The pointwise existence and properties of heat kernel
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Alexander Grigor’yan
Abstract
We consider a semigroup acting on the function space L1 based on a measure space. Assuming that the semigroup satisfies the L1-L∞ ultracontractivity property, we prove that it possesses an integral kernel that is defined pointwise and has some nice properties, including the joint measurability and the continuity in one variable. We apply this result to a heat semigroup associated with a regular Dirichlet form on the space L2.
Abstract
We consider a semigroup acting on the function space L1 based on a measure space. Assuming that the semigroup satisfies the L1-L∞ ultracontractivity property, we prove that it possesses an integral kernel that is defined pointwise and has some nice properties, including the joint measurability and the continuity in one variable. We apply this result to a heat semigroup associated with a regular Dirichlet form on the space L2.
Chapters in this book
- Frontmatter I
- Contents V
- Preface VII
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Part I: Fractals and graphs
- Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes 1
- The pointwise existence and properties of heat kernel 27
- Resistance estimates and critical exponents of Dirichlet forms on fractals 71
- A survey on unbounded Laplacians and intrinsic metrics on graphs 103
- Energy measures for diffusions on fractals: a survey 119
- Hyperbolic graphs induced by iterations and applications in fractals 143
- Geometric implications of fast volume growth and capacity estimates 183
- Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set 201
- Metrics and uniform Harnack inequality on the Strichartz hexacarpet 275
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Part II: Euclidean spaces and manifolds
- Analysis on manifolds and volume growth 299
- Geometric analysis on manifolds with ends 325
- A matrix Harnack estimate for a Kolmogorov type equation 345
- Entropy power concavity inequality on Riemannian manifolds and Ricci flow 359
- Fractional differential operators and divergence equations 385
- Interior gradient estimates for mean curvature type equations and related flows 421
- An alternate induction argument in Simons’ proof of holonomy theorem 443
- Higher integrability for nonlinear nonlocal equations with irregular kernel 459
- On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds 493
- Index 515
Chapters in this book
- Frontmatter I
- Contents V
- Preface VII
-
Part I: Fractals and graphs
- Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes 1
- The pointwise existence and properties of heat kernel 27
- Resistance estimates and critical exponents of Dirichlet forms on fractals 71
- A survey on unbounded Laplacians and intrinsic metrics on graphs 103
- Energy measures for diffusions on fractals: a survey 119
- Hyperbolic graphs induced by iterations and applications in fractals 143
- Geometric implications of fast volume growth and capacity estimates 183
- Parabolic index of an infinite graph and Ahlfors regular conformal dimension of a self-similar set 201
- Metrics and uniform Harnack inequality on the Strichartz hexacarpet 275
-
Part II: Euclidean spaces and manifolds
- Analysis on manifolds and volume growth 299
- Geometric analysis on manifolds with ends 325
- A matrix Harnack estimate for a Kolmogorov type equation 345
- Entropy power concavity inequality on Riemannian manifolds and Ricci flow 359
- Fractional differential operators and divergence equations 385
- Interior gradient estimates for mean curvature type equations and related flows 421
- An alternate induction argument in Simons’ proof of holonomy theorem 443
- Higher integrability for nonlinear nonlocal equations with irregular kernel 459
- On nonexistence results of porous medium type equations and differential inequalities on Riemannian manifolds 493
- Index 515
