Related papers: Sharp estimates for maximal operators associated t…

An explicit formula for the wave operators associated with Schroedinger operators on the discrete half-line is deduced from their stationary expressions. The formula enables us to understand the wave operators as one dimensional…

We study $L^p$-$L^q$ bounds on the spectral projection operator $\Pi_\lambda$ associated to the Hermite operator $H=|x|^2-\Delta$ in $\mathbb R^d$. We are mainly concerned with a localized operator $\chi_E\Pi_\lambda\chi_E$ for a subset…

We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces…

Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this direction set. Our main result provides an essentially sharp uniform bound,…

In this paper we prove that maximal H-monotone operators $T:H^n\rightrightarrows V_1$ whose domain is all the Heisenberg group $H^n$ are locally bounded. This implies that they are upper semicontinuous. As a consequence, maximal…

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's…

We prove that the wave operators of scattering theory for the fourth order Schr\"odinger operators $\Delta^2 + V(x)$ in ${\mathbb R}^4$ are bounded in $L^p({\mathbb R}^4)$ for the set of $p$'s of $(1,\infty)$ depending on the kind of…

We define the Hermite--Sobolev spaces naturally associated to the harmonic oscillator $H= -\Delta +|x|^2.$ Structural properties, relations with the classical Sobolev spaces, boundedness of operators and almost everywhere convergence of…

We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the H\"older relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of…

We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space $H^s$ with $s>3/2$. The main idea is to consider two suitable sequences of…

In this paper, we prove that the oscillation operator, variation operator and maximal differential transform associated with the approximate identities are bounded from ${\rm BMO}({\mathbb R}^n)$ to its subspace ${\rm BLO}({\mathbb R}^n)$.

We prove that the low energy parts of the wave operators $W_\pm$ for Schr\"odinger operators $H = -\lap + V(x)$ on $\R^4$ are bounded in $ L^p(\R^4)$ for $1<p\leq 2$ and are unbounded for $2<p\leq \infty$ if $H$ has resonances at the…

We find the largest linear space of bounded linear operators on L_1(Omega), that being restricted to any L_1(A), A \subset Omega, satisfy the Daugavet equation.

Let $P(D)$ be the Laplacian $\Delta,$ or the wave operator $\square$. The following type of Carleman estimate is known to be true on a certain range of $p,q$: \[ \|e^{v\cdot x}u\|_{L^q(\mathbb{R}^d)} \le C\|e^{v\cdot…

In this paper, we consider the Laplace operator on the half-space with Dirichlet and Neumann boundary conditions. We prove that this operator admits a bounded $H^\infty$-calculus on Sobolev spaces with power weights measuring the distance…

Consider spherical means on the Heisenberg group with a codimension two incidence relation, and associated spherical local maximal functions $M_Ef$ where the dilations are restricted to a set $E$. We prove $L^p\to L^q$ estimates for these…

We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the $L^p$-boundedness of wave operators for two dimensional Schr\"odinger operators and prove that they are bounded in $L^p(\R^2)$ for all $1<p<\infty$ if and only if…

The paper studies the uniqueness problem for the one-dimensional Schr\"{o}dinger operator associated with the formal differential expression \begin{equation*} l[u] =-u''+qu + i[(ru)'+ru'], \end{equation*} in the complex Hilbert space…

We establish $L^{p_1}\times\cdots\times L^{p_k}\to L^r$ and $\ell^{p_1}\times\cdots\times \ell^{p_k}\to \ell^r$ type bounds for multilinear maximal operators associated to averages over isometric copies of a given non-degenerate $k$-simplex…

We study the boundedness problem for maximal operators $\mathcal{M}$ associated to averages along families of hypersurfaces $S$ of finite type in $\mathbb{R}^n.$ In this paper, we prove that if $S$ is a finite type hypersurface which is of…