integrals of radial functions. (English) Zbl 1277.26026

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.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
47G10 Integral operators
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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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
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