Mixed estimates for singular integrals on weighted Hardy spaces
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Mixed estimates for singular integrals on weighted Hardy spaces María Eugenia Cejas1 · Estefanía Dalmasso2 Received: 26 February 2019 / Accepted: 14 October 2019 © Universidad Complutense de Madrid 2019
Abstract In this paper we give quantitative bounds for the norms of different kinds of singular p integral operators on weighted Hardy spaces Hw , where 0 < p ≤ 1 and w is a weight in the Muckenhoupt A∞ class. We deal with Fourier multiplier operators, Calderón– Zygmund operators of homogeneous type which are particular cases of the first ones, and, more generally, we study singular integrals of convolution type. In order to prove mixed estimates in the setting of weighted Hardy spaces, we need to introduce several characterizations of weighted Hardy spaces by means of square functions, Littlewood– Paley functions and the grand maximal function. We also establish explicit quantitative p bounds depending on the weight w when switching between the Hw -norms defined by the Littlewood–Paley–Stein square function and its discrete version, and also by applying the mixed bound Aq −A∞ result for the vector-valued extension of the Hardy– Littlewood maximal operator given in Buckley (Trans Am Math Soc 340(1):253–272, 1993). Keywords Weighted Hardy spaces · Singular integrals · Mixed estimates · Calderón–Zygmund operators · Fourier multipliers Mathematics Subject Classification 42B30 · 42B20 · 42B15 · 42B25
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Estefanía Dalmasso [email protected] María Eugenia Cejas [email protected]
1
Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CONICET, Calle 50 y 115, 1900 La Plata, Buenos Aires, Argentina
2
Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQ., Predio Dr. Alberto Cassano del CCT-CONICET-Santa Fe, Colectora Ruta Nacional 168, S3007ABA Santa Fe, Argentina
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M. E. Cejas, E. Dalmasso
1 Introduction Boundedness of singular integrals in weighted norms arises in the analysis of PDE’s. It p is a classical result that many operators in harmonic analysis are bounded on L w (Rn ), 1 < p < ∞, if the weight w belongs to the so-called A p Muckenhoupt class. The p sharp dependence of the corresponding L w (Rn ) operator norms in terms of the A p characteristic of w has been studied for different operators in the last few years. We recall that a weight w, that is, a non-negative locally integrable function, belongs to the Muckenhoupt A p class for 1 < p < ∞ if
[w] A p
1 = sup |Q| Q
w Q
1 |Q|
w
1 − p−1
p−1 0 : Mw ≤ kw a.e.}. For p = ∞, we say that w ∈ A∞ if 1 [w] A∞ = sup M(wχ Q ) < ∞, Q w(Q) Q where w(Q) = Q w. We call [w] A p the A p constant or characteristic of the weight w. It is well-known that there exists a constant that depends on the dimension n such that [w] A∞ ≤ cn [w] A p , for p ≥ 1. All the above constants can be defined over balls instead of cubes, and they are the same up to a dimensional constant. In [1], Buckley proved that for the Hardy–Littlewood maximal operator M and 1< p 1 in [11]. When considering p = 1, i
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