Two-weight Lp→LqL^p\to L^qLp→Lq bounds for positive dyadic operators in the case 0<q<1≤p<∞0<q< 1 \le p<\infty0<q<1≤p<∞ (original) (raw)

2017, arXiv (Cornell University)

Let σ, ω be measures on R d , and let {λ Q } Q∈D be a family of non-negative reals indexed by the collection D of dyadic cubes in R d. We characterize the two-weight norm inequality, T λ (f σ) L q (ω) ≤ C f L p (σ) for every f ∈ L p (σ), for the positive dyadic operator T λ (f σ) ∶= Q∈D λ Q 1 σ(Q) Q f dσ 1 Q in the difficult range 0 < q < 1 ≤ p < ∞ of integrability exponents. This range of the exponents p, q appeared recently in applications to nonlinear PDE, which was one of the motivations for our study. Furthermore, we introduce a scale of discrete Wolff potential conditions that depends monotonically on an integrability parameter, and prove that such conditions are necessary (but not sufficient) for small parameters, and sufficient (but not necessary) for large parameters. Our characterization applies to Riesz potentials Iα(f σ) = (−∆) − α 2 (f σ) (0 < α < d), since it is known that they can be controlled by model dyadic operators. The weighted norm inequality for Riesz potentials in this range of p, q has been characterized previously only in the special case where σ is Lebesgue measure. Contents Notation 2010 Mathematics Subject Classification. 42B25, 42B35, 47G40. Key words and phrases. Two-weight norm inequalities, positive dyadic potential operators, Wolff potentials, discrete Littlewood-Paley spaces. T.S.H. is supported by the Academy of Finland through funding of his postdoctoral researcher post (Funding Decision No 297929), and by the Jenny and Antti Wihuri Foundation through covering of expenses of his visit to the University of Missouri. He is a member of the Finnish Centre of Excellence in Analysis and Dynamics Research. a −1 For a family a ∶= {a Q }, the family a −1 is defined by a −1 ∶= {a −1 Q }. The least constant in the L p (σ) → L q (ω) two-weight norm inequality for the operator T λ (⋅ σ) is denoted by T λ (⋅ σ) L p (σ)→L q (ω). The uppercase Latin letters P, Q, R, S are reserved for dyadic cubes. When the collection of the cubes is clear from the context, the indexations 'Q ∈ D' and 'Q ∈ Q' are both abbreviated as 'Q' in the indexation of summations, and omitted in the indexation of families (and similarly for the cubes P, R, S). The lowercase Latin letters a, b, c, d are reserved for various families a ∶= {a Q }, b ∶= {b Q },. .. of non-negative reals, and λ ∶= {λ Q } for the fixed family of non-negative reals associated with the operator T λ (⋅ σ). Throughout this paper, we follow the usual convention 0 0 ∶= 0.