Weak-type inequalities for maximal operators acting on Lorentz spaces

2014 ◽  
Vol 101 ◽  
pp. 145-162
Author(s):  
Adam Osękowski
Keyword(s):  
Weak Type ◽  
Lorentz Spaces ◽  
Maximal Operators

scholarly journals Sawyer-type inequalities for Lorentz spaces

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.

Weak type estimates for maximal operators with a cylindric distance function

2006 ◽  
Vol 253 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Sunggeum Hong ◽  
Paul Taylor ◽  
Chan Woo Yang
Keyword(s):  
Distance Function ◽  
Weak Type ◽  
Maximal Operators ◽  
Weak Type Estimates

A Unified Version of Weighted Weak Type Inequality for Martingale Maximal Operators

2021 ◽  
Vol 58 (2) ◽  
pp. 216-229
Author(s):  
Yanbo Ren ◽  
Congbian Ma
Keyword(s):  
Weak Type ◽  
Type Inequality ◽  
Maximal Inequality ◽  
Complete Characterization ◽  
Maximal Operators ◽  
Weak Type Inequality

Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the formfor martingale maximal operators f ∗ is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

scholarly journals BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 655-663
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
Borel Measure ◽  
Weak Type ◽  
Integrable Function ◽  
Maximal Operators ◽  
Weak Type Estimates ◽  
Locally Integrable Function ◽  
Image Position ◽  
Strong Maximal Operator ◽  
General Product

AbstractLet μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

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