Hardy's inequality and Green function on metric measure spaces

Abstract We prove an abstract form of Hardy's inequality for local and non-local regular Dirichlet forms on metric measure spaces, using the Green operator of the Dirichlet form in question. Under additional assumptions such as the volume doubling, the reverse volume doubling, and certain natural estimates of the Green function, we obtain the “classical” form of Hardy's inequality containing distance to a reference point or set.

[1]  S. Tasena.  Heat Kernal Analysis On Weighted Dirichlet Spaces , 2011 .

[2]  A. Grigor’yan.  THE HEAT EQUATION ON NONCOMPACT RIEMANNIAN MANIFOLDS , 1992 .

[3]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[4]  Nicholas T. Varopoulos,et al.  Analysis and Geometry on Groups , 1993 .

[5]  Bartlomiej Dyda,et al.  Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities , 2017, 1705.01360.

[6]  Heat kernel estimates on connected sums of parabolic manifolds , 2016, 1608.01596.

[7]  Farit G. Avkhadiev,et al.  Unified Poincaré and Hardy inequalities with sharp constants for convex domains , 2007 .

[8]  A. Grigor’yan,et al.  Sub-Gaussian estimates of heat kernels on infinite graphs, to appera in Duke Math , 2001 .

[9]  Matthias Keller,et al.  Intrinsic Metrics on Graphs: A Survey , 2014, 1407.7453.

[10]  Richard F. Bass,et al.  Brownian Motion and Harmonic Analysis on Sierpinski Carpets , 1999, Canadian Journal of Mathematics.

[11]  Surgery of the Faber–Krahn inequality and applications to heat kernel bounds , 2016 .

[12]  A. Grigor’yan.  Heat Kernel and Analysis on Manifolds , 2012 .

[13]  Martin T. Barlow,et al.  Brownian motion on the Sierpinski gasket , 1988 .

[14]  A. Ben Amor,et al.  Hardy's inequality in the scope of Dirichlet forms , 2009, 0904.0179.

[15]  Ronald R. Coifman,et al.  Analyse harmonique non-commutative sur certains espaces homogènes : étude de certaines intégrales singulières , 1971 .

[16]  W. D. Evans,et al.  The Analysis and Geometry of Hardy's Inequality , 2015 .

[17]  M. Gromov.  Sign and geometric meaning of curvature , 1991 .

[18]  HARDY-TYPE INEQUALITIES , 2000 .

[19]  Alexander Grigor'yan,et al.  Heat kernel upper bounds on a complete non-compact manifold. , 1994 .

[20]  Lars-Erik Persson,et al.  The Prehistory of the Hardy Inequality , 2006, Am. Math. Mon..

[21]  G. Carron,et al.  INÉGALITÉS ISOPÉRIMÉTRIQUES DE FABER-KRAHN ET CONSÉQUENCES , 2002 .

[22]  Donna Marie Bennett.  An extension of Rellich’s inequality , 1989 .

[23]  Martin L. Silverstein,et al.  Symmetric Markov Processes , 1975 .

[24]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .

[25]  Jiaxin Hu,et al.  Heat kernels on metric measure spaces and an application to semilinear elliptic equations , 2003 .

[26]  P. Sobolevskiĭ,et al.  The best possible constant in generalized Hardy's inequality for convex domain in R n , 1997 .

[27]  D. Edmunds,et al.  Weighted Hardy inequalities , 2005 .

[28]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[29]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[30]  R. Strichartz.  ANALYSIS ON FRACTALS , 1999 .

[31]  S. Dipierro,et al.  Hardy inequalities on Riemannian manifolds and applications , 2012, 1210.5723.

[32]  J. Kigami,et al.  Local Nash Inequality and Inhomogeneity of Heat Kernels , 2004 .

[33]  Martin T. Barlow,et al.  Diffusions on fractals , 1998 .

[34]  Which values of the volume growth and escape time exponent are possible for a graph , 2004 .

[35]  E. Davies.  A review of Hardy inequalities , 1998, math/9809159.

[36]  William Beckner,et al.  Pitt’s inequality with sharp convolution estimates , 2006, math/0612486.

[37]  Yehuda Pinchover,et al.  An Improved Discrete Hardy Inequality , 2016, Am. Math. Mon..

