The Tb-theorem on non-homogeneous spaces

The sharp weighted bound for general Calderón--Zygmund operators

“…These auxiliary operators are already implicit in the original Nazarov-Treil-Volberg argument [22], and their more explicit form was identified in my extension of their result to the vectorvalued situation [9], where this explicit structure became more decisive. Here, it will be checked that these new operators are precisely the dyadic shifts in the generality defined by Lacey, Petermichl and Reguera [16].…”

“…This is once again provided by the proof of a T (1) theorem -this time, the one for nonhomogeneous spaces due to Nazarov, Treil and Volberg [22]. (A variant of the same proof, from a more recent Nazarov-Treil-Volberg preprint [24], is also behind the reduction of Pérez, Treil and Volberg [26].)…”

“…This number, π bad , may easily be estimated as π bad γ,N 2 −rγ . (Thanks to the above mentioned independence of position and goodness, the computation is only slightly different from the case of two independent random systems considered in [22].) In much of the earlier work based on good and bad cubes, it was important that this number could be made as small as one liked by fixing r large enough, and the requirements for its magnitude depended on the implicit constants in certain square function estimates.…”

Условия $C^m$-приближаемости функций решениями эллиптических уравнений

“…Изучение множеств устранимых особенностей решений однородных эл-липтических уравнений и связанных с ними емкостей в различных классах функций (в основном классических: C m , L p , BM O и др.). Кроме цитируемых выше работ К. Толсы [47], [5] и [41] мы выделим ряд наиболее ярких (на наш взгляд) работ последних лет: [102]- [108], открывающих глубокие связи теории емкостей, геометрической теории меры и теории сингулярных интегралов. От-метим также работы [109], [110], в которых изучаются геометрические свойства некоторых емкостей, обсуждавшихся в связи с рассматриваемыми аппроксима-ционными задачами.…”

What is a cube?

“…Another situation is the following: After the seminal work of Nazarov-TreilVolberg [10], it is now standard to treat singular integrals with respect to a nondoubling measure on R n with the help of a random choice of the system of dyadic cubes. Since any cube of R n can arise as a random dyadic cube in their construction, it is necessary to impose certain assumptions, such as the 'accretivity'…”