## Heat kernels on metric-measure spaces and an application to semilinear elliptic equations (2003)

Venue: | Trans. Amer. Math. Soc |

Citations: | 22 - 7 self |

### BibTeX

@ARTICLE{Hu03heatkernels,

author = {Jiaxin Hu and Ka-sing Lau},

title = {Heat kernels on metric-measure spaces and an application to semilinear elliptic equations},

journal = {Trans. Amer. Math. Soc},

year = {2003},

pages = {2065--2095}

}

### OpenURL

### Abstract

Abstract. We consider a metric measure space (M, d,µ) andaheat kernel pt(x, y) on M satisfying certain upper and lower estimates, which depend on two parameters α and β. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, µ). Namely, α is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M. Moreover, the parameters α and β are related by the inequalities 2 ≤ β ≤ α +1. We prove also the embedding theorems for the space W β/2,2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form −Lu + f(x, u) =g(x), where L is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiński carpet in Rn. 1.

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Citation Context ...a posteriori they happen to be the invariants of the space (M,d,µ) itself, provided the function Φ2 decays at ∞ sufficiently fast. The parameter α happens to be the Hausdorff dimension of M (see also =-=[10]-=-). The nature of the parameter β is more complicated. We will call it the walk dimension of the heat kernel pt. This terminology is motivated by the following observation: if the heat kernel pt is the... |

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Citation Context ...is true without any additional assumptions about Φ1 and Φ2 (Theorem 4.12). Note that the classical existence results for equation (1.12) in Rn, n > 2, depend on the critical parameter 2∗ = 2nn−2 (see =-=[19]-=-), which matches (1.9) since α = n and β = 2. Notation. The letters C, c are used to denote positive constants whose values are unimportant but depend only on the hypotheses. The values of C, c may be... |

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Citation Context ...H1) canbe replaced by (H0). Furthermore, (iii) follows from (3.2) alone, in which case β may be any number larger than α. With this understanding, the statement of (iii) is not new (see, for example, =-=[6]-=-, [15], [21]) and is included for completeness. However, if the hypothesis (H1) does hold, then it follows from Theorem 4.2 that any function in D (E) isHölder continuous. Remark. If (M,d) satisfies t... |

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Citation Context ...the other hand, the domain of the energy form associated with the operator − (−∆)β/2 is known to be another Besov space Lip (β/2, 2, 2), which is smaller than Lip (β/2, 2,∞) = W β/2,2 (see [2], [23], =-=[25]-=-). M. Barlow studied in [3] heat kernels on geodesic metric spaces, assuming that a heat kernel pt is the transition density of a diffusion process and that it satisfies (1.5) with functions Φ1,Φ2 of ... |

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Citation Context ...orm on the Sierpiński gasket in terms of Besov spaces, as well as the relation β = β∗, was first obtained by A. Jonsson [18]. K. Pietruska-PaÃluba obtained the similar conclusion for nested fractals =-=[22]-=-. Observe that the condition (H1) under which we prove the equivalence of D (E) and W β/2,2 is optimal. Indeed, for the heat kernel generated by the operator − (−∆)β/2 in Rn the function Φ2 is given b... |

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Citation Context ...s = ∞. On the other hand, the domain of the energy form associated with the operator − (−∆) β/2 is known to be another Besov space Lip (β/2, 2, 2), which is smaller than Lip (β/2, 2, ∞) =W β/2,2 (see =-=[2]-=-, [23], [25]). M. Barlow studied in [3] heat kernels on geodesic metric spaces, assuming that a heat kernel pt is the transition density of a diffusion process and that it satisfies (1.5) with functio... |

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Citation Context ...] and [14], using techniques that are not available for general metric spaces. Our contribution is that we prove (1.11) under the hypothesis (1.10), which seems to be nearly optimal. Barlow proved in =-=[4]-=- that for every pair α, β satisfying (1.11) there exists a random walk satisfying a discrete time version of (1.5) with these parameters. There is no doubt that the same is true for continuous time he... |

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Citation Context .... In general, it may happen that β < β∗, as one can see from the example of the Cauchy-Poisson heat kernel in Rn where β = 1 and β∗ = 2. A theorem below provides conditions to ensure β = β∗ (see also =-=[17]-=-). Definition 4.5. We say that a heat kernel pt on a metric measure space (M,d, µ) satisfies the hypothesis (H2) if pt satisfies the estimate (1.5) with some positive HEAT KERNELS ON METRIC MEASURE SP... |

1 |
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Citation Context ...rnel is defined on a metric measure space, and show that this implies many interesting consequences for analysis on such a space. A similar approach was used by M. Barlow [3] and K. Pietruska-PaÃluba =-=[23]-=-, although in their works a heat kernel was assumed to be the transition density of a diffusion process on M , and in [23] the underlying space M was a subset of Rn. Let a heat kernel pt on (M,d, µ) s... |

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