×

Reverse Stein-Weiss inequalities on the upper half space and the existence of their extremals. (English) Zbl 1422.42036

Summary: The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein-Weiss inequality on the upper half space: \[\mathop{\int}\limits_{\mathbb{R}^n_+}\mathop{\int}\limits_{\partial\mathbb{R}^n_+}\vert x|^{\alpha}|x -y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\| f\|_{L^{q^{\prime}}(\mathbb{R}^n_+)}\|g\|_{L^p(\partial\mathbb{R}^n_ {+})}\] for any nonnegative functions \({f\in L^{q^{\prime}}(\mathbb{R}^n_+)}\), \({g\in L^p(\partial\mathbb{R}^n_+)}\), and \({p,q^{\prime}\in(0,1)}\), \({\beta<\frac{1-n}{p^{\prime}}}\) or \({\alpha<-\frac{n}{q}}\), \({\lambda>0}\) satisfying \[\frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n} =2.\] Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler-Lagrange equations of the reverse Stein-Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein-Weiss inequality and reverse Stein-Weiss inequality on the upper half space \({\mathbb{R}^n_+}\).

MSC:

42B37 Harmonic analysis and PDEs
42B35 Function spaces arising in harmonic analysis
35B40 Asymptotic behavior of solutions to PDEs
45G15 Systems of nonlinear integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[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.; Beckner, W., Geometric inequalities in Fourier anaylsis, Essays on Fourier Analysis in Honor of Elias M. Stein, 36-68 (1995) · Zbl 0888.42006
[2] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897-1905.; Beckner, W., Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123, 6, 1897-1905 (1995) · Zbl 0842.58077
[3] W. Beckner, Sharp inequalities and geometric manifolds, J. Fourier Anal. Appl. 3 (1997), 825-836.; Beckner, W., Sharp inequalities and geometric manifolds, J. Fourier Anal. Appl., 3, 825-836 (1997) · Zbl 0895.58059
[4] W. Beckner, Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1871-1885.; Beckner, W., Pitt’s inequality with sharp convolution estimates, Proc. Amer. Math. Soc., 136, 5, 1871-1885 (2008) · Zbl 1221.42016
[5] W. Beckner, Weighted inequalities and Stein-Weiss potentials, Forum Math. 20 (2008), no. 4, 587-606.; Beckner, W., Weighted inequalities and Stein-Weiss potentials, Forum Math., 20, 4, 587-606 (2008) · Zbl 1149.42006
[6] W. Beckner, Multilinear embedding estimates for the fractional Laplacian, Math. Res. Lett. 19 (2012), no. 1, 175-189.; Beckner, W., Multilinear embedding estimates for the fractional Laplacian, Math. Res. Lett., 19, 1, 175-189 (2012) · Zbl 1291.35426
[7] W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 1, 1-28.; Beckner, W., Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.), 31, 1, 1-28 (2015) · Zbl 1345.46027
[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.; Carlen, E. A.; Carrillo, J. A.; Loss, M., Hardy-Littlewood-Sobolev inequalities via fast diffusion flows, Proc. Natl. Acad. Sci. USA, 107, 46, 19696-19701 (2010) · Zbl 1256.42028
[9] E. Carneiro, A sharp inequality for the Strichartz norm, Int. Math. Res. Not. IMRN 2009 (2009), no. 16, 3127-3145.; Carneiro, E., A sharp inequality for the Strichartz norm, Int. Math. Res. Not. IMRN, 2009, 16, 3127-3145 (2009) · Zbl 1178.35090
[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.; Chen, L.; Liu, Z.; Lu, G., Symmetry and regularity of solutions to the weighted Hardy-Sobolev type system, Adv. Nonlinear Stud., 16, 1, 1-13 (2016) · Zbl 1382.35326
[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.; Chen, L.; Liu, Z.; Lu, G.; Tao, C., Reverse Stein-Weiss inequalities and existence of their extremal functions, Trans. Amer. Math. Soc., 370, 12, 8429-8450 (2018) · Zbl 1418.42041
[12] L. Chen, Z. Liu, G. Lu and C. Tao, Stein-Weiss inequalities with the fractional Poisson kernel, preprint (2018), ; to appear in Rev. Mat. Iberoam.; Chen, L.; Liu, Z.; Lu, G.; Tao, C., Stein-Weiss inequalities with the fractional Poisson kernel, Preprint (2018) · Zbl 1459.35006
[13] L. Chen, G. Lu and C. Tao, Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, preprint (2018), .; Chen, L.; Lu, G.; Tao, C., Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, Preprint (2018) · Zbl 1429.26034
[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.; Chen, L.; Lu, G.; Tao, C., Hardy-Littlewood-Sobolev inequality with fractional Poisson kernel and its appliaction in PDEs, Acta Math. Sin. (Engl. Ser.) · Zbl 1418.35008
[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.; Chen, W.; Jin, C.; Li, C.; Lim, J., Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst., 2005, 164-172 (2005) · Zbl 1147.45301
[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.; Chen, W.; Li, C., The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc. Amer. Math. Soc., 136, 3, 955-962 (2008) · Zbl 1132.35031
[17] J. Dou, Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Commun. Contemp. Math. 18 (2016), no. 5, Article ID 1550067.; Dou, J., Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Commun. Contemp. Math., 18, 5 (2016) · Zbl 1347.35016
