ASPECTS OF SOBOLEV-TYPE INEQUALITIES (London Mathematical Society Lecture Note Series 289) By LAURENT SALOFF-COSTE: 190 pp., £24.95 (LMS members' price £18.71), ISBN 0 521 00607 4 (Cambridge…

By D. Edmunds. 2002

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Smooth metric measure spaces with non-negative curvature (2011, methodology): “5 by the Moser iteration argument as in [30], Chapter II.”
On the equivalence of parabolic Harnack inequalities and heat kernel estimates (2012, unknown): “Then, for any t ∈ (0, T ), u(t, ·) ≥ PΩt f in Ω. (2.5)”
Neumann and Dirichlet Heat Kernels in Inner Uniform Domains (2011, background): “The following theorem due independently to Grigor’yan [47] and Saloff-Coste [76] shows that all these properties are very strongly interrelated.”
Analysis of weighted Laplacian and applications to Ricci solitons (2011, methodology): “3, we may apply the Moser iteration argument (see [14, 25]) to |∇u|(2) to conclude that (3.”
Upper Bounds of Heat Kernels on Doubling Spaces (2014, background): “There is also a vast literature devoted to upper and lower bounds of the heat kernel in connection with the geometry of M (see, for example, [8], [11], [16], [17], [26], [28], [29], [30]).”
Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “The remaining part of the proof follows the arguments in [35], Theorem 2.”