Papers

In several papers, Stochel and Szafraniec studied moment problems on algebraic sets from an operator-theoretic perspective, investigating when positive definite sequences satisfying polynomial relations admit representing measures. Within this framework, Stochel introduced type A sets, and Bisgaard classified the plane curves defined by relations between two monomials that have this property. Curto and Fialkow introduced a stronger, truncated version of the type A property, requiring that the existence of a positive semidefinite extension of prescribed degree guarantees the existence of a representing measure. Motivated by Bisgaard's classification, we determine which plane curves defined by relations between two monomials satisfy this extension property. In the affirmative cases, we obtain explicit bounds on the required extension degree. In the negative cases, we construct truncated sequences that admit positive semidefinite extensions of arbitrarily high order but have no representing measure supported on the curve. These constructions yield explicit polynomials that are nonnegative on the corresponding curves but are not sums of squares in their coordinate rings. In the affirmative cases, we also derive explicit degree bounds for sums-of-squares certificates of strictly positive polynomials.

We study the problem of stable spectral differentiation of functions in Sobolev spaces from noisy data. We introduce a class of admissible Fourier multipliers under simple and directly verifiable conditions and show that the corresponding regularized differentiation operators achieve minimax optimal stability rates. The results extend the previous $L^2$ based results to Sobolev spaces $H^{s,p}(\mathbb{R}^n)$, $1<p<\infty$. The analysis relies on multiplier estimates and applies to a wide class of multipliers, including Gaussian, spectral cutoff, and Tikhonov-type regularizations. Numerical examples demonstrate the behavior of several admissible spectral multipliers.

Inverse inequalities are an important tool in the stability and convergence analysis of kernel approximation methods. In a multiscale setting, however, the trial space changes with the kernel scale $\delta$, and inverse estimates for a fixed kernel are not sufficient. The constants must remain controlled as $\delta\to0$. In this paper, we study scale-uniform inverse inequalities for spaces generated by scaled positive definite kernels whose native spaces are Sobolev spaces. We first establish inverse estimates in scale-dependent Sobolev spaces. On bounded domains, this leads to a scale-uniform inverse inequality in standard Sobolev norms with $L^2$ as the weaker norm. The estimate requires only that the separation distance of the centers be bounded above by a fixed multiple of the kernel scale. This allows the kernel scale to decrease more slowly than the separation distance, as is relevant in multiscale refinement. We then establish more general scale-uniform Bernstein inequalities on the whole space. Our result covers a broad range of weaker and stronger Sobolev indices under the same relation between the separation distance and the kernel scale. The proof works directly with the Fourier representation of functions in the kernel space and combines a low-high frequency decomposition with frame estimates for separated exponential polynomials. This avoids the additional scale factors that arise from separate comparisons of scaled and standard Sobolev norms.

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-\Delta)^{\alpha}$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to $\ell^{2}(\mathbb{N})$. For all $\alpha>0$, we prove that the fractional Hardy inequality $$(-\Delta)^{\alpha}\geq\frac{4^{\alpha}\Gamma^2(\alpha+1/2)}{\pi}\frac{\Gamma(2\,\cdot\,-1)}{\Gamma(2\,\cdot\,-1+2\alpha)}$$ holds on $\ell^{2}(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of $\ell^{2}(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.

We introduce a centroid-based block decomposition of the left-regular representation and use it to prove operator-valued Haagerup inequalities for finitely generated groups with centroid maps. The decomposition separates the three centroid growth conditions into distinct operator-norm contributions and, under a factorization hypothesis for the associated Schur multipliers, yields complementary lower bounds. For median groups of rank $\nu$, the centroid blocks are related to spherical operators, giving two-sided estimates with the explicit factor $\binom{\nu+r}{r}$; this applies in particular to right-angled Artin groups and to groups acting on finite-rank $\operatorname{CAT}(0)$ cube complexes. For finite-rank coarse median groups, we construct a coarse iterate median with error independent of the cardinality of the input set and obtain spherical-block estimates with factor $o(r^{1+\nu/2+\varepsilon})$ for every $\varepsilon>0$.

We investigate the topology generated by modular convergence on vector spaces endowed with a convex modular. Although modular convergence has long played a central role in the theory of modular function spaces, the topology it induces has, to the authors' best knowledge, not previously been studied as an independent mathematical object. Our principal result establishes that the modular topology is compatible with the vector space structure if and only if the underlying modular satisfies the $\Delta_2$-condition. Consequently, the $\Delta_2$-condition admits a purely topological characterization. We also show that the Luxemburg norm topology is the weakest first-countable topology containing all scaled modular topologies. We apply the theory to variable exponent sequence and Lebesgue spaces beyond the classical $\Delta_2$-theory and present an application of our results to the minimization of the Dirichlet integral in $W^{1,p(x)}$ with unbounded exponent $p$, which allows us to solve an open boundary value problem.

