# zbMATH — the first resource for mathematics

Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. (English) Zbl 0906.46026
Let $$X= \{X_1,\dots, X_n\}$$ be a system of locally Lipschitz real-valued vector fields in $$\mathbb{R}^n$$, $$d(x, y)$$ be the Carnot-Carathéodory distance associated to $$X$$, $$B(x_0, R)= \{x\in\mathbb{R}^n\mid d(x, x_0)< R\}$$. The authors introduce the following basic assumptions: the openness of the balls $$B(x_0, R)$$ in the Euclidean topology, the so-called “doubling condition”, and some Poincaré type inequality. Under these not very restrictive conditions some extension properties of Sobolev spaces are investigated and approximation of Sobolev functions by functions which are smooth up to the boundary of a domain is obtained. In particular, for the so-called $$(\varepsilon, \delta)$$-domain $$\Omega\subset\mathbb{R}^n$$ with the radius of $$\Omega>0$$ the following extension result is established.
Let $$1\leq p<\infty$$, let $${\mathcal L}^{1,p}(\Omega)= \{f\in L^p(\Omega)\mid X_jf\in L^p(\Omega), j= 1,\dots,m\}$$ be the weak Sobolev space endowed with the norm $$\| f\|_{{\mathcal L}^{1,p}(\Omega)}= \| f\|_{L^p(\Omega)}+ \sum^m_{j= 1}\| X_jf\|_{L^p(\Omega)}$$. Then there exists a linear operator $${\mathcal E}: {\mathcal L}^{1,p}(\Omega)\to {\mathcal L}^{1,p}(\mathbb{R}^n)$$ such that for some $$C>0$$ one has for $$f\in {\mathcal L}^{1,p}(\Omega)$$,
(i) $${\mathcal E}f(x)= f(x)$$ for a.e. $$x\in\Omega$$,
(ii) $$\|{\mathcal E}f\|_{{\mathcal L}^{1,p}(\mathbb{R}^n)}\leq C\| f\|_{{\mathcal L}^{1,p}(\Omega)}$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35S15 Boundary value problems for PDEs with pseudodifferential operators
##### MathOverflow Questions:
Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?
Full Text:
##### References:
  A. Bellaïche,Sub-Riemannian Geometry, Birkhäuser, Boston, 1996. · Zbl 0862.53031  L. Capogna, D. Danielli and N. Garofalo,An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations18 (1993), 1765–1794. · Zbl 0802.35024 · doi:10.1080/03605309308820992  L. Capogna, D. Danielli and N. Garofalo,Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math.118 (1996), 1153–1196. · Zbl 0878.35020 · doi:10.1353/ajm.1996.0046  L. Capogna and N. Garofalo,NTA domains for Carnot-Carathéodory metrics and a Fatou type theorem, preprint, 1995. · Zbl 0872.35030  W. L. Chow,Uber System von linearen partiellen Differentialgleichungen erster Ordnug, Math. Ann.117 (1939), 98–105. · JFM 65.0398.01 · doi:10.1007/BF01450011  M. Christ,The extension problem for certain function spaces involving fractional orders of differentiability, Ark. Mat.22 (1984), 63–81. · Zbl 0548.46025 · doi:10.1007/BF02384371  S. K. Chua,Extension theorems on weighted sobolev spaces, Indiana Univ. Math. J.4 (1992), 1027–1076. · Zbl 0767.46025 · doi:10.1512/iumj.1992.41.41053  R. Coifman and G. Weiss,Analyse harmonique non-commutative sur certains espaces homogenes, Springer-Verlag, Berlin, 1971. · Zbl 0224.43006  D. Danielli,Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. J.44 (1995), 269–286. · Zbl 0828.35022 · doi:10.1512/iumj.1995.44.1988  D. Danielli,A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, preprint. · Zbl 0940.35057  D. Danielli, N. Garofalo and D. M. Nhieu,Trace inequalities for Carnot-Carathéodory spaces and applications to quasilinear subelliptic equations, preprint. · Zbl 0938.46036  J. Deny and J. L. Lions,Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble)5 (1953), 305–370. · Zbl 0065.09903  L. C. Evans and R. F. Gariepy,Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. · Zbl 0804.28001  C. Fefferman and D. H. Phong,Subelliptic eigenvalue problems, inProceedings of the Conference in Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Ser., Belmont, CA, 1981, pp. 530–606.  