A Dimension-Free Weak-Type Estimate for Operators on UMD-Valued Functions

AbstractLet T denote the unit circle in the complex plane, and let X be a Banach space that satisfies Burkholder’s UMD condition. Fix a natural number, N ∈ . Let P denote the reverse lexicographical order on ZN. For each f ∈ L1(TN, X), there exists a strongly measurable function such that formally, for all . In this paper, we present a summation method for this conjugate function directly analogous to the martingale methods developed by Asmar and Montgomery-Smith for scalar-valued functions. Using a stochastic integral representation and an application of Garling’s characterization of UMD spaces, we prove that the associated maximal operator satisfies a weak-type (1, 1) inequality with a constant independent of the dimension N.

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Hermite Conjugate Functions and Rearrangement Invariant Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-059-5 ◽

1973 ◽

Vol 16 (3) ◽

pp. 377-380 ◽

Cited By ~ 1

Author(s):

Kenneth F. Andersen

Keyword(s):

Weak Type ◽

Conjugate Function ◽

Poisson Integral ◽

Conjugate Functions ◽

Rearrangement Invariant Spaces ◽

Image Position ◽

Invariant Spaces

The Hermite conjugate Poisson integral of a given f ∊ L1(μ), dμ(y)= exp(—y2) dy, was defined by Muckenhoupt [5, p. 247] aswhereIf the Hermite conjugate function operator T is defined by (Tf) a.e., then one of the main results of [5] is that T is of weak-type (1, 1) and strongtype (p,p) for all p>l.

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Boundedness of integral operators on decreasing functions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210515000098 ◽

2015 ◽

Vol 145 (4) ◽

pp. 725-744 ◽

Cited By ~ 2

Author(s):

María J. Carro ◽

Carmen Ortiz-Caraballo

Keyword(s):

Harmonic Analysis ◽

Weak Type ◽

Integral Operators ◽

Lorentz Spaces ◽

Type Estimate ◽

Weighted Lorentz Spaces ◽

Decreasing Functions

We continue the study of the boundedness of the operatoron the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

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Sawyer-type inequalities for Lorentz spaces

Mathematische Annalen ◽

10.1007/s00208-021-02240-4 ◽

2021 ◽

Author(s):

Carlos Pérez ◽

Eduard Roure-Perdices

Keyword(s):

Large Class ◽

Maximal Operator ◽

Weak Type ◽

Lorentz Spaces ◽

Maximal Operators ◽

Type Estimate ◽

Sparse Operators ◽

Extrapolation Theorem ◽

Littlewood Maximal Operator

AbstractThe Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{1,\infty }(uv)} \le C_{u,v} \Vert f \Vert _{L^{1}(u)}, \end{aligned}$$ Mf v L 1 , ∞ ( u v ) ≤ C u , v ‖ f ‖ L 1 ( u ) , where $$u\in A_1$$ u ∈ A 1 and $$uv\in A_{\infty }$$ u v ∈ A ∞ . We prove a novel extension of this result to the general restricted weak type case. That is, for $$p>1$$ p > 1 , $$u\in A_p^{{\mathcal {R}}}$$ u ∈ A p R , and $$uv^p \in A_\infty $$ u v p ∈ A ∞ , $$\begin{aligned} \left\| \frac{Mf}{v}\right\| _{L^{p,\infty }(uv^p)} \le C_{u,v} \Vert f \Vert _{L^{p,1}(u)}. \end{aligned}$$ Mf v L p , ∞ ( u v p ) ≤ C u , v ‖ f ‖ L p , 1 ( u ) . From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including m-linear Calderón-Zygmund operators, avoiding the $$A_\infty $$ A ∞ extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of $$A_p^{{\mathcal {R}}}$$ A p R . Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator $${\mathcal {M}}$$ M , denoted by $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R , establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, $$A_p^{{\mathcal {R}}}$$ A p R and $$A_{\mathbf {P}}^{{\mathcal {R}}}$$ A P R weights, and Lorentz spaces.

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