We give a short complex-analytic proof of a square-restricted form of local stable phase retrieval at the Gaussian in one-dimensional Fock space. The main estimate is a coercivity inequality for the map $F\mapsto F^2$: \[ \|F^2-F(0)^2|_{\mathcal{F}^2(\mathbb{C})} \lesssim \inf_{c\in\mathbb{R}}\||F|^2-c\|_{L^2(d \gamma)}. \] The proof uses a weighted derivative norm, two integrations by parts, and a weighted Cauchy inequality. The proof has been completely verified in Lean with the aid of Large Language Models.

We identify the critical boundary operator-norm profile of finite-prime composition operators on the Hardy–Hilbert space \(\mathcal H^2\) of Dirichlet series. For \[ \varphi_{\delta,\boldsymbol\rho}(s) = \frac12+\delta + \delta\sum_{j=1}^d\rho_jp_j^{-s}, \qquad \boldsymbol\rho\in B_d, \] the renormalized positive coefficient operators converge uniformly in operator norm, with \(O(\delta)\) error, to an explicit multivariate weighted Hankel operator \(\mathcal H_{\boldsymbol\rho}\); consequently, \[ 2\delta\|C_{\varphi_{\delta,\boldsymbol\rho}}\|^2 = \|\mathcal H_{\boldsymbol\rho}\| + O(\delta) \] uniformly over \(B_d\). We show that the limiting operator admits the total-degree reduction \[ \mathcal H_{\boldsymbol\rho} \simeq D_{\boldsymbol\rho} H_{R_{\boldsymbol\rho}/2} D_{\boldsymbol\rho}\oplus\mathbf{0}, \] where the diagonal factors are convolution-collision norms of the normalized prime weights. This structure, together with the affine comparison principle of Brevig and Perfekt, yields an explicit concentration inequality for \(\|\mathcal H_{\boldsymbol\rho}\|\), identifies the one-prime configurations as the exact equality cases in the limiting norm estimate, and gives a quantitative deficit away from them. For fixed \(\sigma>\frac12\), we also obtain a second-order expansion of the squared norm and fully finite-dimensional approximations with explicit total-degree and Dirichlet-sum truncation errors. Together, these results show that a single coefficient-operator structure governs the singular boundary profile, the fixed-\(\sigma\) perturbative regime, and certified finite-dimensional approximation.

Sparse domination is a central tool in modern harmonic analysis, offering a unified approach to weighted inequalities for Calderón–Zygmund operators and related operators such as commutator operators, rough singular integrals, square functions etc. In this expository note, we briefly survey the main ideas behind sparse bounds on $L^p$ and weighted $L^p$ spaces, and discuss boundedness results for sparse operators on Morrey and generalized Morrey spaces. We establish a model result on the boundedness of sparse operators on Komori–Shirai type weighted Morrey spaces $L^{p,\kappa}(w)$, $w\in A_p$. These bounds offer a simpler framework for deriving Morrey-space estimates for Calderón–Zygmund operators and related operators via sparse domination. We conclude with remarks about extensions and connections to other Morrey space settings.

We compare the sharp constants in Lee-type Schatten norm inequalities for finite families of complex matrices with the corresponding constants obtained after applying an arbitrary nonnegative concave function to the relevant absolute values. We prove that the nonlinear problem has exactly the same best constant as the underlying linear problem for every finite matrix dimension, every number of summands, and every Schatten exponent, including the operator-norm endpoint. For finite exponents, the proof proceeds through a lossless single-cap reduction, an exact representation as a positive mixture for finite cap combinations, and a noncommutative reassembly argument based on a weighted Schatten contraction and the Araki-Lieb-Thirring inequality. Finite-spectrum cap interpolation and a perturbation argument at zero then yield the general concave case. Thus every sharp linear result transfers without loss to the full class of nonnegative concave functions.

Let $\mathbf T$ denote the $C^*$-algebra generated by all bounded Toeplitz operators on the Hardy space of the unit disk. Barría and Halmos [Trans. Amer. Math. Soc. 273 (1982), no.~2, 621–630] asked whether the Cesàro operator belongs to $\mathbf T$. We answer this question affirmatively.

We construct a categorical framework in which we can view the direct integral as a functor taking as input objects in a certain category of Abstract Banach bundles and outputting a Banach space. We do this by constructing a quasi-abelian category of abstract Banach bundles, and we prove certain foundational and theoretical results about this category and its objects. As a consequence, we are able to define a notion of direct integral of sheaves.

In our previous work \cite{SZ2026}, we established a class of abstract Orlicz-Morrey spaces endowed with a ball-basis and introduced the notion of \(\Psi\)-bounded oscillation operators. The present paper further develops this theory by investigating sparse domination and weighted estimates for such operators and their commutators within the framework of abstract Orlicz-Morrey spaces endowed with a ball-basis. We obtain pointwise sparse domination results for the operators and establish their boundedness on weighted Orlicz-Morrey spaces. The hypotheses of these conclusions differ from those in the classical Orlicz-Morrey setting and do not rely on dyadic decompositions of Euclidean structure. Moreover, we study the duality of these spaces and prove that the dual of the associated block space is exactly the Orlicz-Morrey space under consideration.

