Gauged Sobolev inequalities

References

• Adams , DR and Hedberg , L.I . 1996 . Function Spaces and Potential Theory , Berlin : Springer .
   Google Scholar
• Alkhutov , Yu.A. and Zhikov , VV . 2001 . On the Hölder property of solutions of degenerate elliptic equations. . Doklady Akademii Nauk , 378 : 583 – 588 .
   Google Scholar
• Bauer , H . 2001 . Measure and Integration Theory , Berlin-New York : Walter de Gruyter .
   Google Scholar
• Barlow , MT and Hambly , BM . 1997 . Transition density estimates for Brownian motion on scale irregular Sierpiński gaskets. . Annales de l'Institut Henri Poincaré , 33 ( 5 ) : 531 – 557 .
   Google Scholar
• Bergh , J and Löfström , L . 1976 . Interpolation Spaces. An Introduction , Berlin–Heidelberg–New York : Springer .
   Google Scholar
• Biroli , M and Mosco , U . 1993 . “ Sobolev inequalities for Dirichlet forms on homogeneous spaces ” . In Boundary Value Problems for Partial Differential Operators, RMA Research Notes in Applied Mathematics , Edited by: Baiocchi , C and Lions , JL . Vol. 29 , 305 – 311 . Paris : Masson .
   Google Scholar
• Biroli , M and Mosco , U . 1995 . Sobolev inequalities on homogeneous spaces. . Potential Analysis , 4 ( 4 ) : 311 – 324 .
   Web of Science ®Google Scholar
• Biroli , M and Mosco , U . 1995 . Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces . Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni , 6 ( 1 ) : 37 – 44 .
   Google Scholar
• Biroli , M and Mosco , U . 1995 . A Saint-Venant type principle for Dirichlet forms on discontinuous media . Annali de Mathematica Pura ed Applicata , 169 ( 4 ) : 125 – 181 .
   Google Scholar
• Bourgain , J , Brezis , H and Mironescu , P . 2001 . “ Another look at Sobolev spaces ” . In Optimal Control and Partial Differential Equations – Innovations and Applications. In honor of Professor Alain Bensoussan s 60th anniversary , Edited by: Menaldi , JL , Rofman , E and Sulem , A . 439 – 455 . Paris : Editors IOS Press .
   Google Scholar
• Capitanelli , R . 2001 . Lagrangeans on Homogeneous Spaces , U. of Rome La Sapienza : Roma .
   Google Scholar
• Chanillo , S and Wheeden , RL . 1985 . Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions . American Journal of Mathematics , 107 ( 5 ) : 1191 – 1226 .
   Web of Science ®Google Scholar
• Coifman , RR and Weiss , G . 1971 . Analyse Harmonique Non Commutative Sur Certains Espaces Homogenes, Lecture Notes in Mathematics , Vol. 242 , Berlin-New York : Springer .
   Google Scholar
• Dal Maso , G , Mosco , U and Vivaldi , MA . 1989 . A pointwise regularity theory for the two-obstacle problem . Acta Mathematica , 63 : 57 – 107 .
   Google Scholar
• Eells , J and Fuglede , B . 2001 . “ Harmonic Maps between Riemannian Polyhedra ” . In Cambridge Tracts in Mathematics , Vol. 142 , Cambridge : Cambridge University Press .
   Google Scholar
• Fabes , E , Kenig , C and Serapioni , R . 1982 . The local regularity of solutions of degenerate elliptic equations . Communications in Partial Differential Equations , 7 ( 1 ) : 77 – 116 .
   Web of Science ®Google Scholar
• Falconer , K.J . 1986 . Random graphs . Mathematical Proceedings of the Cambridge Philosophical Society , 100 : 559 – 582 .
   Google Scholar
• Fefferman , CL and Phong , DH . 1983 . “ Subelliptic eigenvalue problems ” . In Conference on Harmonic analysis in Honor of Antoni Zygmund Chicago, 1981, Proceedings , Edited by: Becker , W . 590 – 606 . Belmout, CA : Wadsworth .
   Google Scholar
• Fefferman , CL and Sanchez-Calle , A . 1986 . Fundamental solutions for second order subelliptic operators . Annals of Mathematics , 124 ( 2 ) : 247 – 272 .
   Web of Science ®Google Scholar
• Franchi , B , Lu , G and Wheeden , RL . 1995 . Representation formulas and weighted Poincaré inequalities for Hörmander vector fields . Annales de l'Institute Fourier , 45 : 577 – 604 .
   Web of Science ®Google Scholar
• Franchi , B , Perez , C and Wheeden , R.L. 1998 . Self-improving properties of John–Nirenberg and Poincaré inequalities on spaces of homogeneous Type . Journal of Functional Analysis , 153 ( 1 ) : 108 – 146 .
