arXiv subjects

Nageswari Shanmugalingam

Publications and source records attributed to Nageswari Shanmugalingam.

At least 19 recordsLinked to original sources

arXiv2026

Warped product spaces: Gromov hyperbolicity and identification of the visual boundary

In this paper, we consider warped product spaces $X\times_φY$, where $X$ is a complete geodesic Gromov hyperbolic space, $Y$ is a compact geodesic metric space, and the warping function $φ$ satisfies suitable exponential growth conditions. We prove that the warped product is Gromov hyperbolic and derive an explicit estimate for its hyperbolicity constant. We further establish a homeomorphism between its Gromov boundary and $\partial_GX\times Y$, with an explicit comparison formula for the visual metric on the Gromov boundary of $X\times_φY$ in terms of the visual metric $d_{\varepsilon, X}$ on $\partial_GX$ and $d_Y$ for suitable $\varepsilon>0$.

math.MG↗

arXiv2026

On the equivalence of BV notions in metric measure spaces

The aim of the paper is to compare in detail several notions of $BV$ space of functions of bounded variation in metric measure spaces $({\rm X},\mathsf{d},\mathfrak{m})$. Informally, they can be grouped in two classes, either by a relaxation procedure starting from a class of nicer functions (and with different notions of pseudo-gradient in the relaxation procedure) or by requiring good behaviour along a rich class of absolutely continuous curves. In the second approach, richness can be understood according to the notion of approximation modulus of [O. Martio, Adv. Calc. Var., 9 (2016)] or according to the notion of test plan introduced in [L. Ambrosio, N. Gigli, and G. Savaré, Invent. Math., 195 (2014)]. Extending [L. Ambrosio and S. Di Marino, J. Funct. Anal., 266 (2014)], we prove that all these approaches are isometrically equivalent in any locally complete metric measure space.

math.FA↗

arXiv2026

Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs

Given a complete doubling metric measure space $X$ that supports a $2$-Poincaré inequality, we approximate harmonic functions on a bounded domain $Ω$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the approximation of the underlying space $X$ by a family of graphs. This approximated harmonic function is realized as the weak limit of a sequence of functions obtained from the graph minimizers. We prove that such a function is a minimizer with respect to a nonlinear energy form on $N^{1,2}_0(Ω)$, which is in turn, majorized by the upper gradient energy on $N^{1,2}(X)$. This energy form on $N^{1,2}_0(Ω)$ is obtained as a $Γ$-limit of a sequence of induced energy forms projected from the discrete energy form on the approximating graphs.

math.AP↗

arXiv2026

Fractional $p$-Laplacians via Neumann problems in unbounded metric measure spaces

We prove well-posedness, Harnack inequality and sharp regularity of solutions to a fractional $p$-Laplace non-homogeneous equation $(-Δ_p)^su =f$, with $0<s<1$, $1<p<\infty$, for data $f$ satisfying a weighted $L^{p'}$ condition in a doubling metric measure space $(Z,d_Z,ν)$ that is possibly unbounded. Our approach is inspired by the work of Caffarelli and Silvestre \cite{CS} (see also Mol{č}anov and Ostrovski{ĭ} \cite{MO}), and extends the techniques developed in \cite{CKKSS}, where the bounded case is studied. Unlike in \cite{EbGKSS}, we do not assume that $Z$ supports a Poincaré inequality. The proof is based on the well-posedness of the Neumann problem on a Gromov hyperbolic space $(X,d_X, μ)$ that arises as an hyperbolic filling of $Z$.

math.AP↗

arXiv2026

Traces of Newton-Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$-Poincaré inequality

We consider the question of whether a domain with uniformly thick boundary at all locations and at all scales has a large portion of its boundary visible from the interior; here, "visibility" indicates the existence of John curves connecting the interior point to the points on the "visible boundary". In this paper, we provide an affirmative answer in the setting of a doubling metric measure space supporting a $p$-Poincaré inequality for $1<p<\infty$, thus extending the results of [20,2,9] to non-Ahlfors regular spaces. We show that $t$-codimensional thickness of the boundary for $0<t<p$ implies $p$-codimensional thickness of the visible boundary. For such domains we prove that traces of Sobolev functions on the domain belong to the Besov class of the visible boundary.

math.MG↗

arXiv2026

Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques

Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincaré inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct methods of calculus of variation and the use of a Maz'ya type inequality, which is a consequence of the Poincaré inequality. However, when the domain and its boundary are unbounded, such a method is unavailable. In this paper, using the technique of sphericalization developed in the prior paper~[32], we establish the existence of solutions to the Dirichlet boundary value problem for $p$-harmonic functions in unbounded uniform domains with unbounded boundary when $1<p<\infty$. We also explore the issue of whether such solutions are unique by considering $p$-parabolicity and $p$-hyperbolicity properties of the domain.

