Some one-sided estimates for oscillatory singular integrals

The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\in A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the oscillatory singular integral is given.Comment: 24 pages, Nonlinear Anal. 201

Wiener type regularity for non-linear integro-differential equations

The primary purpose of this paper is to study the Wiener-type regularity criteria for non-linear equations driven by integro-differential operators, whose model is the fractional $p-$Laplace equation. In doing so, with the help of tools from potential analysis, such as fractional relative Sobolev capacities, Wiener type integrals, Wolff potentials, $(\alpha,p)-$barriers, and $(\alpha,p)-$balayages, we first prove the characterizations of the fractional thinness and the Perron boundary regularity. Then, we establish a Wiener test and a generalized fractional Wiener criterion. Furthermore, we also prove the continuity of the fractional superharmonic function, the fractional resolutivity, a connection between $(\alpha,p)-$potentials and $(\alpha,p)-$Perron solutions, and the existence of a capacitary function for an arbitrary condenser.Comment: 27 pages, any comments are welcom

Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures

Local well-posedness for the dispersion generalized periodic KdV equation

AbstractIn this paper, we set up the local well-posedness of the initial value problem for the dispersion generalized periodic KdV equation: ∂tu+∂x|Dx|αu=∂xu2, u(0)=φ for α>2, s⩾−α4 and φ∈Hs(T). And we show that the −α4 is a lower endpoint to obtain the bilinear estimates (1.2) and (1.3) which are the crucial steps to obtain the local well-posedness by Picard iteration. The case α=2 was studied in Kenig et al. (1996) [10]

Boundedness of Sublinear Operators with Rough Kernels on Weighted Morrey Spaces

The aim of this paper is to get the boundedness of a class of sublinear operators with rough kernels on weighted Morrey spaces under generic size conditions, which are satisfied by most of the operators in classical harmonic analysis. Applications to the corresponding commutators formed by certain operators and BMO functions are also obtained.</jats:p

On weighted weak type norm inequalities for one-sided oscillatory singular integrals

We consider one-sided weight classes of Muckenhoupt type and study the weighted weak type (1, 1) norm inequalities for a class of one-sided oscillatory singular integrals with smooth kernel. © Instytut Matematyczny PAN, 2011