Open AccessWeak and Strong Type $$A_1$$–$$A_\infty $$ Estimates for Sparsely Dominated Operators
Author(s) -
Dorothée Frey,
Zoe Nieraeth
Publication year - 2018
Publication title -
journal of geometric analysis/the journal of geometric analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.156
H-Index - 47
eISSN - 1559-002X
pISSN - 1050-6926
DOI - 10.1007/s12220-018-9989-2
Subject(s) - type (biology) , mathematics , operator (biology) , pure mathematics , dual (grammatical number) , property (philosophy) , combinatorics , ecology , biochemistry , chemistry , philosophy , epistemology , repressor , biology , transcription factor , gene , art , literature
We consider operators T satisfying a sparse domination property| ⟨ T f , g ⟩ | ≤ c ∑ Q ∈ S⟨ f ⟩p 0 , Q⟨ g ⟩q 0 ' , Q| Q |with averaging exponents 1 ≤ p 0 < q 0 ≤ ∞ . We prove weighted strong type boundedness for p 0 < p < q 0 and use new techniques to prove weighted weak type ( p 0 , p 0 ) boundedness with quantitative mixed A 1 - A ∞ estimates, generalizing results of Lerner, Ombrosi, and Pérez and Hytönen and Pérez. Even in the case p 0 = 1 we improve upon their results as we do not make use of a Hörmander condition of the operator T . Moreover, we also establish a dual weak type ( q 0 ' , q 0 ' ) estimate. In a last part, we give a result on the optimality of the weighted strong type bounds including those previously obtained by Bernicot, Frey, and Petermichl.