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Boundedness of Lusin-area and functions on localized BMO spaces over doubling metric measure spaces. (English) Zbl 1220.42014

Let (X,d) be a metric space endowed with a regular Borel measure μ such that 0<μ(B(x,r))< (xX, r>0). (X,d,μ) is called a doubling metric measure space if μ(B(x,2r))C 1 μ(B(x,r)). A positive function ρ on X is called admissible if, for some C 0 , k 0 >0, ρ(x) -1 C 0 ρ(y) -1 (1+ρ(y) -1 d(x,y)) k 0 (x,yX). For an admissible function ρ and q[1,), a function fL loc q (X) is said to be in BMO ρ q (X) if

f BMO ρ q (X) =sup B=B(x,r);xX,r<ρ(x) μ (B) -1 B |f(y)-f B | q d μ (y) 1/q +sup B=B(x,r);xX,rρ(x) μ (B) -1 B |f(y)| q d μ (y) 1/q <·

Similarly, a real valued function fL loc q (X) is said to be in BLO ρ q (X) if, in the definition of BMO ρ q (X), f B is replaced by essinf uB f(u). In the case (X,d,μ)=( n ,|·|,dx) and ρ1, BMO ρ q (X) coincides with the Goldberg’s local BMO space bmo. If ρ is an admissible function, BMO ρ q (X)BMO ρ 1 (X)=:BMO ρ (X).

Now let ρ be an admissible function and {Q t } t0 be a family of operators bounded on L 2 (X) with integral kernels {Q t (x,y)} t0 satisfying that there exist C, δ 1 >0, δ 2 (0,1), γ>0 such that |Q t (x,y)|C(μ(B(x,t))+μ(B(x,d(x,y))) -1 (1+d(x,y)/t) -γ (1+t/ρ(x)) -δ 1 , and | X Q t (x,z)dμ(z)|C(1+ρ(x)/t) -δ 2 . Using this family of operators, the authors define the Littlewood-Paley g-function g(f), Lusin’s area function S(f) and the g λ * function g λ * (f) by

g(f)(x)= 0 |Q t (f)(x)| 2 d t / t 1/2 ,S(f)(x)= 0 d(x,y)<t |Q t (f)(y)| 2 μ (B(y,t)) -1 d μ (y) d t / t 1/2 ,g λ * (f)(x)= X×(0,) |Q t (f)(y)| 2 (1+d(x,y)/t) -λ μ (B(y,t)) -1 d μ (y) d t / t 1/2 ·

Their main result is the following: Let X be a doubling metric measure space satisfying one more condition, “the δ-annular decay property”. Let ρ be an admissible function. Assume that the Littlewood-Paley g-function is bounded on L 2 (X). Then S(f) 2 BLO ρ (X) Cf BMO ρ (X) 2 .

If 3n<λ<, the same result holds for the g λ * function without assuming “the δ-annular decay property”.

Relating to their boundedness results, the authors give a nonnegative fbmo() which is not in blo().

MSC:
42B25Maximal functions, Littlewood-Paley theory
42B30H p -spaces (Fourier analysis)
51F99Metric geometry
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