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Hardy–Sobolev–Maz’ya inequalities for polyharmonic operators

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  • 1.

    Adimurthi, Filippas, S., Tertikas, A.: On the best constant of Hardy–Sobolev inequalities. Nonlinear Anal. 70, 2826–2833 (2009)

    MathSciNet 
    Article 

    Google Scholar
     

  • 2.

    Ahlfors, L.V.: Möbius Transformations in Several Dimensions, Ordway Professorship Lectures in Mathematics. University of Minnesota, School of Mathematics, Minneapolis, MN (1981)

  • 3.

    Anker, J.-P., Ji, L.: Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9, 1035–1091 (1999)

    MathSciNet 
    Article 

    Google Scholar
     

  • 4.

    Beckner, W.: On the Grushin operator and hyperbolic symmetry. Proc. Am. Math. Soc. 129, 1233–1246 (2001)

    MathSciNet 
    Article 

    Google Scholar
     

  • 5.

    Beckner, W.: On Lie groups and hyperbolic symmetry-from Kunze–Stein phenomena to Riesz potentials. Nonlinear Anal. 126, 394–414 (2015)

    MathSciNet 
    Article 

    Google Scholar
     

  • 6.

    Benguria, R.D., Frank, R.L., Loss, M.: The sharp constant in the Hardy–Sobolev–Maz’ya inequality in the three dimensional upper half space. Math. Res. Lett. 15, 613–622 (2008)

    MathSciNet 
    Article 

    Google Scholar
     

  • 7.

    Berchio, E., Ganguly, D.: Improved higher order Poincaré inequalities on the hyperbolic space via Hardy-type remainder terms. Comm. Pure Appl. Anal. 15(5), 1871–1892 (2015)

    MATH 

    Google Scholar
     

  • 8.

    Berchio, E., Ganguly, D., Grillo, G.: Sharp Poincaré–Hardy and Poincaré–Rellich inequalities on the hyperbolic space. J. Func. Anal. 272, 1661–1703 (2017)

    Article 

    Google Scholar
     

  • 9.

    Cowling, M., Giulini, S., Meda, S.: (L^{p}-L^{q}) estimates for functions of the Laplace–Beltrami operator on noncompact symmetric spaces. I. Duke Math. J. 72, 109–150 (1993)

    MathSciNet 
    Article 

    Google Scholar
     

  • 10.

    Davies, E.B., Hinz, A.M.: Explicit constants for Rellich inequalities in (L^{p}(Omega )). Math. Z. 227, 511–523 (1998)

    MathSciNet 
    Article 

    Google Scholar
     

  • 11.

    Davies, E.B., Mandouvalos, N.: Heat kernel bounds on hyperbolic space and Kleinian groups. Proc. Lond. Math. Soc. 52(3), 182–208 (1988)

    MathSciNet 
    Article 

    Google Scholar
     

  • 12.

    Dyda, B., Frank, R.L.: Fractional Hardy–Sobolev–Maz’ya inequality for domains. Stud. Math. 208, 151–166 (2012)

    MathSciNet 
    Article 

    Google Scholar
     

  • 13.

    Filippas, S., Maz’ya, V.G., Tertikas, A.: Sharp Hardy–Sobolev inequalities, C. R. Acad. Sci. Paris Ser. I 339(7), 483–486 (2004)

    MathSciNet 
    Article 

    Google Scholar
     

  • 14.

    Filippas, S., Maz’ya, V.G., Tertikas, A.: Critical Hardy–Sobolev inequalities. J. Math. Pures Appl. 87(1), 37–56 (2007)

    MathSciNet 
    Article 

    Google Scholar
     

  • 15.

    Filippas, S., Moschini, L., Tertikas, A.: Sharp trace Hardy–Sobolev–Maz’ya inequalities and the fractional Laplacian. Arch. Ration. Mech. Anal. 208, 109–161 (2013)

    MathSciNet 
    Article 

    Google Scholar
     

  • 16.

