Related papers: $L^p$ estimates for fractional schrodinger operato…

We will explain how to compute the exact $L^p$ operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates…

We propose a counterpart of the classical Rollnik-class of potentials for fractional and massive relativistic Laplacians, and describe this space in terms of appropriate Riesz potentials. These definitions rely on precise resolvent…

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim…

We present and study a novel numerical algorithm to approximate the action of $T^\beta:=L^{-\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a…

In this manuscript, we provide a point-wise estimate for the $3$-commutators involving fractional powers of the sub-Laplacian on Carnot groups of homogeneous dimension $Q$. This can be seen as a fractional Leibniz rule in the sub-elliptic…

Let $n\in\mathbb{N}$ and ${\alpha}\in(0,\min\{2,n\})$. For any $a\in[a^\ast,\infty)$, the fractional Schr\"odinger operator $L_\alpha$ is defined by \begin{equation*} L_\alpha:=(-\Delta)^{{\alpha}/2}+a{|x|}^{-{\alpha}}, \end{equation*}…

We use the polynomial partitioning method of Guth to prove weighted Fourier restriction estimates in $\Bbb R^3$ with exponents $p$ that range between $3$ and $3.25$, depending on the weight. As a corollary to our main theorem, we obtain new…

We discuss exponential decay in $L^p(R^N)$, $1\leq p \leq \infty$, of solutions of a fractional Schr\"odinger parabolic equation with a locally uniformly integrable potential. The exponential type of the semigroup of solutions is considered…

In this paper we consider fractional Kolmogorov operators defined, in $\mathbb{R}^d$, by \[\Lambda_\kappa=(-\Delta)^{\alpha/2}+\frac{\kappa}{|x|^\alpha} x\cdot \nabla,\] with $\alpha\in (1,2)$, $\alpha<(d+2)/2$ and $\kappa\in \mathbb{R}$.…

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(|x|) - E$ in dimension $n \ge 2$, where $h, \, E > 0$, and $V: [0, \infty) \to \mathbb{R}$ is $L^\infty$…

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive…

Let $\Delta_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{it\Delta_{\mathbb{T}^m\times…

In this paper, we shall prove the uniform sharp $L^p$ decay estimates for a class of oscillatory integral operators with polynomial phases. By this one-dimensional result, we can use the rotation method to obtain uniform sharp $L^p$…

We consider an abstract non-negative self-adjoint operator $H$ on an $L^2$-space. We derive a characterization for the restriction estimate $\| dE_H(\lambda) \|_{L^p \to L^{p'}} \le C \lambda^{\frac{d}{2}(\frac{1}{p} - \frac{1}{p'}) -1}$ in…

We establish the $L^p(\mathbb{R}^3)$ boundedness of the helical maximal function for the sharp range $p>3$. Our results improve the previous known bounds for $p>4$. The key ingredient is a new microlocal smoothing estimate for averages…

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is…

We prove optimal regularity results in $L_p$-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is…

We establish sharp $L^p$ integral mean estimates for $(\alpha,\beta)$-harmonic functions on the unit disk. Explicit bounds for the functions and their partial derivatives are obtained in terms of boundary data, by means of the associated…

In this paper, we are concerned with the following equation involving higher-order fractional Lapalacian \begin{equation*} \left\{\begin{aligned} &(-\Delta)^{p+{\frac{\alpha}{2}}}u(x)=u_+^\gamma~~ \mbox{ in }\mathbb{R}^n,\\…

We consider a non-local operator $L_{{ \alpha}}$ which is the sum of a fractional Laplacian $\triangle^{\alpha/2} $, $\alpha \in (0,1)$, plus a first order term which is measurable in the time variable and locally $\beta$-H\"older…