Boundedness of commutators of an oscillatory integral operator

Xia Xia; Shanzhen Lu

Displaying similar documents to “Boundedness of commutators of an oscillatory integral operator”

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

Huoxiong Wu (2005)

Studia Mathematica

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The author studies the commutators generated by a suitable function a(x) on ℝⁿ and the oscillatory singular integral with rough kernel Ω(x)|x|ⁿ and polynomial phase, where Ω is homogeneous of degree zero on ℝⁿ, and a(x) is a BMO function or a Lipschitz function. Some mapping properties of these higher order commutators on $L^{p} (ℝ ⁿ)$ , which are essential improvements of some well known results, are given.

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.

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.

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.

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...

Oscillation of second-order quasilinear retarded difference equations via canonical transform

George E. Chatzarakis, Deepalakshmi Rajasekar, Saravanan Sivagandhi, Ethiraju Thandapani (2024)

Mathematica Bohemica

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We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $Δ (p (n) {(Δ y (n))}^{α}) + η (n) y^{β} (n - k) = 0$ under the condition $\sum_{n = n_{0}}^{\infty} p^{- \frac{1}{α}} (n) < \infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

Oscillation in deviating differential equations using an iterative method

George E. Chatzarakis, Irena Jadlovská (2019)

Communications in Mathematics

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Sufficient oscillation conditions involving $lim sup$ and $lim inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

Forced oscillation of third order nonlinear dynamic equations on time scales

Baoguo Jia (2010)

Annales Polonici Mathematici

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Consider the third order nonlinear dynamic equation $x^{Δ Δ Δ} (t) + p (t) f (x) = g (t)$ , (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $Δ ³ x (n) + n^{α} {| x |}^{γ} s g n (n) = (- 1) ⁿ n^{c}$ , where α ≥ -1, γ > 0, c > 3, is oscillatory.

Two-point oscillatory solutions to system with relay hysteresis and nonperiodic external disturbance

Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva (2024)

Applications of Mathematics

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We study an $n$ -dimensional system of ordinary differential equations with a constant matrix, a relay-type nonlinearity, and an external disturbance in the right-hand side. We consider a nonideal relay characteristic. The external disturbance is described by the product of an exponential function and a sine function with an initial phase as a parameter. We assume the matrix of the linear part and the vector at the relay characteristic such that, by a nonsingular transformation, the system...

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.