We improve on several weighted inequalities of recent interest by replacing a part of the Ap bounds by weaker A∞ estimates involving Wilson’s A∞ constant [ w ] A ∞ ′ : = sup Q 1 w ( Q ) ∫ Q M ( w χ Q ) . In particular, we show the following improvement of the first author’s A2 theorem for Calderon–Zygmund operators T: ∥ T ∥ ℬ ( L 2 ( w ) ) ≤ c T [ w ] A 2 1 ∕ 2 ( [ w ] A ∞ ′ + [ w − 1 ] A ∞ ′ ) 1 ∕ 2 . Corresponding Ap type results are obtained from a new extrapolation theorem with appropriate mixed Ap-A∞ bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley’s classical bound. We also derive mixed A1-A∞ type results of Lerner, Ombrosi and Perez (2009) of the form ∥ T ∥ ℬ ( L p ( w ) ) ≤ c p p ′ [ w ] A 1 1 ∕ p ( [ w ] A ∞ ′ ) 1 ∕ p ′ , 1 < p < ∞ , ∥ T f ∥ L 1 , ∞ ( w ) ≤ c [ w ] A 1 log ( e + [ w ] A ∞ ′ ) ∥ f ∥ L 1 ( w ) . An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.

/pdf/sharp-weighted-bounds-involving-a-3ugkjiqjth.pdf

Sharp weighted bounds involving A

http://grupo.us.es/anaresba/trabajos/carlosperez/HPR1_corrected.pdf

Sharp Reverse Hölder property for A∞ weights on spaces of homogeneous type

We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents $(p_0,q_0)$ where $1\le p_0<2< q_0 \le \infty $. The norm estimates obtained are powers $\alpha$ of the characteristic used by Auscher and Martell. The critical exponent in this case is $\mathfrak{p}=1+\frac{p_0}{q'_0}$. We prove $\alpha=\frac1{p-p_0}$ when $p_0 < p \le \mathfrak{p}$ and $\alpha= \frac{q_0-1}{q_0-p}$ when $\mathfrak{p} \le p < q_0$.
In particular, we are able to obtain the sharp $A_2$ estimates for non-integral singular operators which do not fit into the class of Calder\'on-Zygmund operators. These results are new even in the Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.

/pdf/sharp-weighted-norm-estimates-beyond-calderon-zygmund-theory-1m24sem3yw.pdf

Sharp weighted norm estimates beyond Calderón–Zygmund theory

A martingale transform $ T$, applied to an integrable locally supported function $ f$, is pointwise dominated by a positive sparse operator applied to $ \lvert f\rvert $, the choice of sparse operator being a function of $ T$ and $ f$ As a corollary, one derives the sharp $ A_p$ bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms The (very easy) method of proof (a) only depends upon the weak-$L^1$ norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the $ A_2$ bounds in that setting

An elementary proof of the $A_2$ Bound

We prove generalized Fefferman-Stein type theorems on sharp functions with $A_p$ weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixed-norm weighted $L_p$-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains.

On _{}-estimates for elliptic and parabolic equations with _{} weights

A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.

Systems of dyadic cubes in a doubling metric space

We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hytonen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions into spaces of homogeneous type.

/pdf/two-weight-norm-inequalities-for-potential-type-and-maximal-4u22haruvo.pdf

Two-weight norm inequalities for potential type and maximal operators in a metric space

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type ☆

We give an intrinsic characterization of all subsets of a doubling metric space that can arise as a member of some system of dyadic cubes on the underlying space, as constructed by Christ.

Anna Kairema

Papers

Systems of dyadic cubes in a doubling metric space

Systems of dyadic cubes in a doubling metric space

Two-weight norm inequalities for potential type and maximal operators in a metric space

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type ☆

What is a cube