Boundedness of higher order commutators of oscillatory singular integrals with rough kernels

Huoxiong Wu

Displaying similar documents to “Boundedness of higher order commutators of oscillatory singular integrals with rough kernels”

Non-homogeneous strongly singular integrals

Bassam Shayya (2008)

Studia Mathematica

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We study the $L^{p}$ mapping properties of a family of strongly singular oscillatory integral operators on ℝⁿ which are non-homogeneous in the sense that their kernels have isotropic oscillations but non-isotropic singularities.

Boundedness of commutators of an oscillatory integral operator

Xia Xia, Shanzhen Lu (2008)

Studia Mathematica

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We obtain a necessary and sufficient condition for $L^{p}$ boundedness of commutators of certain oscillatory integral operators and Lipschitz functions.

Boundedness of certain oscillatory singular integrals

Dashan Fan, Yibiao Pan (1995)

Studia Mathematica

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We prove the $L^{p}$ and $H^{1}$ boundedness of oscillatory singular integral operators defined by Tf = p.v.Ω∗f, where $Ω (x) = e^{i Φ (x)} K (x)$ , K(x) is a Calderón-Zygmund kernel, and Φ satisfies certain growth conditions.

Ψ-pseudodifferential operators and estimates for maximal oscillatory integrals

Carlos E. Kenig, Wolfgang Staubach (2007)

Studia Mathematica

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We define a class of pseudodifferential operators with symbols a(x,ξ) without any regularity assumptions in the x variable and explore their $L^{p}$ boundedness properties. The results are applied to obtain estimates for certain maximal operators associated with oscillatory singular integrals.

On the Existence of Oscillatory Solutions of the Second Order Nonlinear ODE

Martin Rohleder (2012)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The paper investigates the singular initial problem[4pt] ${(p (t) u^{'} (t))}^{'} + q (t) f (u (t)) = 0, u (0) = u_{0}, u^{'} (0) = 0$ [4pt] on the half-line $[0, \infty)$ . Here $u_{0} \in [L_{0}, L]$ , where $L_{0}$ , $0$ and $L$ are zeros of $f$ , which is locally Lipschitz continuous on $ℝ$ . Function $p$ is continuous on $[0, \infty)$ , has a positive continuous derivative on $(0, \infty)$ and $p (0) = 0$ . Function $q$ is continuous on $[0, \infty)$ and positive on $(0, \infty)$ . For specific values $u_{0}$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$ , $p$ and $q$ it is shown that the problem has for each specified...

Singular integrals with highly oscillating kernels on product spaces

Elena Prestini (2000)

Colloquium Mathematicae

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We prove the $L^{2} (^{2})$ boundedness of the oscillatory singular integrals $P_{0} f (x, y) = \int_{D_{x}} e^{i (M_{2} (x) y^{'} + M_{1} (x) x^{'})} ο v e r x^{'} y^{'} f (x - x^{'}, y - y^{'}) d x^{'} d y^{'}$ for arbitrary real-valued $L^{\infty}$ functions $M_{1} (x), M_{2} (x)$ and for rather general domains $D_{x} \subseteq^{2}$ whose dependence upon x satisfies no regularity assumptions.

Oscillation of third order differential equation with damping term

Miroslav Bartušek, Zuzana Došlá (2015)

Czechoslovak Mathematical Journal

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We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $x^{'''} (t) + q (t) x^{'} (t) + r (t) {| x |}^{λ} (t) sgn x (t) = 0, t \geq 0 .$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $λ \leq 1$ and if the corresponding second order differential equation $h^{''} + q (t) h = 0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

Generalized Hörmander conditions and weighted endpoint estimates

María Lorente, José María Martell, Carlos Pérez, María Silvina Riveros (2009)

Studia Mathematica

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We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded...

On the oscillation of a class of linear homogeneous third order differential equations

N. Parhi, P. Das (1998)

Archivum Mathematicum

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In this paper we have considered completely the equation $y^{'''} + a (t) y^{''} + b (t) y^{'} + c (t) y = 0, (*)$ where $a \in C^{2} ([σ, \infty), R)$ , $b \in C^{1} ([σ, \infty), R)$ , $c \in C ([σ, \infty), R)$ and $σ \in R$ such that $a (t) \leq 0$ , $b (t) \leq 0$ and $c (t) \leq 0$ . It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A. C. Lazer earlier.

Hardy spaces and oscillatory singular integrals.

Yibiao Pan (1991)

Revista Matemática Iberoamericana

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Positive coefficients case and oscillation

Ján Ohriska (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and $\int^{\infty} [1 / r (t)] d t$ converges.