    Filippas, S., Moschini, L., Tertikas, A.: Trace Hardy–Sobolev–Mazy’a inequalities for the half fractional Laplacian. Comm. Pure Appl. Anal. 14, 373–382 (2014)

    Article 

    Google Scholar
     

  • 17.

    Filippas, S., Tertikas, A.: Optimizing improved Hardy inequalities. J. Funct. Anal. 192, 186–233. Corrigendum; J. Funct. Anal. 255(2008), 2095 (2002)

  • 18.

    Filippas, S., Tertikas, A., Tidblom, J.: On the structure of Hardy– Sobolev–Maz’ya inequalities. J. Eur. Math. Soc. 11, 1165–1185 (2008)

    MathSciNet 
    MATH 

    Google Scholar
     

  • 19.

    Filippas, S., Tertikas, A., Tidblom, J.: Optimal Hardy–Sobolev–Maz’ya inequalities with multiple interior singularities. In: Laptev, A. (eds.) Around the Research of Vladimir Maz’ya I. International Mathematical Series, vol. 11, pp. 137–160 (2010)

  • 20.

    Frank, R.L., Loss, M.: Hardy–Sobolev–Maz’ya inequalities for arbitrary domains. J. Math. Pures Appl. 97, 39–54 (2011)

    MathSciNet 
    Article 

    Google Scholar
     

  • 21.

    Frank, R.L., Seiringer, R.: Sharp fractional Hardy inequalities in half-spaces. In: Laptev, A. (eds.) Around the Research of Vladimir Maz’ya. International Mathematical Series, vol. II, pp. 161–167 (2010)

  • 22.

    Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.J.: Conformally invariant powers of the Laplacian. I. Existence. J. Lond. Math. Soc. 46(2), 557–565 (1992)

    MathSciNet 
    Article 

    Google Scholar
     

  • 23.

    Grigoryan, A., Noguchi, M.: The heat kernel on hyperbolic space. Bull. Lond. Math. Soc. 30, 643–650 (1998)

    MathSciNet 
    Article 

    Google Scholar
     

  • 24.

    Gover, A.R.: Laplacian operators and Q-curvature on conformally Einstein manifolds. Math. Ann. 336, 311–334 (2006)

    MathSciNet 
    Article 

    Google Scholar
     

  • 25.

    Helgason, S.: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Pure and Applied Mathematics. 113 Academic Press (1984)

  • 26.

    Helgason, S.: Geometric analysis on symmetric spaces, 2nd edition. Mathematical Surveys and Monographs, 39. American Mathematical Society, Providence, RI (2008)

  • 27.

    Juhl, A.: Explicit formulas for GJMS-operators and Q-curvatures. Geom. Funct. Anal. 23, 1278–1370 (2013)

    MathSciNet 
    Article 

    Google Scholar
     

  • 28.

    Li, J., Lu, G., Yang, Q.: Fourier analysis and optimal Hardy–Adams inequalities on hyperbolic spaces of any even dimension. Adv. Math. 333, 350–385 (2018)

    MathSciNet 
    Article 

    Google Scholar
     

  • 29.

    Li, J., Lu, G., Yang, Q.: Sharp Adams and Hardy–Adams inequalities of any fractional order on hyperbolic spaces of all dimensions. Trans. Am. Math. Soc. 373(5), 3483–3513 (2020)

    MathSciNet 
    Article 

    Google Scholar
     

  • 30.

    Liu, C., Peng, L.: Generalized Helgason–Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in ({mathbb{R}}^{n}). Indiana Univ. Math. J. 58(3), 1457–1492 (2009)

    MathSciNet 
    Article 

    Google Scholar
     

  • 31.

    Liu, C., Shi, J.: Invariant mean-value property and ({cal{M}})-harmonicity in the unit ball of ({mathbb{R}}^{n}). Acta Math. Sin. 19, 187–200 (2003)

    MathSciNet 
    Article 

    Google Scholar
     

  • 32.

    Liu, G.: Sharp higher-order Sobolev inequalities in the hyperbolic space ({mathbb{H}}^{n}). Calc. Var. Partial Differ. Equ. 47(3–4), 567–588 (2013)

    Article 

    Google Scholar
     

  • 33.

