Related papers: Almost-orthogonality principles for certain direct…

The Menchov-Rademacher inequality is an inequality in harmonic analysis that bounds the $L_2$ norm of a certain maximal operator. It was first established in order to prove almost everywhere convergence of a one-parameter series of…

This paper derives some discrete maximum principles for $P1$-conforming finite element approximations for quasi-linear second order elliptic equations. The results are extensions of the classical maximum principles in the theory of partial…

In this paper, we study the $L^p(\mathbb{R}^2)$-improving bounds, i.e., $L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ estimates, of the maximal function $M_{\gamma}$ along a plane curve $(t,\gamma(t))$, where…

Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to…

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces…

In this note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed.…

We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First, we obtain an $H^1$ to $L^{1,\infty}$ bound for lacunary maximal operators under a dimensional assumption on the…

Let $\Omega $ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_\Omega f(x)=\sup_{\delta >0, \omega \in \Omega} \frac{1}{2\delta}\int_{-\delta}^\delta |f(x+t\omega)|dt. \emd for the…

For 24 years, it has been an open problem to obtain improved bounds, for the maximal function over a sparse sequence of discrete spherical averages, going beyond the range for the full discrete spherical maximal function. I formulate a…

We consider the maximal function of oscillatory integrals and prove a global estimate for radial test functions which is almost sharp with respect to the Sobolev regularity.

We study vector functions of ${\mathbb R}^n$ into itself, which are of the form $x \mapsto g(|x|)x\,,$ where $g : (0,\infty) \to (0,\infty) $ is a continuous function and call these radial functions. In the case when $g(t) = t^c$ for some…

We exhibit a class of "relatively curved" $\vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of H\"{o}lder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r}…

In this paper we study the regularity properties of fractional maximal operators acting on $BV$-functions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.

For weighted $L^1$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal…

We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded…

Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data, the initial conditions and the operator coefficients.…

We consider the problem of finding an optimal piecewise linear path (polygonal line) connecting two given points with the possibility of making n turns at some points (the absolute value of each turn angle does not exceed a prescribed…

We introduce functions for relative maximization in a general context: the beta and alpha applications. After a systematic study concerning regularities, we investigate how to approximate certain values of these functions using periodic…

We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines…

In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory in optimization. Then we…