A pointwise estimate for positive dyadic shifts and some applications

Abstract

We prove a pointwise estimate for positive dyadic shifts of complexity $m$ which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.

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• Quantitative estimates and extrapolation for multilinear weight classes
• Extrapolation in general quasi-Banach function spaces
• Sparse domination theorem for multilinear singular integral operators with $L^r$-Hörmander condition
• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Sparse bilinear forms for Bochner Riesz multipliers and applications
• Sparse Bounds for the Discrete Cubic Hilbert Transform
• Three observations on commutators of Singular Integral Operators with BMO functions
• Operator-free sparse domination
• Multi-scale sparse domination
• Sparse domination implies vector-valued sparse domination
• Quadratic sparse domination and Weighted Estimates for non-integral Square Functions
• On logarithmic bounds of maximal sparse operators
• On pointwise ${\ell }^r$-sparse domination in a space of homogeneous type
• Sparse bounds for pseudo-multipliers associated to Grushin operators, I
• On a weak type estimate for sparse operators of strong type
• Sparse gradient bounds for divergence form elliptic equations
• Endpoint sparse domination for classes of multiplier transformations