[1]

W. Beckner,
Geometric inequalities in Fourier anaylsis
Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton 1991),
Princeton Math. Ser. 42,
Princeton University, Princeton (1995), 36–68.
Google Scholar

[2]

W. Beckner,
Pitt’s inequality and the uncertainty principle,
Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897–1905.
Google Scholar

[3]

W. Beckner,
Sharp inequalities and geometric manifolds,
J. Fourier Anal. Appl. 3 (1997), 825–836.
CrossrefGoogle Scholar

[4]

W. Beckner,
Pitt’s inequality with sharp convolution estimates,
Proc. Amer. Math. Soc. 136 (2008), no. 5, 1871–1885.
Google Scholar

[5]

W. Beckner,
Weighted inequalities and Stein–Weiss potentials,
Forum Math. 20 (2008), no. 4, 587–606.
Web of ScienceGoogle Scholar

[6]

W. Beckner,
Multilinear embedding estimates for the fractional Laplacian,
Math. Res. Lett. 19 (2012), no. 1, 175–189.
CrossrefGoogle Scholar

[7]

W. Beckner,
Functionals for multilinear fractional embedding,
Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 1, 1–28.
CrossrefGoogle Scholar

[8]

E. A. Carlen, J. A. Carrillo and M. Loss,
Hardy–Littlewood–Sobolev inequalities via fast diffusion flows,
Proc. Natl. Acad. Sci. USA 107 (2010), no. 46, 19696–19701.
CrossrefGoogle Scholar

[9]

E. Carneiro,
A sharp inequality for the Strichartz norm,
Int. Math. Res. Not. IMRN 2009 (2009), no. 16, 3127–3145.
CrossrefGoogle Scholar

[10]

L. Chen, Z. Liu and G. Lu,
Symmetry and regularity of solutions to the weighted Hardy–Sobolev type system,
Adv. Nonlinear Stud. 16 (2016), no. 1, 1–13.
CrossrefWeb of ScienceGoogle Scholar

[11]

L. Chen, Z. Liu, G. Lu and C. Tao,
Reverse Stein–Weiss inequalities and existence of their extremal functions,
Trans. Amer. Math. Soc. 370 (2018), no. 12, 8429–8450.
CrossrefGoogle Scholar

[12]

L. Chen, Z. Liu, G. Lu and C. Tao,
Stein–Weiss inequalities with the fractional Poisson kernel,
preprint (2018), https://arxiv.org/abs/1807.04906;
to appear in Rev. Mat. Iberoam.

[13]

L. Chen, G. Lu and C. Tao,
Existence of extremal functions for the Stein–Weiss inequalities on the Heisenberg group,
preprint (2018), https://arxiv.org/abs/1807.04699.

[14]

L. Chen, G. Lu and C. Tao,
Hardy–Littlewood–Sobolev inequality with fractional Poisson kernel and its appliaction in PDEs,
Acta Math. Sin. (Engl. Ser.), to appear.
Google Scholar

[15]

W. Chen, C. Jin, C. Li and J. Lim,
Weighted Hardy–Littlewood–Sobolev inequalities and systems of integral equations,
Discrete Contin. Dyn. Syst. 2005 (2005), 164–172.
Google Scholar

[16]

W. Chen and C. Li,
The best constant in a weighted Hardy–Littlewood–Sobolev inequality,
Proc. Amer. Math. Soc. 136 (2008), no. 3, 955–962.
Google Scholar

[17]

J. Dou,
Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space,
Commun. Contemp. Math. 18 (2016), no. 5, Article ID 1550067.
Web of ScienceGoogle Scholar

[18]

J. Dou and M. Zhu,
Reversed Hardy–Littewood–Sobolev inequality,
Int. Math. Res. Not. IMRN 2015 (2015), no. 19, 9696–9726.
CrossrefGoogle Scholar

[19]

J. Dou and M. Zhu,
Sharp Hardy–Littlewood–Sobolev inequality on the upper half space,
Int. Math. Res. Not. IMRN 2015 (2015), no. 3, 651–687.
CrossrefGoogle Scholar

[20]

P. Drábek, H. P. Heinig and A. Kufner,
Higher-dimensional Hardy inequality,
General Inequalities. 7 (Oberwolfach 1995),
Internat. Ser. Numer. Math. 123,
Birkhäuser, Basel (1997), 3–16.
Google Scholar