math.AP↗

arXiv2025

Warped products, solid hyperbolic fillings, and the identity $D^{1,p} = N^{1,p} + \mathbb{R}$

We construct a large class of metric measure spaces $Z$ which satisfy the identity $D^{1,p}(Z) = N^{1,p}(Z) + \mathbb{R}$, i.e.\ any measurable function $u \colon Z \to \mathbb{R}$ with an $L^p$-integrable upper gradient is a constant term away from being $L^p$-integrable. To do so, we construct a family of hyperbolic fillings $\mathbb{H}_{α, β}(Y)$, $α, β\in (0, \infty)$, of a metric measure space $Y$, via a warped product of $Y$ with an exponentially weighted positive real line. We then show that for certain classes of $Y$, the above identity is satisfied for $Z=\mathbb{H}_{α,β}(Y)$ when $1\le p\le β/α$. We also show that under mild assumptions on $Y$, the warped product $\mathbb{H}_{α, β}(Y)$ is Gromov hyperbolic as a metric space and the Gromov boundary of $\mathbb{H}_{α, β}(Y)$ is quasisymmetric to $Y$.

math.MG↗

arXiv2025

Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces

In the context of a metric measure space $(X,d,μ)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^θ_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $θ$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents $θ_p(X)$ and $θ_p^{\ast}(X)$ for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces $X$ formed by glueing copies of $n$-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet.

math.FA↗

arXiv2025

Sharp Hölder regularity of weak solutions of the Neumann problem and applications to nonlocal PDE in metric measure spaces

We prove global Hölder regularity result for weak solutions $u\in N^{1,p}(Ω, μ)$ to a PDE of $p$-Laplacian type with a measure as non-homogeneous term: \[ -\text{div}\!\left( |\nabla u|^{p-2}\nabla u \right)=\overlineν, \] where $1<p<\infty$ and $\overlineν\in (N^{1,p}(Ω,μ))^*$ is a signed Radon measure supported in $\overline Ω$. Here, $Ω$ is a John domain in a metric measure space satisfying a doubling condition and a $p$-Poincaré inequality, and $\nabla u$ is the Cheeger gradient. The regularity results obtained in this paper improve on earlier estimates proved by the authors in \cite{CGKS} for the study of the Neumann problem, and have applications to the regularity of solutions of nonlocal PDE in doubling metric spaces. Moreover, the obtained Hölder exponent matches with the known sharp result in the Euclidean case \cite{CSt,BLS,BT}.

math.AP↗

arXiv2025

On homogeneous Newton-Sobolev spaces of functions in metric measure spaces of uniformly locally controlled geometry

We study the large-scale behavior of Newton-Sobolev functions on complete, connected, proper, separable metric measure spaces equipped with a Borel measure $μ$ with $μ(X) = \infty$ and $0 < μ(B(x, r)) < \infty$ for all $x \in X$ and $r \in (0, \infty)$ Our objective is to understand the relationship between the Dirichlet space $D^{1,p}(X)$, defined using upper gradients, and the Newton-Sobolev space $N^{1,p}(X)+\mathbb{R}$, for $1\le p<\infty$. We show that when $X$ is of uniformly locally $p$-controlled geometry, these two spaces do not coincide under a wide variety of geometric and potential theoretic conditions. We also show that when the metric measure space is the standard hyperbolic space $\mathbb{H}^n$ with $n\ge 2$, these two spaces coincide precisely when $1\le p\le n-1$. We also provide additional characterizations of when a function in $D^{1,p}(X)$ is in $N^{1,p}(X)+\mathbb{R}$ in the case that the two spaces do not coincide.

math.MG↗

arXiv2025

Well-posedness of Dirichlet boundary value problems for reflected fractional $p$-Laplace-type inhomogeneous equations in compact doubling metric measure spaces

In this paper we consider the setting of a locally compact, non-complete metric measure space $(Z,d,ν)$ equipped with a doubling measure $ν$, under the condition that the boundary $\partial Z:=\overline{Z}\setminus Z$ (obtained by considering the completion of $Z$) supports a Radon measure $π$ which is in a $σ$-codimensional relationship to $ν$ for some $σ>0$. We explore existence, uniqueness, comparison property, and stability properties of solutions to inhomogeneous Dirichlet problems associated with certain nonlinear nonlocal operators on $Z$. We also establish interior regularity of solutions when the inhomogeneity data is in an $L^q$-class for sufficiently large $q>1$, and verify a Kellogg-type property when the inhomogeneity data vanishes and the Dirichlet data is continuous.

math.AP↗

arXiv2025

Sharp conditions for preserving uniformity, doubling measure and Poincaré inequality under sphericalization

