We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a... more We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Strömberg and A. Torchinsky; however , we must overcome several technical difficulties associated with considering one-sided Calderón-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.
Publ. Mat. 54 (2010), 53–71 A BOUNDEDNESS CRITERION FOR GENERAL MAXIMAL OPERATORS
We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a rever... more We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a reversed weak type inequality, we obtain a boundedness criterion for MB on an arbitrary quasi-Banach function space X. Being applied to specific B and X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M+ on Lpw. 1.
In this paper we improve several weighted weak type bounds for the commutator $[b,T]$ of a Calder... more In this paper we improve several weighted weak type bounds for the commutator $[b,T]$ of a Calder\'on-Zygmund operator $T$ with a $BMO$ function $b$. Our key tool is a new pointwise estimate for $[b,T]$, which is close in the spirit to recent results establishing pointwise estimates of Calder\'on-Zygmund operators by sparse operators.
For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that χ Q XQ ≃ 1, a... more For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that χ Q XQ ≃ 1, and for X = {X Q } define f BMO X := sup Q f − 1 |Q| Q f XQ , f BMO * X := sup Q inf c f − c XQ. We study necessary and sufficient conditions on X such that BMO = BMO X = BMO * X. In particular, we give a full characterization of the embedding BMO ֒→ BMO X in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding BMO * X ֒→ BMO. Our main theorems recover and improve all previously known results in this area.
We construct an example showing that the upper bound [w] A1 log(e + [w] A1) for the L 1 (w) → L 1... more We construct an example showing that the upper bound [w] A1 log(e + [w] A1) for the L 1 (w) → L 1,∞ (w) norm of the Hilbert transform cannot be improved in general.
We prove the sharp mixed Ap − A∞ weighted estimate for the Hardy-Littlewood maximal function in t... more We prove the sharp mixed Ap − A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely ‖M‖Lp,q (w) .p,q,n [w] 1 p Ap [σ] 1 min(p,q) A∞ , where σ = w 1 1−p . Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting.
We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$ , where $x\... more We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$ , where $x\in {\mathbb R}^{n}$ and Q is a cube in ${\mathbb R}^{n}$ . When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré–Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vector-valued square functions.
In this paper we present the results announced in the recent work by the first, second, and fourt... more In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing that, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding "finite" case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calder\'on-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm class. The same occurs with the multilinear Calder\...
We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on ℓ^r-v... more We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on ℓ^r-valued extensions of linear operators. We show that for certain 1 ≤ p, q_1, ..., q_m, r ≤∞, there is a constant C≥ 0 such that for every bounded multilinear operator T L^q_1(μ_1) ×...× L^q_m(μ_m) → L^p(ν) and functions {f_k_1^1}_k_1=1^n_1⊂ L^q_1(μ_1), ..., {f_k_m^m}_k_m=1^n_m⊂ L^q_m(μ_m), the following inequality holds (1) (∑_k_1, ..., k_m |T(f_k_1^1, ..., f_k_m^m)|^r)^1/r_L^p(ν)≤ C T∏_i=1^m (∑_k_i=1^n_i |f_k_i^i|^r)^1/r_L^q_i(μ_i). In some cases we also calculate the best constant C≥ 0 satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.
We show that the Hilbert transform does not map L^1(M_Φw) to L^1,∞(w) for every Young function Φ ... more We show that the Hilbert transform does not map L^1(M_Φw) to L^1,∞(w) for every Young function Φ growing more slowly than t ( e^ e+t). Our proof is based on a construction of M.C. Reguera and C. Thiele.
We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators o... more We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
In this paper, building upon ideas of Naor and Tao and continuing the study initiated in by the a... more In this paper, building upon ideas of Naor and Tao and continuing the study initiated in by the authors and Safe, sufficient conditions are provided for weighted weak type and strong type (p,p) estimates with p>1 for the centered maximal function on the infinite rooted k-ary tree to hold. Consequently a wider class of weights for those strong and weak type (p,p) estimates than the one obtained in by the authors and Safe in a previous work is provided. Examples showing that the Sawyer type testing condition and the A_p condition do not seem precise in this context are supplied as well. We also prove that strong and weak type estimates are not equivalent, highlighting the pathological nature of the theory of weights in this setting. Two weight counterparts of our conditions will be obtained as well.