[38]  A. Grigor’yan,et al.  TWO-SIDED ESTIMATES OF HEAT KERNELS ON METRIC MEASURE SPACES , 2012, 1205.5627.

[39]  P. Fitzsimmons.  Hardy's Inequality for Dirichlet Forms , 2000 .

[40]  Gerard Kerkyacharian,et al.  Heat Kernel Generated Frames in the Setting of Dirichlet Spaces , 2012, 1206.0463.

[41]  A. Grigor’yan,et al.  On the equivalence of parabolic Harnack inequalities and heat kernel estimates , 2012 .

[42]  Nicolas Victoir,et al.  Analysis on local Dirichlet spaces , 2010 .

[43]  Jiaxin Hu,et al.  Heat Kernels and Green Functions on Metric Measure Spaces , 2014, Canadian Journal of Mathematics.

[44]  Jiaxin Hu,et al.  Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces , 2008 .

[45]  Poincaré inequality: From remote balls to all balls ☆ , 2014 .

[46]  L. Saloff-Coste,et al.  Stability results for Harnack inequalities , 2005 .

[47]  N. Ghoussoub,et al.  Bessel pairs and optimal Hardy and Hardy–Rellich inequalities , 2011 .

[48]  J. Deny,et al.  Espaces de dirichlet , 1958 .

[49]  A. Tertikas,et al.  A unified approach to improved L p Hardy inequalities with best constants , 2003, math/0302326.

[50]  A. Grigor’yan.  Heat kernels on metric measure spaces with regular volume growth , 2009 .

[51]  福島 正俊,et al.  Symmetric Markov Processes, Time Change, and Boundary Theory , 2011 .

[52]  G. Lu,et al.  Geometric Hardy's inequalities with general distance functions , 2020 .

[53]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[54]  Potential-theoretic nature of Hardy's inequality for Dirichlet forms , 2006 .

[55]  N. Masmoudi.  About the Hardy Inequality , 2011 .

[56]  J. Sarvas,et al.  Injectivity theorems in plane and space , 1979 .

[57]  I. McGillivray.  A recurrence condition for some subordinated strongly local Dirichlet forms , 1997 .

[58]  Alexander Grigor'yan,et al.  Heat kernels and function theory on metric measure spaces , 2003 .

[59]  R. Strichartz,et al.  Spectral analysis on infinite Sierpiński fractafolds , 2010, 1011.1049.

[60]  J. Xiao.  p-capacity vs surface-area , 2015, 1506.03827.

[61]  I. Kombe,et al.  Improved Hardy and Rellich inequalities on Riemannian manifolds , 2007, math/0702047.

[62]  Jiaxin Hu,et al.  Upper Bounds of Heat Kernels on Doubling Spaces , 2014 .

[63]  W. D. Evans,et al.  Hardy Operators, Function Spaces and Embeddings , 2004 .

[64]  M. Keller,et al.  Optimal Hardy inequalities for Schrödinger operators on graphs , 2016, 1612.04051.

[65]  L. Saloff-Coste,et al.  HEAT KERNEL ON MANIFOLDS WITH ENDS , 2009 .

[66]  Ronald R. Coifman,et al.  Analyse Hamonique Non-Commutative sur Certains Espaces Homogenes , 1971 .

[67]  Jiaxin Hu,et al.  Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces , 2015 .

[68]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties. , 1994 .

[69]  L. Peter,et al.  On the parabolic kernel of the Schr 6 dinger operator , 2006 .

[70]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[71]  D. Cruz-Uribe,et al.  Corrigendum to “Gaussian bounds for degenerate parabolic equations” [J. Funct. Anal. 255 (2) (2008) 283–312] , 2014 .

[72]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality , 1996 .

[73]  R. T. Lewis,et al.  Singular elliptic operators of second order with purely discrete spectra , 1982 .

[74]  Zhen-Qing Chen,et al.  Stability of heat kernel estimates for symmetric jump processes on metric measure spaces , 2016 .

[75]  Yongsheng Han,et al.  A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces , 2008 .

[76]  R. Strichartz.  Review: E. B. Davies, Heat kernels and spectral theory , 1990 .

[77]  A. M. Hinz,et al.  Explicit constants for Rellich inequalities in $L_p(\Omega)$ , 1998 .