Related papers: An Overview on Laakso Spaces

In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We…

We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the same…

Boundedness of an abstract formulation of Hardy operators between Lebesgue spaces over general measure spaces is studied and, when the domain is L^1, shown to be equivalent to the existence of a Hardy inequality on the half line with…

Let $X$ be a metric measure space with an $s$-regular measure $\mu$. We prove that if $A\subset X$ is $\varrho$-porous, then $\dim_{\mathrm{p}}(A)\le s-c\varrho^s$ where $\dim_{\mathrm{p}}$ is the packing dimension and $c$ is a positive…

A famous theorem of Weyl states that if $M$ is a compact submanifold of euclidean space, then the volumes of small tubes about $M$ are given by a polynomial in the radius $r$, with coefficients that are expressible as integrals of certain…

We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space $X$ supports a $p$-Poincar\'e inequality, then the $N^{1,p}(X)$ Sobolev space is reflexive and separable whenever $p\in…

Using elementary arguments based on the Fourier transform we prove that for $1 \leq q < p < \infty$ and $s \geq 0$ with $s > n(1/2-1/p)$, if $f \in L^{q,\infty}(\R^n) \cap \dot{H}^s(\R^n)$ then $f \in L^p(\R^n)$ and there exists a constant…

We prove an existence and uniqueness theorem for fixed points of contraction maps in the framework of quantum metric spaces, where distinguishability is defined by the $L^2$ norm: $d_Q(\psi_1,\psi_2) = \|\psi_1 - \psi_2\|$. The result…

The Banach-Saks property is an important tool in analysis with applications ranging from partial differential equations (PDEs) to calculus of variations and probability theory. We survey the Banach-Saks property for $L^p$-spaces, with a…

In this article we extend the notion of orthogonal metric space to weak orthogonal metric space. Then we establish fixed point results for a mapping satisfying a more general contraction condition. Several nontrivial examples are given in…

Suppose that $f$ belongs to a suitably defined complete metric space $ {{\cal C}}^{{\alpha}}$ of H\"older $ {\alpha}$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper…

We characterize the real interpolation space between weighted $L^1$ and $W^{1,1}$ spaces on arbitrary domains different from $\mathbb{R}^n$, when the weights are positive powers of the distance to the boundary multiplied by an $A_1$ weight.…

We prove a fractional version of Poincar\'e inequalities in the context of $\R^n$ endowed with a fairly general measure. Namely we prove a control of an $L^2$ norm by a non local quantity, which plays the role of the gradient in the…

This is the second paper in a series on intrinsic Donaldson-Thomas theory, a framework for studying the enumerative geometry of general algebraic stacks. In this paper, we present the construction of Donaldson-Thomas invariants for general…

The existence of a counterexample to the infinite-dimensional Carleson embedding theorem has been established by Nazarov, Pisier, Treil, and Volberg. We provide an explicit construction of such an example. We also obtain a non-constructive…

Let $\cal M$ be a semi-finite von Neumann algebra equipped with a distinguished faithful, normal, semi-finite trace $\tau$. We introduce the notion of equi-integrability in non-commutative spaces and show that if a rearrangement invariant…

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$.…

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact…

This paper is concerned with certain generalizations of meagreness and their combinatorial equivalents. The simplest example, and the one which motivated further study in this area, comes about by considering the following definition: a set…

We introduce a quantitative version of Property A in order to estimate the L^p-compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to…