Related papers: $L^2$ bounds for a Kakeya type maximal operator in…
We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following:…
We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\Omega$ in $R^d$ under the assumptions that $\Omega$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it…
We study the maximal operator on the variable exponent H\"older spaces in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Let us stress that there are no…
Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate themes in harmonic analysis, discrete geometry and analytic number theory research.…
We prove Kakeya-type estimates for regulus strips. As a result, we obtain another epsilon improvement over the Kakeya conjecture in $\mathbb{R}^3$, by showing that the regulus strips in the ${\rm SL}_2$ example are essentially disjoint. We…
We refine a recent result of Drury concerning the optimal ratio between the norm and numerical radius of a bounded linear operator $T$ with numerical range lying in a sector of a circular disk. In particular, characterization is given to…
We prove new $\ell ^{p} (\mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d \geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of…
The main aim of this paper is to prove that the maximal operator $\sigma_{p}^{\kappa ,\ast }f:=\sup_{n\in \mathbf{P}}\left\vert \sigma_{n}^{\kappa }f\right\vert /\left( n+1\right) ^{1/p-2}$ is bounded from the Hardy space $% H_{p}$ to the…
We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider L\'evy operators that are symmetric and its Fourier symbol…
In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of $\mathcal{O}(|\log(\epsilon)|)$…
The main aim of this paper is to prove that when $0<p<1/2$ the maximal operator $\overset{\sim }{\sigma }_{p}^{\ast }f:=\underset{n\in \mathbb{N}}{% \sup }\frac{\left\vert \sigma_{n}f\right\vert }{\left( n+1\right) ^{1/p-2}}$ is bounded…
In this paper we derive the maximal subspace of positive numbers, for which the restricted maximal operator of Fej\'er means in this subspace is bounded from the Hardy space $H_{p}$ to the space $L_{p}$ for all $0<p\leq 1/2.$ Moreover, we…
We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^{p}(\mathbb{R}^{n},\exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional…
Maximal angular operator sends a function defined in a sector of the complex plane to a Maximal angular operator sends a function defined in a sector of the complex plane with vertex at 0 to the function of modulus obtained by maximizing…
We show that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. A description of this intersection is also given.
We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $\frac{N}{N-1}\leq \lambda_N\leq 2$, to the case of chemical hypergraphs.
The local maximal inequality for the Schr\"{o}dinger operators of order $\a>1$ is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to…
We improve the $L^{p}\rightarrow L^p$ restriction estimate in $\mathbb{R}^3$ to the range $p>3+3/14$, based on some Kakeya type incidence estimates and the refined decoupling theorem.
We show that the lower bound for the optimal directional discrepancy with respect to the class of rectangles in $\mathbb{R}^2$ rotated in a restricted interval of directions $[-\theta, \theta]$ with $\theta < \frac{\pi}{4}$ is of the order…
We obtain a necessary and sufficient condition on an exponent $p(\cdot)$ for which the Hardy--Littlewood maximal operator is bounded on the variable $L^{p(\cdot)}$ space. It is formulated in terms of the Muckenhoupt-type condition…