A Proof of the Fefferman-Stein-Stromberg Inequality for the Sharp Maximal Functions

We prove sharp L p (w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the A p characteristic of w for all 1 < p < ∞. This implies the same sharp inequalities for the classical Lusin area integral S(f ), the Littlewood-Paley g-function, and their continuous analogs S ψ and g ψ . Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón-Zygmund operator for all 1 < p ≤ 3/2 and 3 ≤ p < ∞, and for its maximal truncations for 3 ≤ p < ∞.2000 Mathematics Subject Classification. 42B20,42B25. Key words and phrases. Littlewood-Paley operators, singular integrals, sharp weighted inequalities. 1 2 ANDREI K. LERNER inequality (1.1) for p = 2 implies (1.1) for all p > 1; analogously, it is enough to prove (1.2) for p = 3.Currently conjecture (1.1) is proved for: the Hilbert transform and Riesz transforms (Petermichl [16,17]), the Ahlfors-Beurling operator (Petermichl and Volberg [18]), any one-dimensional Calderón-Zygmund convolution operator with sufficiently smooth kernel (Vagharshakyan [20]). The proofs in [16,17,18] are based on the so-called Haar shift operators combined with the Bellman function technique. The main idea in [20] is also based on Haar shifts.Recently, Lacey, Petermichl and Reguera [11] have established sharp weighted estimates for Haar shift operators without the use of Bellman functions; their proof uses a two-weight "T b theorem" for Haar shift operators due to Nazarov, Treil and Volberg [15]. This provides a unified approach to works [16,17,18,20]. Very recently, a new, more elementary proof of this result, avoiding the T b theorem, was given by Cruz-Uribe, Martell and Pérez [5]; a key ingredient in [5] was a decomposition of an arbitrary measurable function in terms of local mean oscillations obtained in [14].It was shown in [5] that such a decomposition is very convenient when dealing with certain dyadic type operators. In particular, using these ideas, the authors proved in [5] conjecture (1.2) for the dyadic square function. Note that this is the first result establishing (1.2) for all p > 1. Previously (1.2) was obtained for the dyadic square function in the case p = 2 by Hukovic, Treil and Volberg [9], and, independently by Wittwer [24]. Also, (1.2) in the case p = 2 was proved by Wittwer [25] for the continuous square function. By the extrapolation argument [6], the linear w A 2 bound implies the bound by w max(1,1/(p−1)) Ap

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“…Note that Theorem 4.1 is a development of ideas going back to works of Carleson [3], Garnett-Jones [8] and Fujii [7].…”

Section: Local Mean Oscillationsmentioning

confidence: 99%

Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals

Lerner¹

2011

Advances in Mathematics

128

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“…For instance, see [29,49,13,27,47] and references therein. In [27], Fujii proved a Fefferman-Stein type inequality as in (1.1) when the underlying space is R d with the Lebesgue measure and the weighted norms on both sides of the inequality have two different weights. If two weights are the same, they belong to the class of Muchenhoupt weights A ∞ .…”

Section: Introductionmentioning

confidence: 99%

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Dong

Kim

2018

Trans. Amer. Math. Soc.

115

155

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Abstract. We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixednorm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains. IntroductionThe objective of this paper is two-fold. The first is to present a few generalized versions of the Fefferman-Stein theorem on sharp functions. One of our main theorems in this direction proveswhere (X , µ) is a space of homogeneous type, w is a Muckenhoupt weight, f # dy is the sharp function of f using a dyadic filtration of partitions of X , and L p,q (X , w dµ) is a mixed norm. A space X of homogeneous type is equipped with a quasi-distance and a doubling measure µ. A brief description of spaces of homogeneous type is given in Section 2. For more discussions, see [45,44,11]. If X is the product of two spaces (X 1 , µ 1 ) and (X 2 , µ 2 ) of homogeneous type, w = w 1 (x ′ )w 2 (x ′′ ), and µ(x) = µ 1 (x ′ )µ 2 (x ′′ ), where x ′ ∈ X 1 and x ′′ ∈ X 2 , then the L p,q (X , w dµ) norm is defined asSee (2.13) for a precise definition of the mixed norm. If X is the Euclidean space R d , µ is the Lebesgue measure on R d , w ≡ 1, and p = q, the inequality (1.1) is the celebrated Fefferman-Stein theorem on sharp functions. See, for instance, [26,49], where the measure of the underlying space is clearly infinite. Here we deal with both the cases of a finite measure µ(X ) < ∞ and an infinite measure µ(X ) = ∞. We also present a different form of the Fefferman-Stein theorem. See Theorems 2.3 and 2.4 and Corollaries 2.6 and 2.7.The second objective, which is in fact a motivation of writing this paper, is to apply the generalized versions of the Fefferman-Stein theorem to establishing 2010 Mathematics Subject Classification. 35R05, 42B37, 35B45, 35K25, 35J48. Key words and phrases. sharp/maximal functions, elliptic and parabolic equations, weighted Sobolev spaces, measurable coefficients.H.

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“…Regarding the Fefferman-Stein theorem on sharp functions, there has been considerable study on its generalizations and applications. For instance, see [29,49,13,27,47] and references therein. In [27], Fujii proved a Fefferman-Stein type inequality as in (1.1) when the underlying space is R d with the Lebesgue measure and the weighted norms on both sides of the inequality have two different weights.…”

Section: Introductionmentioning

confidence: 99%