Related papers: On directional maximal operators associated with g…

We study the planar Nikod\'ym maximal operator $\mathcal{N}_{\Theta;\delta}$ associated to a direction set $\Theta \subset \mathbb{S}^{1}$. We show that the quasi-Assouad dimension $s := \dim_{\mathrm{qA}} \Theta$ characterises the…

Maximum Principles on unbounded domains play a crucial r\^ole in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators $\mathcal{L}$ in…

In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schr\"odinger maximal operator $e^{it\Delta}f(x)$ enjoys certain localization property in $t$ for frequency localized functions. In this…

The purpose of the present paper is to place a number of geometric (and hands-on) configurations relating to spectrum and geometry inside a general framework for the {\it Fuglede conjecture}. Note that in its general form, the Fuglede…

In this paper, we will give the weighted bounds for multilinear fractional maximal type operators $\mathcal{M}_{\Omega,\alpha}$ with rough homogeneous kernels. We obtain a mixed $A_{(\vec{P},q)}-A_\infty$ bound and a $A_{\vec{P}}$ type…

Let $\Omega \subset \mathbb{R}^N$ be an open set. In this work we consider solutions of the following gradient elliptic system \[ -\text{div}(A(x)\nabla u_{i,\beta}) = f_i(x,u_{i,\beta}) + a(x)\beta |u_{i, \beta}|^{\gamma -1}u_{i, \beta}…

In this paper we obtain a new boundedness criterion for the maximal operator $M$ on variable exponent spaces $L^{p(\cdot)}$. It is formulated in terms of the variable exponent analogue of the well known weighted $A_{\infty}$ condition.

In this article, we introduce the fractional medians, give an expression of the set of all fractional medians in terms of non-increasing rearrangements and then investigate mapping properties of the fractional maximal operators defined by…

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $\Omega$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -\Delta)^s u = f(u)…

We prove optimal bounds in L^2(R^2) for the maximal oper- ator obtained by taking a singular integral along N arbitrary directions in the plane. We also give a new proof for the optimal L^2 bound for the single scale Kakeya maximal…

We analyze spectral properties of the Hilbert $L$-matrix $$\left(\frac{1}{\max(m,n)+\nu}\right)_{m,n=0}^{\infty}$$ regarded as an operator $L_{\nu}$ acting on $\ell^{2}(\mathbb{N}_{0})$, for $\nu\in\mathbb{R}$, $\nu\neq0,-1,-2,\dots$. The…

Given a Lipschitz function $f:\{1,...,d\}^\mathbb{N} \to \mathbb{R}$, for each $\beta>0$ we denote by $\mu_\beta$ the equilibrium measure of $\beta f$ and by $h_\beta$ the main eigenfunction of the Ruelle Operator $L_{\beta f}$. Assuming…

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $\Omega$, where $\Omega$ is either in $\mathbb{R}^n$ or in a Riemannian manifold. For linear systems of equations arising from low-order…

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on uniformly Banach spaces with uniformly convex duals, or their domains have…

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel $|K_{(i\tau+\alpha)/2}(x)|^2, \alpha \ge 0,\ x…

Let $v,~\omega_1, ~\omega_2$ be weights and $1<p_1, ~p_2<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(\omega_1, \omega_2)\in RH(p_1, p_2).$ For the multisublinear maximal operator $\mathfrak{M}$ in martingale…

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[\Delta - \delta,\Delta + \delta]$ at asymptotically large $\Delta$ in 2d unitary modular invariant CFTs. These bounds…

We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced from those…

If a linear differential operator with rational function coefficients is reducible, its factors may have coefficients with numerators and denominatorsof very high degree. When the base field is $\mathbb C$, we give a completely explicit…

If mu is a smooth density on a hypersurface in R^d whose curvature never vanishes to infinite order, and A is a d-by-d matrix whose eigenvalues all have absolute value greater than 1, then the maximal function given by convolving f with…