We obtain an improved version of the pointwise sparse domination principle established by the fir... more We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator T for which it admits a sparse domination.
In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapola... more In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes A_p⃗, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes A_p⃗ and extrapolate within the classes A_p⃗,r⃗ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range and extend the estimates to the quasi-Banach range. As a direct consequence, we obtain weighted vector-valued inequalities reproving some of the results by Benea and Muscalu. We also extend recent results of Carando et al. on Marcink...
We study mixed weighted weak-type inequalities for families of functions, which can be applied to... more We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.
In this paper we extend the bump conjecture and a particular case of the separated bump conjectur... more In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators T m b. Our results are new even for the first order commutator T 1 b. A new bump type necessary condition for the two-weighted boundedness of T m b is obtained as well. We also provide some results related to a converse to Bloom's theorem.
Abstract. In this paper, we introduce the Hp,+q,α (w) spaces, where 0 < p ≤ 1, 1 < q < ∞... more Abstract. In this paper, we introduce the Hp,+q,α (w) spaces, where 0 < p ≤ 1, 1 < q < ∞, α> 0, and for weights w belonging to the class A+s defined by E. Sawyer. To define these spaces, we consider a one-sided version of the maximal function N+q,α(F, x) defined by A. Calderón. In the case that w ≡ 1, these spaces have been studied by A. Gatto, J. G. Jiménez and C. Segovia. We introduce a notion of p-atom in Hp,+q,α (w), and we prove that we can express the elements of Hp,+q,α (w) in term of series of multiples of p-atoms. On the other side, we prove that the Weyl fractional integral Pα can be extended to a bounded operator from the one-sided Hardy space Hp+ (w) into Hp,+q,α (w). Moreover, we prove that this extension, if α is a natural number, is an isomorphism. 1. Notations, definitions and prerequisites Let f(x) be a Lebesgue measurable function defined on R. The one-sided Hardy-Littlewood maximal functions M+f(x) and M−f(x) are defined as M+f(x) = sup h>0 ∫ x+h ...
Sharp A1 bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and
For any Calderón–Zygmund operator T the following sharp estimate is obtained for 1 < p< ∞:... more For any Calderón–Zygmund operator T the following sharp estimate is obtained for 1 < p< ∞: ‖T‖L p(w) ≤ cνp ‖w‖A1, where νp = p2p−1 log(e+ 1p−1). In the case where p = 2 and T is a classical convolution singular integral, this result is due to R. Fefferman and J. Pipher [7]. Then, we deduce the following weak type (1, 1) estimate related to a problem of Muckenhoupt and Wheeden [11]: sup λ>0 λw{x ∈ Rn: |T f (x) |> λ} ≤ cϕ(‖w‖A1)
We obtain a complex interpolation theorem between weighted Calderon-Hardy spaces for weights in a... more We obtain a complex interpolation theorem between weighted Calderon-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Stromberg and A. Torchinsky; however, we must overcome several technical difficulties associated with considering one-sided Calderon-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.
In this note we introduce a dyadic one-sided maximal function defined as M +,d f (x) = sup Q dyad... more In this note we introduce a dyadic one-sided maximal function defined as M +,d f (x) = sup Q dyadic:x∈Q 1 |Q| Q + |f | , where Q + is a certain cube associated with the dyadic cube Q and f ∈ L 1 loc (R n). We characterize the pair of weights (w, v) for which the maximal operator M +,d applies L p (v) into weak-L p (w) for 1 ≤ p < ∞.
We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a... more We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Strömberg and A. Torchinsky; however , we must overcome several technical difficulties associated with considering one-sided Calderón-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.
Publ. Mat. 54 (2010), 53–71 A BOUNDEDNESS CRITERION FOR GENERAL MAXIMAL OPERATORS
We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a rever... more We consider maximal operators MB with respect to a basis B. In the case when MB satisfies a reversed weak type inequality, we obtain a boundedness criterion for MB on an arbitrary quasi-Banach function space X. Being applied to specific B and X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M+ on Lpw. 1.
