# zbMATH — the first resource for mathematics

Sobolev and isoperimetric inequalities for degenerate metrics. (English) Zbl 0830.46027
Let $$X_1, \dots, X_m$$ be continuous vector fields in an open subset $$\Omega \subseteq \mathbb{R}^N$$. A Sobolev inequality holds for $$p\geq 1$$ if there exist $$q>p$$ and $$C= C(\Omega, p,q)$$ such that $\Biggl( \int_\Omega |u|^q dx\Biggr)^{1/q} \leq C \Biggl( \int_\Omega \biggl(\sum_j \langle X_j, \nabla u\rangle^2 \biggr)^{p/2} dx\Biggr)^{1/p},$ for any $$u\in C_0^\infty (\Omega)$$. A Sobolev inequality with $$p=1$$ and with the corresponding optimal choice of $$q$$ will be called a geometric inequality, it is equivalent to a suitable form of the isoperimetric inequality. In the present paper Sobolev’s inequality is proved, also the geometric inequality for some classes of vector fields.
Reviewer: J.Wloka (Kiel)

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D15 Inequalities for sums, series and integrals 52A38 Length, area, volume and convex sets (aspects of convex geometry) 31B10 Integral representations, integral operators, integral equations methods in higher dimensions
Full Text:
##### References:
 [1] Calderon A.P.: Inequalities for the maximal function relative to a metric. Stud. Math.57 (1976), 297-306 · Zbl 0341.44007 [2] Capogna L., Danielli D., Garofalo N.: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Commun. Partial Differ. Equations18 (1993), 1765-1794 · Zbl 0802.35024 [3] Chanillo, S., Wheeden R.L.: Weighted Poincar? and Sobolev inequalities and estimates for the Peano maximal function. Am. J. Math.107 (1985), 1191-1226 · Zbl 0575.42026 [4] Citti G., Garofalo N., Lanconelli E.: Harnack’s inequality for sums of squares plus a potential. Am. J. Math.115 (1993), 699-734 · Zbl 0795.35018 [5] Couhlon T.: Sobolev inequalities on graphs and on manifolds. Harmonic Analysis and Discrete Potential Theory. Picardello ed., Plenum Press, 1992 [6] Couhlon T.: In?galit? de Gagliardo-Nirenberg pour les semi-groupes d’op?rateurs et applications. Potential theory. (to appear) [7] Couhlon T., Saloff-Coste L.: Isop?rim?trie pour les groupes et les vari?t?s. Rev. Math. Iberoam.9 (1993), 293-314 [8] Danielli D.: Representation formulas and embedding theorem for subelliptic operators. C.R. Acad. Sci., Paris S?r. I, Math.314 (1992), 987-990 · Zbl 0768.46019 [9] David G., Semmes S.: StrongA ? weights, Sobolev inequalities and quasi-conformal mappings. Analysis and Partial Differential equations. Lect. Notes Pure Appl. Math.122 (1990), 101-111, Marcel Dekker, New York · Zbl 0752.46014 [10] Franchi B.: Weighted Sobolev-Poincar? inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans. Am. Math. Soc.327 (1991), 125-158 · Zbl 0751.46023 [11] Franchi B.: In?galit?s de Sobolev pour des champs de vecteurs lipschitziens. C.R. Acad. Sci. Paris S?r. I, Math.311 (1990), 329-332 · Zbl 0734.46023 [12] Franchi B.: Disuguaglianze di Sobolev e disuguaglianze isoperim?triche per metriche degeneri. In: Seminario di Analisi Matematica dell’Universit? di Bologna (January 1993), Bologna, 1993 [13] Federer H.: Geometric Measure Theory Springer, 1969 · Zbl 0176.00801 [14] Franchi B., Gallot S., Wheeden R.L.: In?galit?s isop?rim?triques pour des m?triques d?g?n?r?es. C.R. Acad. Sci. Paris S?r. I, Math.317 (1993), 651-654 [15] Franchi B., Gutierrez C., Wheeden R.L.: Weighted Sobolev-Poincar? inequalities for Grushin type operators. Commun. Partial Differ. Equations.19 (1994), 523-604 · Zbl 0822.46032 [16] Franchi B., Lanconelli E.: H?lder regularity for a class of linear non uniformly elliptic operators with measurable coefficients. Ann. Sc. Norm. Super. Pisa (IV)10 (1983), 523-541 · Zbl 0552.35032 [17] Fefferman C., Phong D.H.: Subelliptic eigenvalue problems. In: Conference on Harmonic Analysis, Chicago 1980, W. Beckner et al. ed. Wadsworth (1981), 590-606 · Zbl 0503.35071 [18] Genebashvili I., Gogatishvili A., Kokilashvili V.: Criteria of general weak type inequalities for integral transforms with positive kernels. Proc. Georgian Acad. Sci. Math.1 (1993), 11-34 · Zbl 0803.42011 [19] Jerison D., Sanchez-Calle A.: Subelliptic second order differential operators. Complex Analysis III. Lect. Notes Math.1277 (1987), Springer, 46-77 · Zbl 0634.35017 [20] Long R.L., Nie F.S.: Weighted Sobolev inequality and eigenvalue estimates of Schr?dinger operators. Harmonic Analysis (Tianjin, 1988). Lect. Notes Math. 1494, Springer, 1991 [21] Lu G.: Weighted Poincar? and Sobolev inequalities for vector fields satisfying H?rmander’s condition and applications. Rev. Mat. Iberoam.8 (1992), 367-439 · Zbl 0804.35015 [22] Nagel A., Stein E.M., Wainger S.: Balls and metrics defined by vector fields I: basic properties. Acta Math.155 (1985), 103-147 · Zbl 0578.32044 [23] Pansu P.: Une in?galit? isop?rim?trique sur le groupe de Heisenberg. C.R. Acad. Sci. Paris S?r. I, Math.295 (1985), 127-130 [24] Saloff Coste L.: In?galit?s de Sobolev produit sur les groupes de Lie nilpotents. J. Funct. Anal., 44-56 · Zbl 0697.22012 [25] Saloff Coste L.: A note on Poincar?, Sobolev and Harnack inequalities. Int. Math. Res. Notices2 (1992), 27-38 · Zbl 0769.58054 [26] Varopoulos N.Th.: Analysis on Lie groups. J. Funct. Anal.76 (1988), 364-410 · Zbl 0634.22008 [27] Varopoulos N.Th.: Sobolev inequalities on Lie groups and symmetric spaces. J. Funct. Anal.86 (1989), 19-40 · Zbl 0697.22013 [28] Varopoulos N.Th., Saloff Coste L., Couhlon T.: Analysis and geometry on groups. Camb. Tracts Math.100, 1993
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.