Alexander Reznikov

Extremizers and sharp weak-type estimates for positive dyadic shifts

We find the exact Bellman function for the weak L1 norm of local positive dyadic shifts. We also describe a sequence of functions, self-similar in nature, which in the limit extremize the local weak-type (1,1) inequality.

On weak weighted estimates of the martingale transform and a dyadic shift

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the A1 conjecture, which stayed open after Muckenhoupt–Wheeden’s conjecture was disproved by Reguera–Thiele. 1. End-point estimates. Notation and facts. The end-point estimates play an important part in the theory of singular integrals (weighted or unweighted). They are usually the most difficult estimates in the theory, and the most interesting of course. It is a general principle that one can extrapolate the estimate from the end-point situation to all other situations. We refer the reader to the book [1] that treats this subject of extrapolation in depth. On the other hand, it happens quite often that the singular integral estimates exhibit a certain “blow-up” near the end point. To catch this blow-up can be a difficult task. We demonstrate this hunt for blow-ups by the examples of weighted dyadic singular integrals and their behavior in Lp(w). The end-point p will be naturally 1 (and sometimes slightly unnaturally 2) depending on the martingale singular operator. The singular integrals in this article are the easiest possible. They are dyadic martingale operators on σ-algebra generated by usual homogeneous dyadic lattice on the real line. We do not consider any non-homogeneous situation, and this standard σ-algebra generated by a dyadic lattice D will be provided with Lebesgue measure. Our goal will be to show how the technique of Bellman function gives the proof of the blow-up of the weighted estimates of the corresponding weighted dyadic singular operators. This blow-up will be demonstrated by certain estimates from below of the Bellman function of a dyadic problem. Interestingly, one can bootstrap then the correct estimates from below of a dyadic operators to the estimate from below of such classical operators as e. g. the Hilbert transform. The same rate of blow-up then persists for the classical operators. But this bootstrapping argument will be carried out in 2010 Mathematics Subject Classification. 42B20, 42B35, 47A30. FN is partially supported by the NSF grant DMS-1265623. AV and VV are partially supported by the program “Research in Pairs” of the Oberwolfach Institute for Mathematics, and by the NSF grant DMS-1600065; AV is also supported by the NSF grant DMS-1265549, and VV is also supported by the RFBR grant 14-0100748. 1 2 F. NAZAROV, A. REZNIKOV, V. VASYUNIN, AND A. VOLBERG a separate note, here, for simplicity, we work only with dyadic martingale operators. As to the Bellman function part of our consideration below, this part will be reduced to the task to find the lower estimate for the solutions of the homogeneous Monge–Ampère differential equation. 1.1. End-point estimates for martingale transform. We will work with a standard dyadic filtration D = ∪kDk on R, where Dk def = { [ m 2k , m+ 1 2k ) : m ∈ Z } . We consider the martingale transform related to this homogeneous dyadic filtration. The symbol 〈φ〉 I denotes average value of φ over the set I i. e., 〈φ〉I = 1 |I| ∫ I φ(t) dt. We consider martingale differences (recall that the symbol ch(J) denotes the dyadic children of J) ∆ J φ def = ∑ I∈ch(J) χ I ( 〈φ〉 I − 〈φ〉 J ). For our case of dyadic lattice on the line we have that |∆Jφ| is constant on J , the set ch(J) consists of two halves of J (J+ and J−), and ∆ J φ = 1 2 ( 〈φ〉 J+ − 〈φ〉 J− )(χ J+ − χ J− ) . We consider the dyadic A1 class of weights, but we skip the word dyadic in what follows, because we consider here only dyadic operators. A positive function w is called an A1 weight if [w]A1 def = sup J∈D 〈w〉 J infJ w <∞ . ByMw we will denote the dyadic maximal function of w, that isMw(x) = sup{〈w〉J : J ∈ D, J ∋ x}. Then w ∈ A1 with “norm” Q means that Mw ≤ Q · w a. e. , and Q = [w]A1 is the best constant in this inequality. Recall that a martingale transform is an operator given by Tεφ = ∑ J∈D εJ∆Jφ . It is convenient to use Haar function h J associated with dyadic interval J ,

On Weak Weighted Estimates of Martingale Transform

We consider several weak type estimates for singular operators using the Bellman function approach. We disprove the

Dimension-Free Properties of Strong Muckenhoupt and Reverse Hölder Weights for Radon Measures

A_1

Covering and separation of Chebyshev points for non-integrable Riesz potentials

conjecture of Muckenhoupt, which stayed open after Muckenhoupt--Wheedens conjecture was disproved by Reguera--Thiele.

Logarithmic bump conditions and the two-weight boundedness of Calderón-Zygmund operators

In this paper, we prove self-improvement properties of strong Muckenhoupt and Reverse Hölder weights with respect to a general Radon measure on

A Bellman function proof of the L2 bump conjecture

\mathbb {R}^n

L

Rn. We derive our result via a Bellman function argument. An important feature of our proof is that it uses only the Bellman function for the one-dimensional problem for Lebesgue measure; with this function in hand, we derive dimension-free results for general measures and dimensions.