2016/05/16 by Mingming Cao, Qingying Xue, Cao, Mingming +1
Mathematics · #42B25 #47G10 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.CA #msc:42B25 #msc:47G10
paper · pdf · doi:10.48550/arxiv.1605.04649
32 pages
arxiv created 2016/05/16 · openalex publication_date 2016/05/16 · arxiv updated 2016/05/17 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
It is well-known that the Lp boundedness and weak (1,1) estiamte (λ>2) of the classical Littlewood-Paley gλ*-function was first studied by Stein, and the weak (p,p) (p>1) estimate was later given by Fefferman for λ=2/p. In this paper, we investigated the Lp(μ) boundedness of the non-homogeneous Littlewood-Paley gλ,μ*-function with non-convolution type kernels and a power bounded measure μ: gλ,μ^*(f)(x) = (\iint_\mathbb Rn+1+ ((t)/(t + |x - y|))m λ |θtμf(y)|2 \fracdμ(y) dttm+1)1/2, x ∈ \mathbb Rn, λ> 1, where θtμf(y) = ∫_\mathbb Rn st(y,z) f(z) dμ(z), and st is a non-convolution type kernel. Based on a big piece prior boundedness, we first gave a sufficient condition for the Lp(μ) boundedness of gλ,μ^*. This was done by means of the non-homogeneous good lambda method. Then, using the methods of dyadic analysis, we demonstrated a big piece global Tb theorem. Finally, we obtaind a sufficient and necessary condition for Lp(μ) boundedness of gλ,μ^*-function. It is worth noting that our testing conditions are weak (1,1) type with respect to measures.