Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function

Dirección General de Investigación Científica y Técnic

Endpoint estimates for commutators of singular integral operators

We prove endpoint estimates for commutatotrs of singular integrals with BMO functions. We first show that they satisfy L(log L) type inequalities and then that there is a Hardy space type estimate where the usual atomic Hardy space is replaced by an appropriate subspace.Dirección General de Investigación Científica y Técnic

Sharp weighted inequalities for the vector-valued maximal function

We prove in this paper some sharp weighted inequalities for the vector-valued maximal function Mq of Fefferman and Stein defined by Mqf(x) = X∞ i=1 (M fi(x))q !1/q, where M is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 <q<p< ∞ there exists a constant C such that Z Rn Mqf(x)p w(x)dx ≤ C Z Rn |f(x)|p q M[ p q ]+1w(x)dx. Furthermore the result is sharp since M[ p q ]+1 cannot be replaced by M[ p q ]. We also show the following endpoint estimate w({x ∈ Rn : Mqf(x) > λ}) ≤ C λ Z Rn |f(x)|q Mw(x)dx, where C is a constant independent of λ.Dirección General de Investigación Científica y Técnic

A remark on weighted inequalities for general maximal operators

Let 1 < p < ∞, and let w, v be two non–negative functions. We give a sufficient condition on w, v for which the general maximal operator MB is bounded from Lp(v) into Lp(w). Our condition is stronger but closely related to the Ap,B condition for two weights

Weighted norm inequalities for singular integral operators

For a Calderón-Zygmund singular integral operator T, we show that the following weighted inequality holds Z Rn|T f(y)|p w(y)dy ≤ C Z Rn |f(y)| p M[p]+1w(y)dy, where Mk is the Hardy-Littlewood maximal operator M iterated k times, and [p] is the integer part of p. Moreover, the result is sharp since it does not hold for M[p]. We also give the following endpoint result: w({y ∈ R n : |T f(y)| > λ}) ≤ C λ Z Rn |f(y)| M2w(y)dy

Sharp weighted bounds involving A∞

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] 0 A∞ := sup Q 1 / w(Q) Z Q M(wχQ). In particular, we show the following improvement of the first author’s A2 theorem for Calderón-Zygmund operators T : kT kB(L2(w)) ≤ cT [w] 1/2 A2 [w] 0 A∞ + [w −1] 0 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 Pérez (2009) of the form kT kB(L p(w)) ≤ cpp0 [w] 1/p A1 ([w] 0 A∞ ) 1/p 0 , 1 < p < ∞, kT f kL 1,∞(w) ≤ c[w]A1 log(e + [w] 0 A∞ )k f kL 1(w). An estimate dual to the last one is also found, as well as new bounds for commutators of singular integralsAcademy of FinlandMinisterio de Ciencia e InnovaciónJunta de Andalucí

L1 → Lq Poincaré inequalities for 0 < q < 1 imply representation formulas

Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B0⊂S be a given ball. It has been a well-known result by now (see [1–4]) that the validity of an L1→L1 Poincaré inequality of the following form: ∫B|f−fB|dv⩽cr(B)∫Bgdμ, for all metric balls B⊂B0⊂S, implies a variant of representation formula of fractional integral type: for ν-a.e. x∈B0, |f(x)−fB0|⩽C∫B0g(y)ρ(x,y)μ(B(x,ρ(x,y)))dμ(y)+Cr(B0)μ(B0)∫B0g(y)dμ(y). One of the main results of this paper shows that an L1 to Lq Poincaré inequality for some 0 < q < 1, i.e., (∫B|f−fB|qdv)1/q⩽cr(B)∫Bgdμ, for all metric balls B⊂B0, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition, supλ>0λν({x∈B:|f(x)−fB|>λ})ν(B)⩽Cr(B)∫Bgdμ, also implies the same formula. Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved.National Science FoundationDirección General de Investigación Científica y TécnicaNorth Atlantic Treaty Organizatio

The L(log L)e endpoint estimate for maximal singular integral operators

We prove in this paper the following estimate for the maximal operator T ∗ associated to the singular integral operator T: kT ∗ fkL 1,∞ (w) . 1 ǫ Z Rn | f(x)| ML(log L) ǫ (w)(x) dx, w ≥ 0, 0 < ǫ ≤ 1. This follows from the sharp L p estimate kT ∗ fkLp (w) . p ′ ( 1 δ ) 1/p ′ kfk L p (ML(log L) p−1+δ (w)), 1 < p < ∞, w ≥ 0, 0 < δ ≤ 1. As as a consequence we deduce that kT ∗ fkL 1,∞ (w) . [w]A1 log(e + [w]A∞ ) Z Rn | f | w dx, extending the endpoint results obtained in [LOP] A. Lerner, S. Ombrosi and C. Pérez, A1 bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Mathematical Research Letters (2009), 16, 149–156 and [HP] T. Hytönen and C. Pérez, Sharp weighted bounds involving A∞, Analysis and P.D.E. 6 (2013), 777–818. DOI 10.2140/apde.2013.6.777 to maximal singular integrals. Another consequence is a quantitative two weight bump estimate.Unión EuropeaMinisterio de Ciencia e Innovació