On the Littlewood-Paley function $g^*$ of multiple Fourier integrals and Hankel multiplier transformations

Singular Integrals and Littlewood–Paley Operators

We prove mixed Ap-Ar inequalities for several basic singular integrals, Littlewood–Paley operators, and the vector-valued maximal function. Our key point is that r can be taken arbitrarily big. Hence, such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the “Ap-A∞” constant in these works involves two independent su...

Pseudo-localization of Singular Integrals and Noncommutative Littlewood-paley Inequalities

Understood in a wide sense, square functions play a central role in classical Littlewood-Paley theory. This entails for instance dyadic type decompositions of Fourier series, Stein’s theory for symmetric diffusion semigroups or Burkholder’s martingale square function. All these topics provide a deep technique when dealing with quasi-orthogonalitymethods, sums of independent variables, Fourier m...

Littlewood-Paley g-Functions and Multipliers for the Laguerre Hypergroup

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón’s reproducing formula and Littlewood–Paley theory with weights to establish the H w → H w (0 < p < ∞) and H w → Lw (0 < p ≤ 1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w ∈ A∞. The bounds will be expressed in terms of ...

A Remark on Littlewood-paley Theory for the Distorted Fourier Transform

We consider the classical theorems of Mikhlin and LittlewoodPaley from Fourier analysis in the context of the distorted Fourier transform. The latter is defined as the analogue of the usual Fourier transform as that transformation which diagonalizes a Schrödinger operator −∆+ V . We show that for such operators which display a zero energy resonance the full range 1 < p < ∞ in the Mikhlin theore...