Related papers: L^p Estimates for Maximal Averages Along One-varia…

A now classical result in the theory of variable Lebesgue spaces due to Lerner [A. K. Lerner, On modular inequalities in variable $L^p$ spaces, Archiv der Math. 85 (2005), no. 6, 538-543] is that a modular inequality for the…

We consider the $L^p$ mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal function, the circular maximal functions of Wolff and Bourgain, and…

We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the…

Let $P(D)$ be the Laplacian $\Delta,$ or the wave operator $\square$. The following type of Carleman estimate is known to be true on a certain range of $p,q$: \[ \|e^{v\cdot x}u\|_{L^q(\mathbb{R}^d)} \le C\|e^{v\cdot…

A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…

Let C be a projective smooth curve of genus g> 1. Let E be a vector bundle of rank r on C. For each integer r'<r, associate to E the invariant s_{r'}(E)=r'deg(E)-rdeg(E') where E'is a subbundle of E of rank r' and maximal degree. For every…

We establish the following universality property in high dimensions: Let $X$ be a random vector with density in $\mathbb{R}^n$. The density function can be arbitrary. We show that there exists a fixed unit vector $\theta \in \mathbb{R}^n$…

We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\|\frac{ T(fv)} {v}\Big\|_{L^{1,\infty}(uv)}\le c\, \|f\|_{L^1(uv)},$$…

The goal of this note is to establish non-tangential convergence results for Schr\"{o}dinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the…

In this paper, we investigate the geometric properties of the variable mixed Lebesgue-sequence space $\ell^{q(\cdot)} (L^{p(\cdot)})$ as a Banach space. We show that, if $ 1<q_-,p_-,q_+,p_+<\infty $, then $\ell^{q(\cdot)} (L^{p(\cdot)})$ is…

We study the $p$-independence of spectra of Laplace operators on graphs arising from regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth condition. Moreover, under a…

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 consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first-order, in the Lebesgue space L^p(R^d;R^m) with p in (1,\infty). Sufficient conditions to prove generation results of an analytic…

For $f\in {\frak S}({\Bbb R}^d)$, we consider the Bochner-Riesz operator ${\frak R}^{\delta}$ of index $\delta>0$ defined by $$\hat {{\frak R}^{\delta}f}(\xi)=(1-|\xi|^2)^{\delta}_+ \hat f (\xi).$$ Then we prove the Bochner-Riesz conjecture…

We prove Fatou type theorem on almost everywhere convergence of convolution integrals in spaces $L^p\,(1<p<\infty)$ for general kernels, forming an approximate identity. For a wide class of kernels we show that obtained convergence regions…

We calculate correlation functions for vertex operators with negative integer exponentials of a periodic Liouville field, and derive the general case by continuing them as distributions. The path-integral based conjectures of Dorn and Otto…

We consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of…

Sharp weighted estimates are obtained for vector-valued extensions of the Hardy-Littlewood maximal operator, Calder\'on-Zygmund operators and Coifman-Rochberg-Weiss commutator. Those estimates will rely upon suitable pointwise estimates in…

We prove $L^p\times L^q\rightarrow L^r$ bounds for certain lacunary bilinear maximal averaging operators with parameters satisfying the H\"older relation $1/p+1/q=1/r$. The boundedness region that we get contains at least the interior of…

We examine the averaging operator $T$ corresponding to the manifold in $\mathbb{R}^{2d}$ of pairs of points $(u,v)$ satisfying $|u| = |v| = |u - v| = 1$, so that $\{0,u,v\}$ is the set of vertices of an equilateral triangle. We establish…