616 research outputs found
BOUNDEDNESS OF HIGHER-ORDER MARCINKIEWICZ-TYPE INTEGRALS
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
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
We control a broad class of singular (or "rough") Fourier multipliers by
geometrically-defined maximal operators via general weighted
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
for both 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
multipliers, oscillatory integrals and dispersive PDE, along with natural
extensions to higher dimensions.Comment: 22 page
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