Isoperimetric Inequalities for Non-Local Dirichlet Forms
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Isoperimetric Inequalities for Non-Local Dirichlet Forms Feng-Yu Wang1,2 · Jian Wang3 Received: 15 July 2017 / Accepted: 9 October 2019 / © Springer Nature B.V. 2019
Abstract Let (E, F , μ) be a σ -finite measure space. For a non-negative symmetric measurable function J (x, y) on E × E, consider the quadratic form 1 E (f, f ) := (f (x) − f (y))2 J (x, y) μ(dx) μ(dy) 2 E×E in L2 (μ). We characterize the relationship between the isoperimetric inequality and the super Poincar´e inequality associated with E . In particular, sharp Orlicz-Sobolev type and Poincar´e type isoperimetric inequalities are derived for stable-like Dirichlet forms on Rn , which include the existing fractional isoperimetric inequality as a special example. Keywords Isoperimetric inequality · Non-local Dirichlet form · Super Poincar´e inequality · Orlicz norm Mathematics Subject Classification (2010) 47G20 · 47D62
1 Introduction For local (i.e. differential) quadratic forms, the isoperimetric inequality is a geometric inequality using the surface area of a set to bound its volume, see, for instance [21, 35, 38] and references therein, for the study of isoperimetric inequalities and applications to symmetric diffusion processes. In this case, the surface area refers to the possibility for the associated diffusion process to exit the set. In the non-local case, the associated process is a jump process which exits a set without hitting the boundary, so it is reasonable to replace the surface area of a set A by the jump Jian Wang
[email protected] Feng-Yu Wang [email protected]; [email protected] 1
Center of Applied Mathematics, Tianjin University, Tianjin 300072, China
2
Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, UK
3
College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA), Fujian Normal University, Fuzhou 350007, China
F.-Y. Wang, J. Wang
rate from A to its complementary Ac . In this spirit, the famous Cheeger inequality [14] for the first eigenvalue was extended in [17, 30] to jump processes (see also [44] for finite Markov chains). See [15, 16, 32, 33, 43, 51, 52, 54] for the study of more general functional inequalities of symmetric jump processes using isoperimetric constants. These references only consider large jumps (i.e. the total jump rate is finite). In this paper, we aim to investigate isoperimetric inequalities for non-local forms with infinite jump rates, for which small jumps will play a key role. To explain our motivation more clearly, let us start from the following classical isoperimetric inequality on Rn : μ∂ (∂A) ≥ nμ(A)
n−1 n
1
ωnn ,
(1.1)
Rn
with finite volume, ∂A is its boundary, ωn is the where A is a measurable subset of volume of the n-dimensional unit ball, μ is the Lebesgue measure and μ∂ is the area measure induced by μ: μ({dist(·, A) ≤ ε}) − μ(A) . μ∂ (∂A) := lim sup ε ε↓0 In particular, the equality in Eq. 1.1 holds for A being a ball. By the co-area formula, Eq. 1.1 is equivalent to the sharp L1 -Sobolev
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