C. Fefferman and A. Sanchez-Calle,Fundamental solutions for second order subelliptic operators, Ann. of Math. (2)124 (1986), 247–272. · Zbl 0613.35002 · doi:10.2307/1971278  G. B. Folland and E. M. Stein,Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, 1982. · Zbl 0508.42025  B. Franchi and E. Lanconelli,Une metrique associée à une classe d’operateurs elliptiques degénérés, inProceedings of the Meeting ”Linear Partial and Pseudo Differential Operators”, Rend. Sem. Mat. Univ. Politec. Torino, 1982.  B. Franchi and E. Lanconelli,Hölder regularity theorem for a class of linear non uniform elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa10 (1983), 523–541. · Zbl 0552.35032  B. Franchi and E. Lanconelli,Une condition géométrique pour l’inégalité de Harnack, J. Math. Pures Appl.64 (1985), 237–256. · Zbl 0599.35134  B. Franchi, R. Serapioni and F. Serra Cassano,Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math.22 (1996), 859–890. · Zbl 0876.49014  B. Franchi, R. Serapioni and F. Serra Cassano,Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7)11 (1997), 83–117. · Zbl 0952.49010  K. O. Friedrichs,The identity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc.55 (1944), 132–151. · Zbl 0061.26201  N. Garofalo,Recent Developments in the Theory of Subelliptic Equations and Its Geometric Aspects, Birkhäuser, Boston, to appear.  N. Garofalo and D. M. Nhieu,Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math.49 (1996), 1081–1144. · Zbl 0880.35032 · doi:10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A  W. Hansen and H. Huber,The Dirichlet problem for sublaplacians on nilpotent groups–Geometric criteria for regularity, Math. Ann.246 (1984), 537–547. · Zbl 0601.31007  P. Hartman,Ordinary Differential Equations, Birkhäuser, Boston, 1982. · Zbl 0476.34002  E. Hille,Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass., 1968. · Zbl 0179.40301  L. Hörmander,Hypoelliptic second-order differential equations, Acta Math.119 (1967), 147–171. · Zbl 0156.10701 · doi:10.1007/BF02392081  D. Jerison,The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J.53 (1986), 503–523. · Zbl 0614.35066 · doi:10.1215/S0012-7094-86-05329-9  D. Jerison and C. E. Kenig,Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math.46 (1982), 80–147. · Zbl 0514.31003 · doi:10.1016/0001-8708(82)90055-X  P. W. Jones,Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math.147 (1981), 71–88. · Zbl 0489.30017 · doi:10.1007/BF02392869  G. Lu,Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Mat.40 (1996), 301–329. · Zbl 0873.35006  O. Martio and J. Sarvas,Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I4 (1979), 384–401. · Zbl 0406.30013  N. Meyers and J. Serrin,H=W, Proc. Nat. Acad. Sci. U.S.A.51 (1964), 1055–1056. · Zbl 0123.30501 · doi:10.1073/pnas.51.6.1055  C. B. Morrey Jr.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. · Zbl 0142.38701  A. Nagel, E. M. Stein and S. Wainger,Balls and metrics defined by vector fields I: basic properties, Acta Math.155 (1985), 103–147. · Zbl 0578.32044 · doi:10.1007/BF02392539  D. M. Nhieu,The extension problem for Sobolev spaces on the Heisenberg group, Ph.D Thesis, Purdue University, 1996. · Zbl 0880.35032  O. A. Oleinik and E. V. Radkevich,Second order equations with non-negative characteristic form, inMathematical Analysis 1969, Itogi Nauki, Moscow, 1971 [Russian]; English translation: Amer. Math. Soc., Providence, R.I., 1973.  R. S. Phillips and L. Sarason,Elliptic-parabolic equations of the second order, J. Math. Mech.17 (1967/8), 891–917. · Zbl 0163.34402  L. Saloff-Coste,A note on Poincaré, Sobolev, and Harnack inequalities, Duke Math. J., I.M.R.N.2 (1992), 27–38. · Zbl 0769.58054  N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon,Analysis and Geometry on Groups, Cambridge Tracts in Mathematics 100, Cambridge University Press, 1992. · Zbl 0813.22003  S. K. Vodop’yanov and A. V. Greshnov,On extension of functions of bounded mean oscillation from domains in a space of homogeneous type with intrinsic metric, Siberian Math. J. 36,5 (1995), 873–901. · Zbl 0865.30029 · doi:10.1007/BF02112531
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.