Related papers: On a Conjecture of EM Stein on the Hilbert Transfo…

Let $\psi:\mathbb{N}\rightarrow\mathbb{R}_+$ be a monotonically non-increasing function, and let $\psi_v:\mathbb{N}\rightarrow\mathbb{R}_+$ be defined by $\psi_v(q)=1/q^v$. In this article, we consider self-similar sets whose iterated…

Vector calculus in three dimensions with a Euclidian metric is the lingua franca of classical physics, including classical electrodynamics. This article corrects some long-standing imprecision in a fundamental result. Some textbooks assert…

We obtain a necessary and sufficient condition on a polynomial $P(t_1,t_2)$ for the $\ell^{p}$ boundedness of the discrete double Hilbert transforms associated with $P(t)$ for $1 < p < \infty$. The proof is based on the multi-parameter…

It is shown here that if $(Y,\|\cdot\|_Y)$ is a Banach space in which martingale differences are unconditional (a UMD Banach space) then there exists $c=c(Y)\in (0,\infty)$ with the following property. For every $n\in \mathbb{N}$ and…

We study here limit spaces $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_\alpha$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known…

In this paper, the main aim is to consider the mapping properties of the maximal or nonlinear commutator for the fractional maximal operator with the symbols belong to the Lipschitz spaces on variable Lebesgue spaces in the context of…

In this paper, the $L^p_v(\R)$-boundedness of the Dunkl-Hausdorff operator $\displaystyle H_{\al,\phi} f(x)=\ent\frac{ |\phi(t)|}{|t|^{2\al+2}}f\lf(\frac{x}{t}\rh) dt $ has been characterized and for a certain type of weight $v$, the…

In Learning Theory, the smoothness assumption on the target function (known as source condition) is a key factor in establishing theoretical convergence rates for an estimator. The existing general form of the source condition, as discussed…

Here we obtain bounds on the spectrum of that operator whose inverse, when it exists, gives the Green's function. We consider the wide of physical problems that can be cast in a form where a constitutive equation ${\bf J}({\bf x})={\bf…

This paper is concerned with the question of reconstructing a vector in a finite-dimensional real Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We analyze various Lipschitz…

Let $M$ be a closed smooth connected spin manifold of even dimension $n$, let $g$ be a Riemannian metric of regularity $W^{1,p}$, $p > n$, on $M$ whose distributional scalar curvature in the sense of Lee-LeFloch is bounded below by…

We consider a Higgs bundle over a compact K\"ahler manifold with a smooth, non-holomorphic Higgs field. We assume that the holomorphic vector bundle decomposes into a direct sum of holomorphic line bundles. Under an assumption on the zero…

We prove $L^p$ boundedness results, $p > 2$, for local maximal averaging operators over a smooth 2D hypersurface $S$ with either a $C^1$ density function or a density function with a singularity that grows as $|(x,y)|^{-\beta}$ for $\beta <…

We prove that the existence of a Mihlin-H\"ormander functional calculus for an operator $L$ implies the boundedness on $L^p$ of both the maximal operators and the continuous square functions build on spectral multipliers of $L.$ The…

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $\mathfrak{X}$ we let $\Omega^p_{\rm HL}(\mathfrak{X}) \subset…

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both…

By explicitly constructing the Hilbert space, Higuchi showed that there is a lower bound on the mass of a minimally-coupled free spin-2 field on a curved background \cite{HiguchiBound}. Using the vacuum persistence amplitude, we show that…

Let $H^n\cong \Bbb R^{2n}\ltimes \Bbb R$ be the Heisenberg group and let $\mu_t$ be the normalized surface measure for the sphere of radius $t$ in $\Bbb R^{2n}$. Consider the maximal function defined by $Mf=\sup_{t>0} |f*\mu_t|$. We prove…

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover,…

We study the $L^p$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on $L^p$ for $p > 2$ in all dimensions $n…