Related papers: Sharp $L^p$ decay estimates for degenerate and sin…

We study singular integral operators with variable Calder\'on--Zygmund kernels and their commutators with $VMO$ functions in the framework of Orlicz spaces. After revisiting the classical $L^p$ theory, we establish boundedness results in…

Oscillatory integral operators with $1$-homogeneous phase functions satisfying a convexity condition are considered. For these we show the $L^p - L^p$-estimates for the Fourier extension operator of the cone due to Ou--Wang via polynomial…

We study bilinear rough singular integral operators $\mathcal{L}_{\Omega}$ associated with a function $\Omega$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slav\'ikov\'a (Math. Ann. 376: 431-455, 2020), they…

Consider a non-negative self-adjoint operator $H$ in $L^2(\mathbb{R}^d)$. We suppose that its heat operator $e^{-tH}$ satisfies an off-diagonal algebraic decay estimate, for some exponents $p_0\in[0,2)$. Then we prove sharp $L^p\to L^p$…

Let T be an oscillatory integral operator on L^2(R) with a smooth real phase function S(x,y). We prove that, in all cases but the one described below, after localization to a small neighborhood of the origin the norm of T decays like…

Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type :$$\|\Delta\_{x}^{\frac \alpha 2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v +…

In this paper, we present new proofs for both the sharp $L^p$ estimate and the decoupling theorem for the H\"ormander oscillatory integral operator. The sharp $L^p$ estimate was previously obtained by Stein\;\cite{stein1} and Bourgain-Guth…

We obtain $L^2$ decay estimates in $\lambda$ for oscillatory integral operators whose phase functions are homogeneous polynomials of degree m and satisfy various genericity assumptions. The decay rates obtained are optimal in the case of…

A local median decomposition is used to prove that a weighted local mean of a function is controlled by a weighted local mean of its local sharp maximal function. Together with (a local version of) the estimate $M^{\sharp}_{0,s}(Tf)(x) \le…

In this paper we establish sharp estimates (up to logarithmic losses) for the multilinear oscillatory integral operator studied by Phong, Stein, and Sturm and Carbery and Wright on any product $\prod_{j=1}^d L^{p_j}(\mathbb R)$ with each…

We establish endpoint bounds on a Hardy space $H^1$ for a natural class of multiparameter singular integral operators which do not decay away from the support of rectangular atoms. Hence the usual argument via a Journ\'e-type covering lemma…

It is shown that integral operators of the fully nonlinear type $K(x)(t)=\int_\Omega k(t,s,x(t),x(s))\,ds$ exhibit similar degeneracy phenomena in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In particular, $K$ is…

It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb{R})$ for all $1<p<\infty$. In this note it is shown that $$ \| S_{\mathcal{I}_{E_2}} \|_{L^p (\mathbb{R})…

In this paper, we investigate sharp damping estimates for a class of one dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are…

For a limited range of indices $p$, we obtain $L^p(\mathbb{R}^n)$ boundedness for singular integral operators whose kernels satisfy a condition weaker than the typical H\"ormander smoothness estimate. These operators are assumed to be…

In the paper, we study a kind of Oscillatory singular integral operator with Calder\'{o}n Type Commutators $T_{P,K,A} $ defined by \[T_{P,K,A} f(x)=\text { p.v.} \int_{\mathbb{R}^{n}} f(y) \frac{K(x-y)}{|x-y|}(A(x)-A(y)-\nabla A(y))(x-y)…

Let $\Omega$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator…

In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ, Li, Tao, and Thiele [6]. A key purpose of this work is to determine when such…

We prove certain $L^p$ estimates ($1<p<\infty$) for non-isotropic singular integrals along surfaces of revolution. As an application we obtain $L^p$ boundedness of the singular integrals under a sharp size condition on their kernels.

In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators $L=-(\nabla-i\mathbf{a})^TA(\nabla-i\mathbf{a})+V$. The latter class includes, in…