We study a multilinear oscillatory integral with rough kernel and establish a boundedness criterion.

The purpose of this article is to obtain a multidimensional extension of Lacey and Thiele's result on the boundedness of a model sum which plays a crucial role in the boundedness of the bilinear Hilbert transform in one dimension. This proof is a simplification of the original proof of Lacey and Thiele modeled after the presentation of Bilyk and Grafakos.

Let $G$ be the Lie group ${ℝ}^2⋉{ℝ}^{+}$ endowed with the Riemannian symmetric space structure. Let $X_0,X_1,X_2$ be a distinguished basis of left-invariant vector fields of the Lie algebra of $G$ and define the Laplacian $\Delta =-(X_0^2+X_1^2+X_2^2)$. In this paper we consider the first order Riesz transforms $R_i=X_i{\Delta }^{-1/2}$ and $S_i={\Delta }^{-1/2}{X}_i$, for $i=0,1,2$. We prove that the operators $R_i$, but not the $S_i$, are bounded from the Hardy space $H^1$ to $L^1$. We also show that the second-order Riesz transforms $T_{ij}=X_i{\Delta }^{-1}{X}_j$ are bounded from $H^1$ to $L^1$, while the transforms $S_{ij}={\Delta }^{-1}{X}_i{X}_j$ and $R_{ij}=X_i{X}_j{\Delta }^{-1}$, for $i,j=0,1,2$, are not.

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

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

In the setting of spaces of homogeneous type, it is shown that the commutator of Calderón-Zygmund type operators as well as the commutator of a potential operator with a BMO function are bounded in a generalized grand Morrey space. Interior estimates for solutions of elliptic equations are also given in the framework of generalized grand Morrey spaces.

The author investigates the boundedness of the higher order commutator of strongly singular convolution operator, $T_b^m$, on Herz spaces $K{̇}_q^{\alpha ,p}\left(ℝⁿ\right)$ and $K_q^{\alpha ,p}\left(ℝⁿ\right)$, and on a new class of Herz-type Hardy spaces $HK{̇}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$ and $H{K}_{q,b,m}^{\alpha ,p,0}\left(ℝⁿ\right)$, where 0 < p ≤ 1 < q < ∞, α = n(1-1/q) and b ∈ BMO(ℝⁿ).

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 $L^p\left(ℝⁿ\right)$, which are essential improvements of some well known results, are given.