[21]

R. L. Frank and E. H. Lieb,
Inversion positivity and the sharp Hardy–Littlewood–Sobolev inequality,
Calc. Var. Partial Differential Equations 39 (2010), no. 1–2, 85–99.
CrossrefWeb of ScienceGoogle Scholar

[22]

R. L. Frank and E. H. Lieb,
A new, rearrangement-free proof of the sharp Hardy–Littlewood–Sobolev inequality,
Spectral Theory, Function Spaces and Inequalities,
Oper. Theory Adv. Appl. 219,
Birkhäuser, Basel (2012), 55–67.
Google Scholar

[23]

R. L. Frank and E. H. Lieb,
Sharp constants in several inequalities on the Heisenberg group,
Ann. of Math. (2) 176 (2012), no. 1, 349–381.
Web of ScienceCrossrefGoogle Scholar

[24]

X. Han,
Existence of maximizers for Hardy–Littlewood–Sobolev inequalities on the Heisenberg group,
Indiana Univ. Math. J. 62 (2013), no. 3,737–751.
Web of ScienceCrossrefGoogle Scholar

[25]

X. Han, G. Lu and J. Zhu,
Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group,
Nonlinear Anal. 75 (2012), no. 11, 4296–4314.
CrossrefWeb of ScienceGoogle Scholar

[26]

F. Hang, X. Wang and X. Yan,
Sharp integral inequalities for harmonic functions,
Comm. Pure Appl. Math. 61 (2008), no. 1, 54–95.
CrossrefGoogle Scholar

[27]

G. H. Hardy and J. E. Littlewood,
Some properties of fractional integrals. I,
Math. Z. 27 (1928), no. 1, 565–606.
CrossrefGoogle Scholar

[28]

I. W. Herbst,
Spectral theory of the operator ${({p}^{2}+{m}^{2})}^{1/2}-Z{e}^{2}/r$,
Comm. Math. Phys. 53 (1977), no. 3, 285–294.
Google Scholar

[29]

E. H. Lieb,
Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities,
Ann. of Math. (2) 118 (1983), no. 2, 349–374.
CrossrefGoogle Scholar

[30]

E. H. Lieb and M. Loss,
Analysis, 2nd ed.,
Grad. Stud. Math. 14,
American Mathematical Society, Providence, 2001.
Google Scholar

[31]

P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. I,
Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201.
Google Scholar

[32]

P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. II,
Rev. Mat. Iberoam. 1 (1985), no. 2, 45–121.
Google Scholar

[33]

G. Lu and C. Tao,
Reverse Hardy–Littlewood–Sobolev and Stein–Weiss inequalities on the Heisenberg group,
preprint (2018).

[34]

G. Lu and J. Zhu,
Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality,
Calc. Var. Partial Differential Equations 42 (2011), no. 3–4, 563–577.
Web of ScienceCrossrefGoogle Scholar

[35]

Q. A. Ngô and V. H. Nguyen,
Sharp reversed Hardy–Littlewood–Sobolev inequality: The case of whole space ${\mathbb{R}}^{n}$,
preprint (2016), https://arxiv.org/abs/1508.02041v2.

[36]

Q. A. Ngô and V. H. Nguyen,
Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space ${\mathbf{R}}_{+}^{n}$,
Int. Math. Res. Not. IMRN 2017 (2017), no. 20, 6187–6230.
Google Scholar

[37]

E. Sawyer and R. L. Wheeden,
Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces,
Amer. J. Math. 114 (1992), no. 4, 813–874.
CrossrefGoogle Scholar

[38]

S. L. Sobolev,
On a theorem in functional analysis (in Russian),
Mat. Sb. 4 (1938), 471–497.
Google Scholar

[39]

E. M. Stein,
Singular Integrals and Differentiability Properties of Functions,
Princeton Math. Ser. 30,
Princeton University, Princeton, 1970.
Google Scholar

[40]

E. M. Stein,
Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals,
Princeton Math. Ser. 43,
Princeton University, Princeton, 1993.
Google Scholar

[41]

E. M. Stein and G. Weiss,
Fractional integrals on *n*-dimensional Euclidean space,
J. Math. Mech. 7 (1958), 503–514.
Google Scholar

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