In this paper we improve several weighted weak type bounds for the commutator $[b,T]$ of a Calder... more In this paper we improve several weighted weak type bounds for the commutator $[b,T]$ of a Calder\'on-Zygmund operator $T$ with a $BMO$ function $b$. Our key tool is a new pointwise estimate for $[b,T]$, which is close in the spirit to recent results establishing pointwise estimates of Calder\'on-Zygmund operators by sparse operators.
For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that χ Q XQ ≃ 1, a... more For every cube Q ⊂ R n we let X Q be a quasi-Banach function space over Q such that χ Q XQ ≃ 1, and for X = {X Q } define f BMO X := sup Q f − 1 |Q| Q f XQ , f BMO * X := sup Q inf c f − c XQ. We study necessary and sufficient conditions on X such that BMO = BMO X = BMO * X. In particular, we give a full characterization of the embedding BMO ֒→ BMO X in terms of so-called sparse collections of cubes and we give easily checkable and rather weak sufficient conditions for the embedding BMO * X ֒→ BMO. Our main theorems recover and improve all previously known results in this area.
We construct an example showing that the upper bound [w] A1 log(e + [w] A1) for the L 1 (w) → L 1... more We construct an example showing that the upper bound [w] A1 log(e + [w] A1) for the L 1 (w) → L 1,∞ (w) norm of the Hilbert transform cannot be improved in general.
We prove the sharp mixed Ap − A∞ weighted estimate for the Hardy-Littlewood maximal function in t... more We prove the sharp mixed Ap − A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely ‖M‖Lp,q (w) .p,q,n [w] 1 p Ap [σ] 1 min(p,q) A∞ , where σ = w 1 1−p . Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting.
We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$ , where $x\... more We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$ , where $x\in {\mathbb R}^{n}$ and Q is a cube in ${\mathbb R}^{n}$ . When applied to operators, this result recovers our recent works [37, 39]. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalised Poincaré–Sobolev inequalities, tent spaces and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localisable in the sense of [39], as we will demonstrate in an application to vector-valued square functions.
In this paper we present the results announced in the recent work by the first, second, and fourt... more In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing that, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding "finite" case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calder\'on-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm class. The same occurs with the multilinear Calder\...
We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on ℓ^r-v... more We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on ℓ^r-valued extensions of linear operators. We show that for certain 1 ≤ p, q_1, ..., q_m, r ≤∞, there is a constant C≥ 0 such that for every bounded multilinear operator T L^q_1(μ_1) ×...× L^q_m(μ_m) → L^p(ν) and functions {f_k_1^1}_k_1=1^n_1⊂ L^q_1(μ_1), ..., {f_k_m^m}_k_m=1^n_m⊂ L^q_m(μ_m), the following inequality holds (1) (∑_k_1, ..., k_m |T(f_k_1^1, ..., f_k_m^m)|^r)^1/r_L^p(ν)≤ C T∏_i=1^m (∑_k_i=1^n_i |f_k_i^i|^r)^1/r_L^q_i(μ_i). In some cases we also calculate the best constant C≥ 0 satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.
We show that the Hilbert transform does not map L^1(M_Φw) to L^1,∞(w) for every Young function Φ ... more We show that the Hilbert transform does not map L^1(M_Φw) to L^1,∞(w) for every Young function Φ growing more slowly than t ( e^ e+t). Our proof is based on a construction of M.C. Reguera and C. Thiele.
We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators o... more We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
In this paper, building upon ideas of Naor and Tao and continuing the study initiated in by the a... more In this paper, building upon ideas of Naor and Tao and continuing the study initiated in by the authors and Safe, sufficient conditions are provided for weighted weak type and strong type (p,p) estimates with p>1 for the centered maximal function on the infinite rooted k-ary tree to hold. Consequently a wider class of weights for those strong and weak type (p,p) estimates than the one obtained in by the authors and Safe in a previous work is provided. Examples showing that the Sawyer type testing condition and the A_p condition do not seem precise in this context are supplied as well. We also prove that strong and weak type estimates are not equivalent, highlighting the pathological nature of the theory of weights in this setting. Two weight counterparts of our conditions will be obtained as well.
