We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group ${\mathbb{R}}_+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the ${\mathbb{R}}_+$-action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of non-trivial and locally trivial principal ${\mathbb{R}}_+$-bundle over the quotient space. Our bundle ${\mathbb{R}}_+ \to \mathcal{X}_* \to \mathcal{X}_*/{\mathbb{R}}_+$ is a curious example of a non-trivial principal fibre bundle with contractible fibre. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the ${\mathbb{R}}_+$-action on the compactification.

Inspired by a result in T. H. Colding. (16). Acta. Math. 209(2) (2012), 229-263 [16] of Colding, the present paper studies the Green function $G$ on a non-parabolic $\operatorname {RCD}(0,\,N)$ space $(X,\, \mathsf {d},\, \mathfrak {m})$ for some finite $N>2$. Defining $\mathsf {b}_x=G(x,\, \cdot )^{\frac {1}{2-N}}$ for a point $x \in X$, which plays a role of a smoothed distance function from $x$, we prove that the gradient $|\nabla \mathsf {b}_x|$ has the canonical pointwise representative with the sharp upper bound in terms of the $N$-volume density $\nu _x=\lim _{r\to 0^+}\frac {\mathfrak {m} (B_r(x))}{r^N}$ of $\mathfrak {m}$ at $x$;

\[ |\nabla \mathsf{b}_x|(y) \le \left(N(N-2)\nu_x\right)^{\frac{1}{N-2}}, \quad \text{for any }y \in X \setminus \{x\}. \]

$y \in X \setminus \{x\}$

$N$

$\operatorname {RCD}(N-2,\, N-1)$

$x$

$N$

$N$

$\mathbb {R}^N$

$\operatorname {RCD}(0,\,N)$

The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $\mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of $\mathbb {R}^n$—where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.

Let $\mathcal {M}$ be an Ahlfors $n$-regular Riemannian manifold such that either the Ricci curvature is non-negative or the Ricci curvature is bounded from below together with a bound on the gradient of the heat kernel. In the paper [IMRN, 2022, no. 2, 1245-1269] of Brazke–Schikorra–Sire, the authors characterised the BMO function $u : \mathcal {M} \to \mathbb {R}$ by a Carleson measure condition of its $\sigma$-harmonic extension $U:\mathcal {M}\times \mathbb {R}_+ \to \mathbb {R}$. This paper is concerned with the similar problem under a more general Dirichlet metric measure space setting, and the limiting behaviours of BMO & Carleson measure, where the heat kernel admits only the so-called diagonal upper estimate. More significantly, without the Ricci curvature condition, we relax the Ahlfors regularity to a doubling property, and remove the pointwise bound on the gradient of the heat kernel. Some similar results for the Lipschitz function are also given, and two open problems related to our main result are considered.

We give Trudinger-type inequalities for Riesz potentials of functions in Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our results are new even for the doubling metric measure setting. In particular, our results improve and extend the previous results in Morrey spaces of an integral form in the Euclidean case.

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

We investigate a dichotomy property for Hardy–Littlewood maximal operators, noncentred $M$ and centred $M^{c}$, that was noticed by Bennett et al. [‘Weak-$L^{\infty }$ and BMO’, Ann. of Math. (2) 113 (1981), 601–611]. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^{c}$ in the context of nondoubling metric measure spaces $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707})$. In addition, if $X=\mathbb{R}^{d}$, $d\geq 1$, and $\unicode[STIX]{x1D70C}$ is the metric induced by an arbitrary norm on $\mathbb{R}^{d}$, then we give the exact characterisation (in terms of $\unicode[STIX]{x1D707}$) of situations in which $M^{c}$ possesses the dichotomy property provided that $\unicode[STIX]{x1D707}$ satisfies some very mild assumptions.

Our aim in this paper is to deal with Sobolev inequalities for Riesz potentials of functions in Lebesgue spaces of variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.