Balls and quasi-metrics: A space of homogeneous type modeling the real analysis related to the Monge-Ampère equation

Aimar, Hugo; Forzani, Liliana; Toledano, R.

doi:10.1007/bf02498215

Cited by 39 publications

(44 citation statements)

References 2 publications

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“…The properties (A) and (B) of the sections imply the following engulfing property proved in [1]: there exists a constant θ > 1, depending only on K 1 and 1 , such that for y ∈ S(x, r) we have (D) S(y, r) ⊂ S(x, θr) and S(x, r) ⊂ S(y, θr).…”

Section: Preliminary Resultsmentioning

confidence: 82%

$BLO$ spaces associated with the sections

Lin

2007

Proc. Amer. Math. Soc.

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Abstract. BLO spaces associated with the sections are introduced. It is shown that some properties which hold for the classical space BLO related to the balls (or cubes) remain valid for the space BLO related to the sections. IntroductionIn 1980, R. R. Coifman and R. Rochberg [5] introduced the space BLO of functions of bounded lower oscillation. More precisely, we say that a locally integrable function f on R n is in BLO ifwhere In this note, we will introduce the spaces BM O F based on sections as the substitutes for the classical spaces BLO defined by R. R. Coifman and R. Rochberg [5]. We will give out a criterion for the new spaces BLO F . Meanwhile, we will also show that the singular integral operator H * (see Section 2) is bounded from L ∞ into BLO F . Preliminary resultsIn this section, we first introduce some notation and then state our main results. The analysis of the Monge-Ampère equation suggests the axiomatic definition of what are called 'sections'. These are defined as follows.For x ∈ R n and t > 0, let S(x, t) denote an open and bounded convex set containing x. We call S(x, t) a section if the family {S(x, t) : x ∈ R n , t > 0} is monotone increasing in t, i.e., S(x, t) ⊂ S(x, t ) for t ≤ t , and satisfies the following conditions:

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Section: Preliminary Resultsmentioning

confidence: 82%

$BLO$ spaces associated with the sections

Lin

2007

Proc. Amer. Math. Soc.

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“…This result was proved by H. Aimar, L. Forzani, and R. Toledano in [1] under the assumption of several geometric conditions on the sections of ϕ, which, later on, turned out to be equivalent to the (DC)-doubling condition, (see Theorem 8 in [12]). Conversely, if Eq.…”

Section: Real Analysis Associated To the Monge-ampère Equationmentioning

confidence: 82%

A Mean-Value Inequality for Non-negative Solutions to the Linearized Monge–Ampère Equation

Forzani

Maldonado

2009

Potential Anal

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“…To prove that the distributional Hessian D 2 D u of an Alexandrov solution is a L 1 function, it is enough to prove an a priori equi-integrability estimate on smooth solutions. 5 To do this, a key observation is that any domain Ω endowed with the Lebesgue measure and the family of "balls" given by the sections {S(x, p, t)} x∈Ω, t∈R of solutions of (2.6) as defined in (2.22) is a space homogenous type in the sense of Coifman and Weiss, see [25,68,1]. In particular Stein's Theorem implies that if…”

Section: 4mentioning

confidence: 99%

“…It says that if we take the function ψx ,y 0 ,y 1 = max −c(·, y 0 ) + c(x, y 0 ), −c(·, y 1 ) + c(x, y 1 ) , then we are able to touch the graph of this function from below atx with the family of functions {−c(·, y t ) + c(x, y t )} t∈ [0,1] . This suggests that we could use this family to regularize the cusp of ψx ,y 0 ,y 1 at the pointx, by slightly moving above the graphs of the functions −c(·, y t ) + c(x, y t ).…”

Section: 1mentioning

confidence: 99%