   Web of Science ®Google Scholar
• Graf , S , Mauldin , RD , Williams , SC and Hambly , BM . 1988 . The exact Hausdorff dimension in random recursive constructions . Memoirs of the American Mathematical Society , 381 : 126
   Google Scholar
• Grigoryan , A . 1999 . “ Isoperimetric inequalities and capacities on Riemannian manifolds ” . In The Maz'ya Anniversary Collection , Edited by: Rossman , J , Tak , P and Wildenhain , G . Vol. I , 139 – 153 . Basel : Birkhäuser .
   Google Scholar
• Hambly , B.M . 1992 . Brownian motion on a homogeneous random fractal . Probablity Theory Relatéd Fields , 94 : 1 – 38 . Theory, Uspekhi Matematicheskoe Nauk, 27, 67–148 (1972); Russian Mathematical Surveys, 27, 71–148
   Google Scholar
• Hajłasz , P . 1996 . Sobolev spaces on an arbitrary metric space . Potential Analysis , 5 : 403 – 415 .
   Web of Science ®Google Scholar
• Hajłasz , P and Koskela , P . 2000 . Sobolev met Poincaré . Memoirs of American Mathematical Society , 145 : 688
   Google Scholar
• Hedberg , L . 1972 . On certain convolution inequalities . Proceedings of the American Mathematical Societý , 36 : 505 – 510 .
   Web of Science ®Google Scholar
• Hedberg , L . 1981 . Spectral synthesis in Sobolev spaces and uniqueness of solutions of the Dirichlet problem . Acta Mathematica , 147 : 237 – 264 .
   Web of Science ®Google Scholar
• Heinonen , J . 2000 . Lecture on Analysis on Metric Spaces , Berlin-New York : Springer V .
   Google Scholar
• Hörmander , L . 1967 . Hypoelliptic second order differential equations . Acta Mathematica , 119 : 147 – 171 .
   Web of Science ®Google Scholar
• Jerison , D . 1986 . The Poincaré inequality for vector fields satisfying an Hörmander's condition . Duke Mathematical Journal , 53 ( 2 ) : 503 – 523 .
   Web of Science ®Google Scholar
• Jerison , D and Sanchez Calle , A . 1987 . “ Subelliptic Second Order Differential Operators ” . In Lectures Notes in Mathematica , Vol. 1277 , 46 – 77 . Berlin-New York : Springer .
   Google Scholar
• Jonsson , A and Wallin , H . 1984 . Function Spaces on Subsets of . Part 1. Math Reports , Vol. 2 , London : Harward Acadamic Publishers .
   Google Scholar
• Jost , J . 1998 . “ Nonlinear Dirichlet forms ” . In New Directions in Dirichlet Forms , Edited by: Jost , J , Kendall , W , Mosco , U , Röckner , M and Sturm , K.-Th . Vol. 8 , 117 – 155 . AMS/IP Stud. Adv. Math. .
   Google Scholar
• Long , RL and Nie , FS . 1988 . “ Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators, Harmonic Analysis, Tianjin ” . In Lecture Notes in Mathematics , Vol. 1494 , 131 – 141 . Berlin-New York : Springer .
   Google Scholar
• Lorentz , G.G . 1950 . Some new functional spaces . Annals of Mathematics , 51 ( 2 ) : 37 – 55 .
   Google Scholar
• Lorentz , G.G . 1951 . On the theory of spaces Λ . Pacific Journal of Mathematics , 1 : 411 – 429 .
   Google Scholar
• Macias , RA and Segovia , C . 1979 . Lipschitz functions on spaces of homogeneous type . Advances in Mathematics , 33 : 257 – 270 .
   Web of Science ®Google Scholar
• MacManus , P and Perez , C . 1998 . Generalized Poincaré inequalities: sharp self-improving properties . International Mathematics Research Notices , 2 : 101 – 116 .
   Google Scholar
• Malý , J and Mosco , U . 1999 . Remarks on measure-valued Lagrangeans on homogeneous spaces . Research Mathematica Napoli , 48 ( Suppl. ) : 217 – 231 .
   Google Scholar
• Mauldin , RD and Williams , SC . 1988 . Hausdorff dimension in graph directed constructions . Transactions of the American Mathematical Society , 309 : 811 – 829 .
   Web of Science ®Google Scholar
• Maz'ya , V.G . 1964 . A theorem on multidimensional Schrödinger operator . Izvestiya Akademii Nauk , 28 : 1145 – 1172 .
   Google Scholar
• Maz'ya , V.G . 1985 . Sobolev Spaces , Berlin-New York : Sringer V .
