We consider various types of Hardy-Sobolev inequalities on a Carnot-Caratheodory space (Ω, d) associated to a system of smooth vector fields Χ = {Χ 1,Χ 2,…,Χm} on Rn satisfying the Hormander finite rank condition rank Lie[Χ 1,…, Ωm] ≡ n. One of our main concerns is the trace inequality \(\int\limits_\Omega ^{} {|\ell \left( x \right)} |^p V\left( x \right)dx \le C\int\limits_\Omega ^{} {|X\ell |^P dx},\ell \in C_0^\infty \left( \Omega \right),\) where V is a general weight, i.e., a nonnegative locally integrable function on Ω, and 1 < p < +∞. Under sharp geometric assumptions on the domain Ω C Rn that can be measured equivalently in terms of subelliptic capacities or Hausdorff contents, we establish various forms of Hardy-Sobolev type inequalities.
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References
Ancona, A.: On strong barriers and an inequality of Hardy for domains in Rn. J. London Math. Soc. 34., 274–290 (1986)
Biroli, M.: Schrödinger type and relaxed Dirichlet problems for the subelliptic p-Laplacian. Potential Anal. 15, 1–16 (2001)
Björn, J., MacManus, P., Shanmugalingam, N.: Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces. J. Anal. Math. 85, 339–369 (2001)
Capogna, L., Danielli, D., Garofalo, N.: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Commun. Partial Differ. Equ. 18, no. 9–10, 1765–1794 (1993)
Capogna, L., Danielli, D., Garofalo, N.: Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations. Am. J. Math. 118, no.6, 1153–1196 (1996)
Capogna, L., Danielli, D., Garofalo, N.: Subelliptic mollifiers and a basic pointwise estimate of Poincaré type. Math. Z. 226, 147–154 (1997)
Capogna, L., Garofalo, N.: Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics. J. Fourier Anal. Appl. 4, no. 4–5, 403–432 (1998)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)
Chow, W.L.: Über systeme von linearen partiellen Differentialgleichungen erster Ord-nung. Math. Ann. 117, 98–105 (1939)
Ciatti, P., Ricci, F., Sundari, M.: Heisenberg-Pauli-Weyl uncertainty inequalities and polynomial volume growth. Adv. Math. 215, 616–625 (2007)
D'Ambrosio, L.: Hardy-type inequalities related to degenerate elliptic differential operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) IV, 451–486 (2005)
Danielli, D.: Regularity at the boundary for solutions of nonlinear subelliptic equations. Indiana Univ. Math. J. 44, 269–285 (1995)
Danielli, D.: A Fefferman-Phong type inequality and applications to quasilinear subel-liptic equations. Potential Anal. 11, 387–413 (1999)
Danielli, D., Garofalo, N.: Green Functions in Nonlinear Potential Theory in Carnot Groups and the Geometry of their Level Sets. Preprint (2003)
Danielli, D., Garofalo, N., Phuc, N.C.: Hardy-Sobolev Inequalities with Sharp Constants in Carnot-Carath´eodory Spaces. Preprint (2008)
Duc, D.M., Phuc, N.C., Nguyen, T.V.: Weighted Sobolev's inequalities for bounded domains and singular elliptic equations. Indiana Univ. Math. J. 56, 615–642 (2007)
Fefferman, C.: The uncertainty principle. Bull. Am. Math. Soc. 9, 129–206 (1983)
Folland, G.B., Stein,E.M.: Hardy Spaces on Homogeneous Groups. Princeton Univ. Press, Princeton, NJ (1982)
Folland, G.B.: A fundamental solution for a subelliptic operator. Bull. Am. Math. Soc. 79, 373–376 (1973)
Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13, 161–207 (1975)
Franchi, B., Serapioni, R., Serra Cassano, F.: Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. Unione Mat. Ital., VII, Ser. B 11, no. 1, 83–117 (1997)
Garofalo, N., Lanconelli, E.: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 40, no. 2, 313–356 (1990)
Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot- Caratheodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)
Garofalo, N., Nhieu, D.M.: Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. J. Anal. Math. 74, 67–97 (1998)
Hajlasz, P.: Pointwise Hardy inequalities, Proc. Am. Math. Soc. 127, 417–423 (1999)
Hansson, K., Maz’ya, V.G., Verbitsky, I.E.: Criteria of solvability for multidimensional Riccati equations. Ark. Mat. 37, 87–120 (1999)
Hardy, G.: Note on a theorem of Hilbert. Math. Z. 6, 314–317 (1920)
Heinonen, J., Holopainen, I.: Quasiregular maps on Carnot groups. J. Geom. Anal. 7, no. 1, 109–148 (1997)
Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)
Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, Oxford (1993)
Hormander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Hunt, R.: On L p,q spaces. Eins. Math. 12, 249–276 (1966)
Jerison, D.: The Poincaré inequality for vector fields satisfying Hormander's condition. Duke Math. J. 53, 503–523 (1986)
Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 147–153 (1980)
Kilpelainen, T., Maly, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)
Kombe, I.: Sharp Weighted Hardy Type Inequalities and Uncertainty Principle Inequality on Carnot Groups. Preprint (2005)
Lehrbäck, J.: Pointwise Hardy inequalities and uniformly fat sets. Proc. Am. Math. Soc. [To appear]
Lewis, J.L.: Uniformly fat sets. Trans. Am. Math. Soc. 308, 177–196 (1988)
Lohoué, N.: Une variante de l’inégalité de Hardy. Manuscr. Math. 123, 73–78 (2007)
Maz'ya, V.G.: The negative spectrum of the n-dimentional Schrödinger operator (Russian). Dokl. Akad. Nauk SSSR 144, 721–722 (1962); English transl.: Sov. Math., Dokl. 3, 808–810 (1962)
Maz'ya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin-Tokyo (1985)
Mikkonen, P.: On the Wolff potential and quasilinear elliptic equations involving measures. Ann. Acad. Sci. Fenn., Ser AI, Math. Dissert. 104 (1996)
Monti, R., Morbidelli, D.: Regular domains in homogeneous groups. Trans. Am. Math. Soc. 357, no. 8, 2975–3011 (2005)
Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields I: Basics properties. Acta Math. 155, 103–147 (1985)
Niu, P., Zhang, H., Wang, Y.: Hardy type and Rellich type inequalities on the Heisenberg group. Proc. Am. Math. Soc. 129, no. 12, 3623–3630 (2001)
Rashevsky, P.K.: Any two points of a totally nonholonomic space may be connected by an admissible line (Russian). Uch. Zap. Ped. Inst. Liebknechta, Ser. Phys. Math. 2, 83–94 (1938)
Rothschild, L.P., Stein, E.M.: Hypoelliptic differential operators and nilpotent groups. Acta. Math. 137, 247–320 (1976)
Sanchez-Calle, A.: Fundamental solutions and geometry of sum of squares of vector fields. Invent. Math. 78, 143–160 (1984)
Sobolev, S.L.: On a theorem of functional analysis (Russian). Mat. Sb. 46, 471–497 (1938); English transl.: Am. Math. Soc., Transl., II. Ser. 34, 39–68 (1963)
Trudinger, N.S., Wang, X.J.: On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124, 369–410 (2002)
Wannebo, A.: Hardy inequalities. Proc. Am. Math. Soc. 109, 85–95 (1990)
Stein, E.M.: Singular Integrals and Differentiability of Functions. Princeton Univ. Press, Princeton, NJ (1970)
Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ (1993)
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Danielli, D., Garofalo, N., Phuc, N.C. (2009). Inequalities of Hardy–Sobolev Type in Carnot–Carathéodory Spaces. In: Maz’ya, V. (eds) Sobolev Spaces In Mathematics I. International Mathematical Series, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-0-387-85648-3_5
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