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

By Andrei K. Lerner. 2010

162 citations recorded by the source corpus.

Abstract 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 Calderon–Zygmund operator for all 1 p ⩽ 3 / 2 and 3 ⩽ p ∞ , and for its maximal truncations for 3 ⩽ p ∞ .

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/3169ea3ba34eef324f06a36ea15a5dc7ea4a3825

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Citations in the corpus

Musielak-Orlicz Campanato Spaces and Applications (2013, unknown): “7 and the boundedness of the square function Sφf(y) := ∫ Γ(y) |φt ∗b2(x)| 2 dx dt tn+1 on the weighted Lebesgue L2(Rn) space with the weight [φ(·, ‖χB0‖ −1 Lφ(Rn))] −1 (see, for example, [15, 36, 24]), we have ∫”
Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents (2014, methodology): “Moreover, the intrinsic Lusin area function has the distinct advantage of being pointwise comparable at different cone openings, which is a property long known not to hold true for the classical Lusin area function; see Wilson [44, 45, 46, 47] and also Lerner [24, 25].”
SHARP BOUNDS FOR GENERAL COMMUTATORS ON WEIGHTED LEBESGUE SPACES (2010, unknown): “A bit earlier, in [22], the sharp L(w) bound for T was obtained for values of p outside the interval (3/2, 3) and the proof is based on the corresponding estimates for the intrinsic square functions.”
Intrinsic square functions on the weighted Morrey spaces (2011, background): “Moreover, in [12], Lerner showed sharp Lw norm inequalities for the intrinsic square functions in terms of the Ap characteristic constant of w for all 1 < p < ∞.”
Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces (2020, background): “Moreover, the intrinsic Lusin-area function has the distinct advantage of being pointwisely comparable at different cone openings, which is a property long known not to hold true for the classical Lusin-area function; see Wilson [29–32] and also Lerner [13,14].”
Borderline weak‐type estimates for singular integrals and square functions (2015, background): “A. K. Lerner, ‘Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals’, Adv.”
Sharp weighted estimates for classical operators (2010, background): “Very recently, Lerner [35] proved the analog of Theorem 1.”
The sharp weighted bound for general Calderón-Zygmund operators (2010, methodology): “f recovering all suﬃciently smooth, odd, convolution-type Calderón–Zygmund operators in dimension N = 1 from dyadic shifts, thereby proving (1.1) for all these operators. By a diﬀerent method, Lerner [16] was able to estimate all standard convolution-type operators in arbitrary dimension by controll”
The A_p-A_infty inequality for general Calderon--Zygmund operators (2011, background): “ +[w1−p′]1/p A∞ kfkLp(w). We note thatanypolynomialdependence oncomplexityκis suﬃcientto conclude our paper. The application of Lerner’sformula [10] to Haarshift operatorsis well-known, see [1], [2], [9], [6]. But all previous applications have given bounds that are exponential in κ. Lerner’s formul”
On Sharp Aperture-Weighted Estimates for Square Functions (2013, methodology): “2) ‖Sψ‖Lp(w) ≤ cp,n,ψ[w] max( 1 2 , 1 p−1 ) Ap , and this estimate is sharp in terms of [w]Ap (we also refer to [13] for a detailed history of closely related results).”