Related papers: $L^2$-Maximal functions on graded Lie groups

In this paper, we verify the $L^2$-boundedness for the jump functions and variations of Calder\'on-Zygmund singular integral operators with the underlying kernels satisfying \begin{align*}\int_{\varepsilon\leq |x-y|\leq N}…

Sharp $L^p$--$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate $L^p$--$L^q$ estimates for…

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\,…

In this paper, we prove \( L^p \) boundedness results for lacunary elliptic maximal operators on the Heisenberg group. Furthermore, we extend these \( L^p \) estimates from skew-symmetric matrices, which naturally arise in Heisenberg group…

Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\,\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that…

For any nonempty set $U\subset\R^+$, we consider the maximal operator $\h^U$ defined as $\h^Uf=\sup_{u\in U}|H^{(u)} f|$, where $H^{(u)}$ represents the Hilbert transform along the monomial curve $u\gamma(s)$. We focus on the…

We establish the $L^p$ boundedness of Hilbert transforms and maximal functions along flat curves in the Heisenberg group. This generalizes the $\mathbb{R}^n$ result by Carbery, Christ, Vance, Wainger, and Watson. What is new about our…

This work studies the Fourier transform of the characteristic function of planar convex bodies averaged over affine transformations. We establish lower and upper bounds on the latter quantities in terms of the geometric properties of the…

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by…

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$. In \cite{biswas6}, a Fourier transform was defined for functions on $X$, and a Fourier inversion formula and Plancherel theorem…

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on $L^p(\mathbb{G})$ $(1<p<+\infty$, $\mathbb{G}$ is a graded Lie group). We show the explicit solution of the…

This contribution presents substantial computational advancements to compare measures even with varying masses. Specifically, we utilize the nonequispaced fast Fourier transform to accelerate the radial kernel convolution in unbalanced…

We establish, for the first time, a Bochner-type integral representation for the logarithmic Laplacian on weighted graphs. Assuming stochastic completeness of the underlying graph, we further derive an explicit pointwise formula for this…

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on…

In this paper, we investigate the behavior of the Fourier transform on finite dimensional 2-step Lie groups and develop a general theory akin to that of the whole space or the torus. We provide a familiar framework in which computations are…

We consider the averages of a function $ f$ on $ \mathbb R ^{n}$ over spheres of radius $ 0< r< \infty $ given by $ A_{r} f (x) = \int_{\mathbb S ^{n-1}} f (x-r y) \; d \sigma (y)$, where $ \sigma $ is the normalized rotation invariant…

In this paper, we study the Lp-bondedness of the spherical maximal function associated to the Dunkl operators.

For $G=\mathrm{SL}_2$ or $\mathrm{GL}_2$, we present explicit formulas for the nonabelian Fourier kernels on $G$, as conjectured by A. Braverman and D. Kazhdan. Additionally, we furnish explicit formulas for the orbital Hankel transform on…

Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions…

We find a sharp combinatorial bound for the metric entropy of sets in R^n and general classes of functions. This solves two basic combinatorial conjectures on the empirical processes. 1. A class of functions satisfies the uniform Central…