We study sphericalization, which is a mapping that conformally deforms the metric and the measure of an unbounded metric measure space so that the deformed space is bounded. The goal of this paper is to study sharp conditions on the deforming density function under which the sphericalization preserves uniformity of the space, the doubling property of the measure and the support of a Poincaré inequality. We also provide examples that demonstrate the sharpness of our conditions.

math.MG↗

arXiv2024

On Carrasco Piaggio's theorem connecting combinatorial modulus and Ahlfors regular conformal dimension

The goal of this paper is to provide an expository description of a result of Carrasco Piaggio connecting the Ahlfors regular conformal dimension of a compact uniformly perfect doubling metric space with the combinatorial $p$-moduli of the metric space. We give detailed construction of a metric associated with the $p$-modulus of the space when the $p$-modulus is zero, so that the constructed metric is in the Ahlfors regular conformal gauge of the metric space. To do so, we utilize the tools of hyperbolic filling, developed first by Gromov and by Bourdon and Pajot.

math.MG↗

arXiv2024

On the Dirichlet-to-Neumann Map for the $p$-Laplacian on a Metric Measure Space

In this note, we construct a Dirichlet-to-Neumann map, from a Besov space of functions, to the dual of this class. The Besov spaces are of functions on the boundary of a bounded, locally compact uniform domain equipped with a doubling measure supporting a $p$-Poincaré inequality so that this boundary is also equipped with a Radon measure that has a codimensional relationship with the measure on the domain. We construct this map via the following recipe. We show first that solutions to Dirichlet problem for the $p$-Laplacian on the domain with prescribed boundary data in the Besov space induce an operator that lives in the dual of the Besov space. Conversely, we show that there is a solution, in the homogeneous Newton-Sobolev space, to the Neumann problem for the $p$-Laplacian with the Neumann boundary data given by a continuous linear functional belonging to the dual of the Besov space. We also obtain bounds on its operator norm in terms of the norms of trace and extension operators that relate Newton-Sobolev functions on the domain to Besov functions on the boundary.

math.AP↗

arXiv2023

Homogeneous Newton-Sobolev spaces in metric measure spaces and their Banach space properties

In this note we prove the Banach space properties of the homogeneous Newton-Sobolev spaces $HN^{1,p}(X)$ of functions on an unbounded metric measure space $X$ equipped with a doubling measure supporting a $p$-Poincaré inequality, and show that when $1<p<\infty$, even with the lack of global $L^p$-integrability of functions in $HN^{1,p}(X)$, we have that every bounded sequence in $HN^{1,p}(X)$ has a strongly convergent convex-combination subsequence. The analogous properties for the inhomogeneous Newton-Sobolev classes $N^{1,p}(X)$ are proven elsewhere in existing literature

math.FA↗

arXiv2023

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

We give a sharp Hausdorff content estimate for the size of the accessible boundary of any domain in a metric measure space of controlled geometry, i.e., a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality for a fixed $1\le p<\infty$. This answers a question posed by Jonas Azzam. In the process, we extend the result to every doubling gauge in metric measure spaces which satisfies a codimension one bound.

math.MG↗

arXiv2023

Construction of a Dirichlet form on metric measure spaces of controlled geometry

Given a compact doubling metric measure space $X$ that supports a $2$-Poincaré inequality, we construct a Dirichlet form on $N^{1,2}(X)$ that is comparable to the upper gradient energy form on $N^{1,2}(X)$. Our approach is based on the approximation of $X$ by a family of graphs that is doubling and supports a $2$-Poincaré inequality. We construct a bilinear form on $N^{1,2}(X)$ using the Dirichlet form on the graph. We show that the $Γ$-limit $\mathcal{E}$ of this family of bilinear forms (by taking a subsequence) exists and that $\mathcal{E}$ is a Dirichlet form on $X$. Properties of $\mathcal{E}$ are established. Moreover, we prove that $\mathcal{E}$ has the property of matching boundary values on a domain $Ω\subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form $\mathcal{E}$) on a domain in $X$ with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.

math.MG↗

arXiv2023

Fine properties of metric space-valued mappings of bounded variation in metric measure spaces

Here we consider two notions of mappings of bounded variation (BV) from the metric measure space into the metric space; one based on relaxations of Newton-Sobolev functions, and the other based on a notion of AM-upper gradients. We show that when the target metric space is a Banach space, these two notions coincide with comparable energies, but for more general target metric spaces, the two notions can give different function-classes. We then consider the fine properties of BV mappings (based on the AM-upper gradient property), and show that when the target space is a proper metric space, then for a BV mapping into the target space, co-dimension $1$-almost every point in the jump set of a BV mapping into the proper space has at least two, and at most $k_0$, number of jump values associated with it, and that the preimage of balls around these jump values have lower density at least $γ$ at that point. Here $k_0$ and $γ$ depend solely on the structural constants associated with the metric measure space, and jump points are points at which the map is not approximately continuous.

math.MG↗