   Google Scholar
• Maz'ya , V.G . 1997 . Lectures on isoperimetric and isocapacitary inequalities in the theory of sobolev spaces . Contemporary Mathematics , (to appear) Functions on Bad Domains. World Scientific
   Google Scholar
• Mosco , U . 1994 . Composite media and asymptotic Dirichlet forms . Journal of Functional Analysis , 123 ( 2 ) : 368 – 421 .
   Web of Science ®Google Scholar
• Mosco , U . 1997 . Variational fractals . Ann. Scuola Norm. Sup. Pisa, Special Volume in Memory of E. De Giorgi , 4 ( XXV ) : 683 – 712 .
   Google Scholar
• Mosco , U . 1998 . “ Dirichlet forms and self-similarity ” . In New directions in Dirichlet Forms, AMS/IP Stud. Adv. Math , Edited by: Jost , J , Kendall Mosco , W , Röckner , M and Sturm , K.-Th. 117 – 155 . Providence, RI : American Mathematical Society .
   Google Scholar
• Mosco , U . 2002 . Harnack inequalities on recurrent metric fractals . Trudy Matem. Inst. imeni V.A. Steklova , 236 : 503 – 508 . Proceedings of the Steklov Institute of Mathematics, 236, 1–6
   Google Scholar
• Mosco , U . 2003 . “ Harnack inequalities on scale irregular Sierpiński gaskets ” . In Nonlinear Problems in Mathematical Physics and Related Topics II, In Honor of Prof. O.A. Ladyzenskaya , Edited by: Birman , M Sh , Hildebrandt , S , Solonnikov , VA and Uraltseva , NN . USA : AMS Intern. Press: Kluwer–Plenum Publisher .
   Google Scholar
• Mosco , U . “ Irregular Similarity and Quasi-Metric scaling ” . In Conf. “Stochastic and Potential Theory” , Edited by: De la Pradelle , A and Feyel , D . 1 – 6 . France : Saint-Priest-de-Gimel . September. Lecture Notes of the Conference
   Google Scholar
• Mosco , U . 2006 . “ Measure-valued Lagrangians ” . In Vol. in memory of L. Amerio, Memorie Mat. Appl. Acc. XL , Roma : Accademia Nazionale Delle Scienze Detta Dei XL .
   Google Scholar
• Mosco , U and Vivaldi , MA . 2003 . Variational problems with fractal layers . Memorie Mat. Appl. Rendiconti Acc. XL, 121⁰, XXVII Fasc 1 , : 237 – 251 .
   Google Scholar
• Nagel , A , Stein , EM and Wainger , S . 1985 . Balls and metrics defined by vector fields. I. Basic properties . Acta Mathematics , 155 ( 1–2 ) : 103 – 147 .
   Web of Science ®Google Scholar
• O'Neil , R . 1963 . Convolution operators and Lp,q spaces . Duke Mathematical Journal , 30 : 129 – 142 .
   Web of Science ®Google Scholar
• Peetre , J . 1966 . Espaces d'interpolation et espaces de Soboleff . Annales de l'Institute Fourier Grenoble , 16 : 279 – 317 .
   Web of Science ®Google Scholar
• Rothschild , LP and Stein , E . 1985 . Hypoelliptic differential operators and nilpotent groups . Acta Mathematics , 137 : 247 – 320 .
   Google Scholar
• Saloff-Coste , L . 1992 . A note on Poincaré, Sobolev and Harnack inequalities . International Mathematical Research Notices , 65 ( 2 ) : 27 – 38 .
   Google Scholar
• Stein , E.M . 1995 . Harmonic Analysis , Princeton : Princeton University Series .
   Google Scholar
• Stein , EM and Weiss , G . 1971 . Introduction to Fourier Analysis on Euclidean Spaces Princeton : Princeton University Series, .
   Google Scholar
• Tartar , L . 1998 . Imbedding theorems in Lorentz spaces and applications . Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 8 , 1 ( 3 ) : 479 – 500 .
   Google Scholar
• Triebel , H . 1992 . Theory of Function Spaces II , Basel : Birkhäuser .
   Google Scholar
• Triebel , H . 1997 . Fractals and Spectra Related to Fourier Analysis and Function Spaces, Monographs in Mathematics , Vol. 91 , Basel : Birkhäuser .
   Google Scholar
• Varopoulos , N.T . 1989 . Sobolev inequalities on Lie groups and symmetric spaces . Journal of Functional Analysis , 86 : 19 – 40 .
   Web of Science ®Google Scholar
• Ziemer , W.P . 1989 . Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variations. Graduate Texts in Mathematics , Vol. 120 , Berlin-New York : Springer .
   Google Scholar