We obtain an improved version of the pointwise sparse domination principle established by the fir... more We obtain an improved version of the pointwise sparse domination principle established by the first author in [19]. This allows us to determine nearly minimal assumptions on a singular integral operator T for which it admits a sparse domination.
In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapola... more In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes A_p⃗, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes A_p⃗ and extrapolate within the classes A_p⃗,r⃗ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range and extend the estimates to the quasi-Banach range. As a direct consequence, we obtain weighted vector-valued inequalities reproving some of the results by Benea and Muscalu. We also extend recent results of Carando et al. on Marcink...
We study mixed weighted weak-type inequalities for families of functions, which can be applied to... more We study mixed weighted weak-type inequalities for families of functions, which can be applied to study classical operators in harmonic analysis. Our main theorem extends the key result from D. Cruz-Uribe, J.M. Martell and C. Perez, Weighted weak-type inequalities and a conjecture of Sawyer, Int. Math. Res. Not., V. 30, 2005, 1849-1871.
In this paper we extend the bump conjecture and a particular case of the separated bump conjectur... more In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators T m b. Our results are new even for the first order commutator T 1 b. A new bump type necessary condition for the two-weighted boundedness of T m b is obtained as well. We also provide some results related to a converse to Bloom's theorem.
Abstract. In this paper, we introduce the Hp,+q,α (w) spaces, where 0 < p ≤ 1, 1 < q < ∞... more Abstract. In this paper, we introduce the Hp,+q,α (w) spaces, where 0 < p ≤ 1, 1 < q < ∞, α> 0, and for weights w belonging to the class A+s defined by E. Sawyer. To define these spaces, we consider a one-sided version of the maximal function N+q,α(F, x) defined by A. Calderón. In the case that w ≡ 1, these spaces have been studied by A. Gatto, J. G. Jiménez and C. Segovia. We introduce a notion of p-atom in Hp,+q,α (w), and we prove that we can express the elements of Hp,+q,α (w) in term of series of multiples of p-atoms. On the other side, we prove that the Weyl fractional integral Pα can be extended to a bounded operator from the one-sided Hardy space Hp+ (w) into Hp,+q,α (w). Moreover, we prove that this extension, if α is a natural number, is an isomorphism. 1. Notations, definitions and prerequisites Let f(x) be a Lebesgue measurable function defined on R. The one-sided Hardy-Littlewood maximal functions M+f(x) and M−f(x) are defined as M+f(x) = sup h>0 ∫ x+h ...
Sharp A1 bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and
For any Calderón–Zygmund operator T the following sharp estimate is obtained for 1 < p< ∞:... more For any Calderón–Zygmund operator T the following sharp estimate is obtained for 1 < p< ∞: ‖T‖L p(w) ≤ cνp ‖w‖A1, where νp = p2p−1 log(e+ 1p−1). In the case where p = 2 and T is a classical convolution singular integral, this result is due to R. Fefferman and J. Pipher [7]. Then, we deduce the following weak type (1, 1) estimate related to a problem of Muckenhoupt and Wheeden [11]: sup λ>0 λw{x ∈ Rn: |T f (x) |> λ} ≤ cϕ(‖w‖A1)
We obtain a complex interpolation theorem between weighted Calderon-Hardy spaces for weights in a... more We obtain a complex interpolation theorem between weighted Calderon-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Stromberg and A. Torchinsky; however, we must overcome several technical difficulties associated with considering one-sided Calderon-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.
In this note we introduce a dyadic one-sided maximal function defined as M +,d f (x) = sup Q dyad... more In this note we introduce a dyadic one-sided maximal function defined as M +,d f (x) = sup Q dyadic:x∈Q 1 |Q| Q + |f | , where Q + is a certain cube associated with the dyadic cube Q and f ∈ L 1 loc (R n). We characterize the pair of weights (w, v) for which the maximal operator M +,d applies L p (v) into weak-L p (w) for 1 ≤ p < ∞.
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