Related papers: $L^2$ Bounds for a maximal directional Hilbert tra…

We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is…

Let $L_{A}=-{\rm div}(A\nabla)$ be an elliptic divergence form operator with bounded complex coefficients subject to mixed boundary conditions on an arbitrary open set $\Omega\subseteq\mathbb{R}^{d}$. We prove that the maximal operator…

In this paper, we introduce new general frameworks for estimating the maximal dimension of Hilbert cubes contained in finite truncations of arbitrary sets. As applications, we investigate Hilbert cubes in a range of arithmetic sets,…

This note concerns bounded derivations on maximal triangular operator algebras on a Hilbert space. Given any bounded derivation $\delta$ on a maximal triangular algebra whose invariant lattice is continuous at 1, an operator which is shown…

Homogenization of a scalar elliptic equation in a bounded domain with Neuman boundary condition is studied. Coefficients of the operator are oscillating over two different groups of variables with different small periods $\varepsilon$ and…

Let $\Omega\subset\mathbb R^n$ be a bounded domain of class $C^{2+\alpha}$, $0<\alpha<1$. We show that if $n\geq 3$ and $u_\Omega$ is the maximal solution of equation $\Delta u = n(n-2)u^{(n+2)/(n-2)}$ in $\Omega$, then the hyperbolic…

We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $\gamma$ be a closed smooth…

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…

By giving a thin width of $0<\delta\ll 1$ to both a unit circle and a unit line, we set an annulus and a tube on the Euclidean plane $\mathbb{R}^2$. Consider the maximal means $M_\delta$ over dilations of the annulus, and $N_\delta$ over…

We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to S. Kwapie\'{n}, from the linear to the multilinear setting. We…

We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz single- and double-layer boundary-integral operators as mappings from $L^2(\partial \Omega)\rightarrow H^1(\partial \Omega)$ (where $\partial\Omega$ is the…

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We…

We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately.…

Let $\Omega$ be a strongly Lipschitz domain of $\reel^n$. Consider an elliptic second order divergence operator $L$ (including a boundary condition on $\partial\Omega$) and define a Hardy space by imposing the non-tangential maximal…

This paper establishes an explicit $L^2$-estimate for weak solutions $u$ to linear elliptic equations in divergence form with general coefficients and external source term $f$, stating that the $L^2$-norm of $u$ over $U$ is bounded by a…

Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\| P T (I-P) \| = \| (I-P) T P \|$ for all orthogonal projections $P$ on $\mathcal{H}$. When $\mathcal{H}$ is finite-dimensional, we also…

A rigorous lower bound is obtained for the average resolution of any estimate of a shift parameter, such as an optical phase shift or a spatial translation. The bound has the asymptotic form k_I/<2|G|> where G is the generator of the shift…

Stein conjectured that the Hilbert transform in the direction of a vector field is bounded on, say, $L^2$ whenever $v$ is Lipschitz. We establish a wide range of $L^p$ estimates for this operator when $v$ is a measurable, non-vanishing,…

$L^p$ to $L^p_{\beta}$ boundedness theorems are proven for translation invariant averaging operators over hypersurfaces in Euclidean space. The operators can either be Radon transforms or averaging operators with multiparameter fractional…

We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators ${\mathscr L}$. We prove that if $-{\mathscr L}$ generates an analytic semigroup on $L^{2}(\gamma_{\infty})$, then…