H¹and BMO for certain locally doubling metric measure spaces of finite measure

Abstract

In a previous paper the authors developed a H 1 −BM O theory for unbounded metric measure spaces (M, ρ, µ) of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and "isoperimetric" property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R d , ρϕ, µϕ), where dµϕ = e −ϕ dx and ρϕ is the Riemannian metric corresponding to the length element ds 2 = (1+|∇ϕ|) 2 ( dx 2 1 +· · ·+ dx 2 d ). This generalizes previous work of the last two authors for the Gauss space.2000 Mathematics Subject Classification. 42B20, 42B30, 46B70, 58C99 .

Estimates for functions of the Laplacian on manifolds with bounded geometry

“…Those estimates will involve the Hardy space H 1 (M ) introduced in [5] and some related spaces, which will be defined in [24].…”

Local maximal operators on measure metric spaces

“…Mauceri and Meda [16] defined and studied the space BMO ρa (R n , d, γ n ), where d, γ n and ρ a are as described earlier. Later on Carbonaro, Mauceri and Meda [4], [5], extended ideas from [10] and [16] to a general context of measure metric spaces with measure satisfying local doubling property (and some additional geometric properties as well) and ρ = ρ b chosen to be ρ ≡ b, b > 0 being a parameter. To be precise in [5], [16] (the case of µ(X) < ∞), in the definition of the local BMO space it was assumed that f ∈ L 1 (R n , γ n ) rather than a weaker condition f ∈ L 1 loc,ρ (R n , γ n ).…”

“…Later on Carbonaro, Mauceri and Meda [4], [5], extended ideas from [10] and [16] to a general context of measure metric spaces with measure satisfying local doubling property (and some additional geometric properties as well) and ρ = ρ b chosen to be ρ ≡ b, b > 0 being a parameter. To be precise in [5], [16] (the case of µ(X) < ∞), in the definition of the local BMO space it was assumed that f ∈ L 1 (R n , γ n ) rather than a weaker condition f ∈ L 1 loc,ρ (R n , γ n ). Similarly, in [4] (the case of µ(X) = ∞), it was assumed that f ∈ L 1 loc (X, µ), which is again a stronger condition than f ∈ L 1 loc,ρ (X, µ) for ρ ≡ ∞.…”

“…To be precise in [5], [16] (the case of µ(X) < ∞), in the definition of the local BMO space it was assumed that f ∈ L 1 (R n , γ n ) rather than a weaker condition f ∈ L 1 loc,ρ (R n , γ n ). Similarly, in [4] (the case of µ(X) = ∞), it was assumed that f ∈ L 1 loc (X, µ), which is again a stronger condition than f ∈ L 1 loc,ρ (X, µ) for ρ ≡ ∞. Recently, Lin and Stempak [14] defined and investigated the spaces BMO ρ (Ω), where Ω and ρ = ρ k are as in Section 5.…”

“…However, it is evident that in order to get a richer theory it is necessary to narrow the setting and assume some additional conditions of geometrical nature on the considered measure metric space. This approach is well represented in the papers by Carbonaro, Mauceri and Meda [4], [5], where, for instance, approximate midpoint or isoperimetric properties were imposed to get satisfactory results. Having defined the local objects we then investigate them from the point of similarities or differences with the global ones.…”