Dyadic Harmonic Analysis and Weighted Inequalities: The Sparse Revolution

Bump conditions and two-weight inequalities for commutators of fractional integrals

“…In this section, we will prove Theorems 1.1-1.3 and Corollary 1.4-1.5. To begin with recalling some notation,definitions and facts related to sparse families (see [18,24] for more details). Given a cube Q ⊂ R n , let D(Q) be the set of cubes obtained by repeatedly subdividing Q and its descendants into 2 n congruent subcubes.…”

Dyadic lower little BMO estimates

“…where the shift coefficients c I KL are complex numbers satisfying the bound c I KL ≤ 2 −(i+j) 2 , for all K ∈ ch i (I), L ∈ ch j (I) and for all I ∈ D. This definition of Haar shifts follows [Per,p. 34].…”

Variational characterizations of weighted Hardy spaces and weighted $BMO$ spaces

“…Now we recall the definitions of dyadic lattice, sparse family and sparse operator; see, for example, [19,20,30]. Given a cube Q ⊂ R n , let D(Q) be the set of cubes obtained by repeatedly subdividing Q and its descendants into 2 n congruent subcubes.…”