Related papers: Carleson Measures and Logvinenko-Sereda sets on co…

In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $\varphi-$transform in the variable exponent settings,…

This is a survey on reverse Carleson measures for various Hilbert spaces of analytic functions. These spaces include Hardy, Bergman, certain harmonically weighted Dirichlet, Paley-Wiener, Fock, model, and de Branges-Rovnyak spaces.

This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4:…

A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes in flat domains while preserving…

In this work we introduce a new method for manufacturing minimal submanifolds in Riemannian geometry. For this we employ the so called complex-valued eigenfunctions. This is particularly interesting in the cases when the Riemannian ambient…

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a nonnegative self-adjoint operator in $L^2(\mathcal{X})$ satisfying the Davies-Gaffney estimates. Let $\varphi:\,\mathcal{X}\times[0,\infty)\to[0,\infty)$ be a function such…

This article deals with maximal operators on ${\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional H\"ormander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1…

In this paper we analyze the eigenvalues and eigenfunctions of the Hodge Laplacian for generic metrics on a closed 3-manifold $M$. In particular, we show that the nonzero eigenvalues are simple and the zero set of the eigenforms of degree 1…

Let \phi\ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {\phi=0}.

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…

Radial eigenfunctions of the Laplace-Beltrami operator on compact rank-one symmetric spaces may be expressed in terms of Jacobi polynomials. We use this fact to prove an identity for Jacobi polynomials which is inspired by results of…

In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…

In the present paper, we study the boundedness and compactness of Toeplitz operators and Berezin-type operators between different weighted Bergman spaces over tubular domains in $\mathbb{C}^n$. We establish their connection with Carleson…

The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute…

This paper establishes Carleson embeddings of M{\"u}ntz spaces $M^q_{\Lambda}$ into weighted Lebesgue spaces $L^p(\mathrm{d}\mu)$, where $\mu$ is a Borel regular measure on $[0,1]$ satisfying $\mu([1-\varepsilon])\lesssim…

We use spectral embeddings to give upper bounds on the spectral function of the Laplace--Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower…

We discuss compactness of the d-bar-Neumann operator in the setting of weighted L^2-spaces on C^n.$ For this purpose we use a description of relatively compact subsets of L^2- spaces. We also point out how to use this method to show that…

A new approach for constructing minimal submanifolds of codimension 1 in the round spheres is proposed. In the case of $\mathbb{S}^3$ two immersions of the Clifford torus and all Lawson $\tau_{n, m}$ surfaces are described in terms of…

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2…