Related papers: Sharp Fourier extension on fractional surfaces

We establish some weighted $L^2$ estimates for the Fourier extension operator in $\mathbb{R}^2$ and discuss several applications to $L^p$ problems. These include estimates for the maximal Schr\"odinger operator and the maximal extension…

In \cite{[NE]} we introduce $\alpha$-expansions a real numbers in $(0,1]$, given by \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $\alpha>1$ and $d_{i}\in\mathbb{N}$ and discuss ergodic theoretical and dimension…

We prove the unique continuation property for the differential inequality $|(-\Delta)^{\alpha/2}u|\leq|V(x)u|$, where $0<\alpha<n$ and $V\in L_{\textrm{loc}}^{n/\alpha,\infty}(\mathbb{R}^n)$, $n\geq3$.

Let $Q$ be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension $Q_{min}$ (the closure of the graph) and the…

We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…

Polyharmonic functions f of infinite order and type {\tau} on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of infinite order and type…

We consider minimal non-negative Jacobi operator with $p\times p-$matrix entries. Using the technique of boundary triplets and the corresponding Weyl functions, we describe the Friedrichs and Krein extensions of the minimal Jacobi operator.…

We relate non integer powers ${\mathcal L}^{s}$, $s>0$ of a given (unbounded) positive self-adjoint operator $\mathcal L$ in a real separable Hilbert space $\mathcal H$ with a certain differential operator of order $2\lceil{s}\rceil$,…

Sharp Moser-Trudinger type inequalities and their extremal functions play an important role in studying nonlinear PDEs and geometry. We establish a new sharp Moser-Trudinger type inequality in the upper half space in two dimensions and…

In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract…

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane…

In this paper we consider the minimization of a novel class of fractional linear growth functionals involving the Riesz fractional gradient. These functionals lack the coercivity properties in the fractional Sobolev spaces needed to apply…

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove…

In this paper we discuss some recent results about extremal contractions of complex algebraic varieties. These are proper surjective maps, $\phi: X\longrightarrow Z$, of normal varieties with connected fibers such that $X$ has mild…

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,\phi(T))$ where $\phi(T)$ is a polynomial of…

We show how a rescaling of fractional operators with bounded kernels may help circumvent their documented deficiencies, for example, the inconsistency at zero or the lack of inverse integral operator. On the other hand, we build a novel…

We prove the range of exponents in the general $L^2$ Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak and Seeger is sharp for a large class of measures on $\mathbb{R}^d$. This extends to higher dimensions the sharpness result of…

In this paper, we consider a class of the Caputo fractional stochastic differential equations of fractional order $\alpha \in (\frac{1}{2},1]$. Our aim is to analyze of the continuous dependence of solutions on the fractional order…

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each…

We consider the variational formulation of the Griffith fracture model in two spatial dimensions and prove existence of strong minimizers, that is deformation fields which are continuously differentiable outside a closed jump set and which…