[18] J. Dou and M. Zhu, Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN 2015 (2015), no. 19, 9696-9726.; Dou, J.; Zhu, M., Reversed Hardy-Littewood-Sobolev inequality, Int. Math. Res. Not. IMRN, 2015, 19, 9696-9726 (2015) · Zbl 1329.26033
[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.; Dou, J.; Zhu, M., Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not. IMRN, 2015, 3, 651-687 (2015) · Zbl 1308.42016
[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.; Drábek, P.; Heinig, H. P.; Kufner, A., Higher-dimensional Hardy inequality, General Inequalities. 7, 3-16 (1997) · Zbl 0883.26013
[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.; Frank, R. L.; Lieb, E. H., Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39, 1-2, 85-99 (2010) · Zbl 1204.39024
[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.; Frank, R. L.; Lieb, E. H., A new, rearrangement-free proof of the sharp Hardy-Littlewood-Sobolev inequality, Spectral Theory, Function Spaces and Inequalities, 55-67 (2012) · Zbl 1297.39023
[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.; Frank, R. L.; Lieb, E. H., Sharp constants in several inequalities on the Heisenberg group, Ann. of Math. (2), 176, 1, 349-381 (2012) · Zbl 1252.42023
[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.; Han, X., Existence of maximizers for Hardy-Littlewood-Sobolev inequalities on the Heisenberg group, Indiana Univ. Math. J., 62, 3, 737-751 (2013) · Zbl 1299.39020
[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.; Han, X.; Lu, G.; Zhu, J., Hardy-Littlewood-Sobolev and Stein-Weiss inequalities and integral systems on the Heisenberg group, Nonlinear Anal., 75, 11, 4296-4314 (2012) · Zbl 1309.42032
[26] F. Hang, X. Wang and X. Yan, Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math. 61 (2008), no. 1, 54-95.; Hang, F.; Wang, X.; Yan, X., Sharp integral inequalities for harmonic functions, Comm. Pure Appl. Math., 61, 1, 54-95 (2008) · Zbl 1173.26321
[27] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), no. 1, 565-606.; Hardy, G. H.; Littlewood, J. E., Some properties of fractional integrals. I, Math. Z., 27, 1, 565-606 (1928) · JFM 54.0275.05
[28] I. W. Herbst, Spectral theory of the operator (p^2+m^2)^{1/2}-Ze^2/r, Comm. Math. Phys. 53 (1977), no. 3, 285-294.; Herbst, I. W., Spectral theory of the operator (p^2+m^2)^{1/2}-Ze^2/r, Comm. Math. Phys., 53, 3, 285-294 (1977) · Zbl 0375.35047
[29] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349-374.; Lieb, E. H., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2), 118, 2, 349-374 (1983) · Zbl 0527.42011
[30] E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001.; Lieb, E. H.; Loss, M., Analysis (2001) · Zbl 0966.26002
[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.; Lions, P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam., 1, 1, 145-201 (1985) · Zbl 0704.49005
[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.; Lions, P.-L., The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoam., 1, 2, 45-121 (1985) · Zbl 0704.49006
[33] G. Lu and C. Tao, Reverse Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on the Heisenberg group, preprint (2018).; Lu, G.; Tao, C., 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.; Lu, G.; Zhu, J., Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42, 3-4, 563-577 (2011) · Zbl 1231.35290
[35] Q. A. Ngô and V. H. Nguyen, Sharp reversed Hardy-Littlewood-Sobolev inequality: The case of whole space \(\mathbb{R}^n\), preprint (2016), .; Ngô, Q. A.; Nguyen, V. H., Sharp reversed Hardy-Littlewood-Sobolev inequality: The case of whole space \(\mathbb{R}^n\), Preprint (2016)
[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.; Ngô, Q. A.; Nguyen, V. H., Sharp reversed Hardy-Littlewood-Sobolev inequality on the half space {\mathbf{R}}^n_+, Int. Math. Res. Not. IMRN, 2017, 20, 6187-6230 (2017) · Zbl 1405.42038
[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.; Sawyer, E.; Wheeden, R. L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114, 4, 813-874 (1992) · Zbl 0783.42011
[38] S. L. Sobolev, On a theorem in functional analysis (in Russian), Mat. Sb. 4 (1938), 471-497.; Sobolev, S. L., On a theorem in functional analysis, Mat. Sb., 4, 471-497 (1938)
[39] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970.; Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970) · Zbl 0207.13501
[40] E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University, Princeton, 1993.; Stein, E. M., Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals (1993) · Zbl 0821.42001
[41] E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503-514.; Stein, E. M.; Weiss, G., Fractional integrals on n-dimensional Euclidean space, J. Math. Mech., 7, 503-514 (1958) · Zbl 0082.27201
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.