    Lu, G., Yang, Q.: A sharp Trudinger–Moser inequality on any bounded and convex planar domain. Calc. Var. Partial. Differ. Equ. 55(153), 1–16 (2016)

    MathSciNet 
    MATH 

    Google Scholar
     

  • 34.

    Lu, G., Yang, Q.: Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four. Adv. Math. 319, 567–598 (2017)

    MathSciNet 
    Article 

    Google Scholar
     

  • 35.

    Lu, G., Yang, Q.: Paneitz operators on hyperbolic spaces and high order Hardy–Sobolev–Maz’ya inequalities on half spaces. Am. J. Math. 141, 1777–1816 (2019)

    MathSciNet 
    Article 

    Google Scholar
     

  • 36.

    Lu, G., Yang, Q.: Green’s functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy–Sobolev–Maz’ya inequalities on half spaces. Preprint https://arxiv.org/abs/1903.10365

  • 37.

    Mancini, G., Sandeep, K.: On a semilinear elliptic equation in ({mathbb{H}}^{n}) , Ann. Scoula Norm. Sup. Pisa Cl. Sci. (5) Vol VII (2008), 635–671

  • 38.

    Mancini, G., Sandeep, K., Tintarev, K.: Trudinger–Moser inequality in the hyperbolic space ({mathbb{H}}^{N}). Adv. Nonlinear Anal. 2(3), 309–324 (2013)

    MathSciNet 
    MATH 

    Google Scholar
     

  • 39.

    Maz’ya, V.G.: Sobolev Spaces, 2nd edn. Springer, Berlin, Heidelberg (2011)

    Book 

    Google Scholar
     

  • 40.

    Maz’ya, V.G., Shaposhnikova, T.: A collection of sharp dilation invariant integral inequalities for differentiable functions. In: Maz’ya, V. (ed.), Sobolev Spaces in Mathematics I: Sobolev Type Inequalities, Int. Math. Ser., vol. 8, pp. 223–247. Springer (2009)

  • 41.

    Nguyen, V.H.: Some trace Hardy type inequalities and trace Hardy–Sobolev–Maz’ya type inequalities. J. Func. Anal. 270, 4117–4151 (2016)

    MathSciNet 
    Article 

    Google Scholar
     

  • 42.

    Pinchover, Y., Tintarev, K.: On the Hardy–Sobolev–Maz’ya inequality and its generalizations. In: Maz’ya, V. (ed.) “Sobolev Spaces in Mathematics I: Sobolev Type Inequalities”, International Mathematical Series 8. Springer, pp. 281–297 (2009)

  • 43.

    Secchi, S., Smets, D., Willem, M.: Remarks on a Hardy–Sobolev inequality. C. R. Acad. Sci. Paris, Ser. I 336, 811–815 (2003)

    MathSciNet 
    Article 

    Google Scholar
     

  • 44.

    Sloane, C.A.: A fractional Hardy–Sobolev–Maz’ya inequality on the upper half space. Proc. Am. Math. Soc. 139(11), 4003–4016 (2011)

    MathSciNet 
    Article 

    Google Scholar
     

  • 45.

    Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications. Springer, New York (1985)

    Book 

    Google Scholar
     

  • 46.

    Tertikas, A., Tintarev, K.: On existence of minimizers for the Hardy–Sobolev–Maz’ya inequality. Ann. Mat. Pura Appl. 186(4), 645–662 (2007)

    MathSciNet 
    Article 

    Google Scholar
     

  • 47.

    Tertikas, A., Zographopoulos, N.B.: Best constants in the Hardy–Rellich inequalities and related improvements. Adv. Math. 209, 407–459 (2007)

    MathSciNet 
    Article 

    Google Scholar
     

  • 48.

    Wang, G., Ye, D.: A Hardy–Moser–Trudinger inequality. Adv. Math. 230, 294–320 (2012)

    MathSciNet 
    Article 

    Google Scholar
     



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