Heat Kernels and Besov Spaces Associated with Second Order Divergence Form Elliptic Operators

Cao, Jun; Grigor’yan, Alexander

doi:10.1007/s00041-019-09708-7

Cited by 10 publications

(3 citation statements)

References 44 publications

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“…Strichartz and in [13] on more general metric measure spaces by A. Grigor'yan. See also [9,14,15] for some equivalent Besov type characterizations of H σ (K), and [5,6,9,11] for related interpolation results. In this paper, we focus on the following three aspects.…”

Section: Introductionmentioning

confidence: 99%

Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Cao,

Qiu

2020

Preprint

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We study the Sobolev spaces H σ (K) and H σ 0 (K) on p.c.f. self-similar sets in terms of the boundary behavior of functions. First, for σ ∈ R + , we make an exact description of the tangents of functions in H σ (K) at the boundary. Second, we characterize H σ 0 (K) as the space of functions in H σ (K) with zero tangent of an appropriate order depending on σ. Last, we extend H σ (K) to σ ∈ R, and obtain various interpolation theorems with σ ∈ R + or σ ∈ R. We illustrate that there is a countable set of critical orders, that arises naturally in the boundary behavior of functions, such that H σ 0 (K) presents a critical phenomenon if σ is critical. These orders will play a crucial role in our study. They are just the values in 1 2 + Z+ in the classical case, but are much more complicated in the fractal case.

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Section: Introductionmentioning

confidence: 99%

Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Cao,

Qiu

2020

Preprint

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“…Recently, Cao and Grigor'yan [8,9] have made much progress showing (1.1) holds on a larger region, under the assumption of Gaussian heat kernel estimates. Their work introduces some new techniques, but the results and ideas are restricted to the d W = 2 case.…”

Section: Introductionmentioning

confidence: 99%

Equivalence of Besov spaces on p.c.f. self-similar sets

Cao¹,

Qiu²

2020

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On p.c.f. self-similar sets, of which the walk dimensions of heat kernels are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces B p,q σ (K) and the Lipschitz-Besov spaces Λ p,q σ (K), are identitical. In particular, we provide concrete examples that B p,q σ (K) = Λ p,q σ (K) with σ > 1. Our method is purely analytical, and does not involve any heat kernel estimate.

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“…fractal K with a regular harmonic structure, H σ N (K) is equivalent to a Besov type space B 2,2 σ (K) for 0 < σ < 1, as well as H 1 N (K) = domE = B 2,∞ 1 (K). We refer to [4,5,9] for Besov type characterizations of Sobolev spaces with 0 < σ < 1 on general metric measure spaces, related to the heat kernel estimates, and [5,7] for related interpolation results.…”

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confidence: 99%

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

Cao,

Qiu

2019

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We consider the Sobolev type spaces H σ (K) with σ ≥ 0, where K is a postcritically finite self-similar set with the natural boundary. Firstly, we compare different classes of Sobolev spaces H σ N (K), H σ D (K) and H σ (K), and observe a sequence of critical orders of σ in our comparison theorem. Secondly, We present a general atomic decomposition theorem of Sobolev spaces H σ (K), where the same critical orders play an important role. At the same time, we provide purely analytic approaches for various Besov type characterizations of Sobolev spaces H σ (K). 22010 Mathematics Subject Classification. Primary 28A80.

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Heat Kernels and Besov Spaces Associated with Second Order Divergence Form Elliptic Operators

Cited by 10 publications

References 44 publications

Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Sobolev spaces on p.c.f. self-similar sets: boundary behavior and interpolation theorems

Equivalence of Besov spaces on p.c.f. self-similar sets

Sobolev spaces on p.c.f. self-similar sets: critical orders and atomic decompositions

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