616 research outputs found

    BOUNDEDNESS OF HIGHER-ORDER MARCINKIEWICZ-TYPE INTEGRALS

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    Let A be a function with derivatives of order m and D γ A ∈Λ β (0 < β < 1, |γ| = m). The authors in the paper proved that if Ω ∈ L s (S n−1 ) (s ≥ n/(n − β)) is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integral

    BOUNDEDNESS OF HIGHER-ORDER MARCINKIEWICZ-TYPE INTEGRALS

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    Let A be a function with derivatives of order m and D γ A ∈Λ β (0 < β < 1, |γ| = m). The authors in the paper proved that if Ω ∈ L s (S n−1 ) (s ≥ n/(n − β)) is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integral

    Optimal control of singular Fourier multipliers by maximal operators

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    We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted L2(R)L^2(\mathbb{R}) norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form ei∣ξ∣αe^{i|\xi|^\alpha} for both α\alpha positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more novel "improper fractional averages" associated with "escape" regions. Some applications are given to the theory of Lp−LqL^p-L^q multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to higher dimensions.Comment: 22 page
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