Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ ( Y , d ) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$ μ on $$\mathrm{Y}$$ Y giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$ ( Y , d , μ ) is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$ W 1 , 2 ( Y , d , μ ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at $$x\in \mathrm{Y}$$ x ∈ Y is the tangent cone at x of $$\mathrm{Y}$$ Y . The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$ x ∈ Y such a cone is a $$\mathrm{CAT}(0)$$ CAT ( 0 ) space and, as such, has a Hilbert-like structure.

Energy of wave and solutions of C-Holm equation under Lagrangian point of view

We deal with the Camassa-Holm equation possesses a global continuous semigroup of weak conservative solutions for initial data. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure µ with . The total energy is preserved by the solution.

Lipschitz metric for the modified two-component Camassa–Holm system

This paper is devoted to the existence and Lipschitz continuity of global conservative weak solutions in time for the modified two-component Camassa–Holm system on the real line. We obtain the global weak solutions via a coordinate transformation into the Lagrangian coordinates. The key ingredients in our analysis are the energy density given by the positive Radon measure and the proposed new distance functions as well.

Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term

In the present paper we study the Dirichlet problem for an equation involving the 1-Laplacian and a total variation term as reaction. We prove a strong multiplicity result. Namely, we show that for any positive Radon measure concentrated in a set away from the boundary and singular with respect to a certain capacity, there exists an unbounded solution, and measures supported on disjoint sets generate different solutions. These results can be viewed as the analogue for the 1-Laplacian operator of some known multiplicity results which were first obtained by Ireneo Peral, to whom this article is dedicated, and his collaborators.

Local Integral Estimates for Quasilinear Equations with Measure Data

Local integral estimates as well as local nonexistence results for a class of quasilinear equations-Δpu=σP(u)+ωforp>1and Hessian equationsFk-u=σP(u)+ωwere established, whereσis a nonnegative locally integrable function or, more generally, a locally finite measure,ωis a positive Radon measure, andP(u)~exp⁡αuβwithα>0andβ≥1orP(u)=up-1.

Bochner algebras and their compact multipliers

Let $\mathfrak{A}$ be a normed algebra with identity, Ω be a locally compact Hausdorf space and λ be a positive Radon measure on Ω with supp(λ) = Ω. In this paper, we establish a necessary and sufficient condition for L 1(Ω, $\mathfrak{A}$) to be an algebra with pointwise multiplication. Under this condition, we then characterize compact and weakly compact left multipliers on L 1(Ω, $\mathfrak{A}$).

Symmetric Hunt Processes and Regular Dirichlet Forms

This chapter studies a symmetric Hunt process associated with a regular Dirichlet form. Without loss of generality, the majority of the chapter assumes that E is a locally compact separable metric space, m is a positive Radon measure on E with supp[m] = E, and X = (Xₜ, Pₓ) is an m-symmetric Hunt process on (E,B(E)) whose Dirichlet form (E,F) is regular on L²(E; m). It adopts without any specific notices those potential theoretic terminologies and notations that are formulated in the previous chapter for the regular Dirichlet form (E,F). Furthermore, throughout this chapter, the convention that any numerical function on E is extended to the one-point compactification E ∂ = E ∪ {∂} by setting its value at δ‎ to be zero is adopted.

A NEW CHARACTERIZATION FOR REGULAR BMO WITH NON-DOUBLING MEASURES

Let $\mu$ be a positive Radon measure on $\mathbb{R}^d$ which satisfies $\mu(B(x,r))\le Cr^{n}$ for any $x\in\mathbb{R}^d$ and $r>0$ and some fixed constants $C>0$ and $n\in(0,d]$. In this paper, a new characterization of the space $\rbmo(\mu)$, which was introduced by Tolsa, is given. As an application, it is proved that the $L^p(\mu)$-boundedness with $p\in(1,\infty)$ of Calderón–Zygmund operators is equivalent to various endpoint estimates.

A new proof of Donoghue's interpolation theorem

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

Approximation of relaxed Dirichlet problems by boundary value problems in perforated domains

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.