In this paper, we study metrically bounded Volterra-type operators $J_g$ and $I_g$ on the area Nevanlinna spaces $N_\alpha^p$, where $1\le p<\infty$ and $\alpha>-1$. Choe, Koo and Smith \cite{CKS} obtained partial characterizations of the corresponding symbol classes, but the exact characterization was still open. We establish a Littlewood–Paley type characterization of $N_\alpha^p$ and use it to resolve this problem affirmatively. More precisely, $J_g$ is metrically bounded on $N_\alpha^p$ if and only if $g$ belongs to the Bloch space $\mathcal B$, while $I_g$ is metrically bounded on $N_\alpha^p$ if and only if $g\in H^\infty$.

We determine the sharp constant in the nearest-neighbor Hardy inequality on $\Z^3$ with the Euclidean inverse-square weight. For every finitely supported function $u:\Z^3\to\C$, we prove \[ \sum_{x\in\Z^3}\sum_{j=1}^3 |u(x+e_j)-u(x)|^2 \geq \frac14\sum_{x\in\Z^3\setminus\{0\}} \frac{|u(x)|^2}{|x|^2}. \] The coefficient $1/4$ is sharp, and equality is not attained by a nonzero finitely supported function. The proof uses an explicit reciprocal edge field and an edgewise completion of squares, together with a concavity argument for the associated vertex weight.

Let $0<p, q<\infty$ with $p \neq q$, and suppose that $p<2$ or $q \leq 1$. We prove that the Lorentz sequence space $\ell_ {p, q}$ does not isomorphically embed into $L_ {p, q}(0,1)$. Consequently, $L_ {p, q}(0, \infty)$ does not isomorphically embed into $L_ {p, q}(0,1)$.

Machine learning learns functions: prompt to response, image to caption. What these functions are mathematically remains hard to say. We present a method to approximate these kinds of transformations using techniques from dynamical systems that fall under the umbrella of Koopmanism. We introduce the use of composition operators, which generalize the Koopman operator and, crucially, can map between distinct spaces, motivating the perspective that LLM transformations are rectangular infinite-dimensional operators. This formalism reveals useful structure: under natural assumptions on the prompt and response distributions, the LLM operator is an isometry, and misalignment between learned representations manifests as spectral pollution of its finite sections. We then outline a method of constructing finite-dimensional approximations of an LLM operator, and demonstrate how the singular value spectrum can be used to compare tasks and models.

The oscillator (or metaplectic) semigroup is classically defined as the two-fold cover of the semigroup of positive complex symplectic matrices. Although geometrically precise, this definition does not provide an intrinsic operator-theoretic characterization of the bounded operators corresponding to positive symplectic matrices. This is in contrast with the real metaplectic group, whose relation with the symplectic group can be expressed directly in terms of the Schrödinger representation of the Heisenberg group through its intertwining property. In the complex setting, such a relation appears in the existing literature mainly in infinitesimal form or on suitable classes of Gaussian functions. In this work we prove that the (complexified) Schrödinger intertwining relation holds for every function in $L^2(\mathbb{R}^d)$. The main point is the converse statement: if an arbitrary complex symplectic matrix $S$ admits a nonzero bounded operator on $L^2(\mathbb{R}^d)$ satisfying this intertwining relation, then $S$ is necessarily positive and the operator coincides, up to a nonzero scalar, with the corresponding element of the oscillator semigroup. Consequently, positive complex symplectic matrices are exactly those admitting a nonzero bounded Schrödinger intertwiner, and the associated intertwining space is one-dimensional. This provides a Schrödinger-representation characterization of the oscillator semigroup, parallel to the classical one for the real metaplectic group.

We study the exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions. It is shown that the sufficient number of nodes for discretization with the minimal number of nodes is equal to $N$ or $N+1$ and depends on the dimension $N$. The connection with Hadamard matrices and the Hadamard conjecture is demonstrated.

In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ by infinite convolutions. Given admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{m}$ and a sequence $\bn=\{n_{k}\}_{k=1}^{\infty}$ of positive integers, for every $\bw\in \Omega$, we write $\mu^{\mathbf{n}}(\bw) = \delta_{N_{\omega_{1}}^{-n_{1}}B_{\omega_{1}}} * \delta_{N_{\omega_{1}}^{-n_{1}}N_{\omega_{2}}^{-n_{2}}B_{\omega_{2}}} * \cdots$. First, we show that the mapping $\mu^{\mathbf{n}}: (\bw, B) \mapsto \mu^{\mathbf{n}}(\bw)(B)$ is a random measure. Next, we introduce the notion of a $t$-equi-positive family, and use it to obtain a general sufficient condition under which an infinite convolution is a spectral measure and possesses a specified set of spectral eigenvalues. We then extend the concepts of spectrality and spectral eigenvalues to random measures, and show that, under the assumption that the corresponding standard infinite convolution is non-degenerate for $\mathbb{P}$-a.e.\ $\boldsymbol{\omega}\in\Omega$, the measures $\mu^{\mathbf{n}}$ are spectral random measures for $\mathbb{P}$-a.e.\ $\boldsymbol{\omega}$, admitting the spectral eigen set \[ \mathcal{E}_m=\{t\in\mathbb{N}_+:\gcd(t,N_k)=1,\,1\le k\le m\}. \] Moreover, for each such $t$, there exist uncountably many spectra $\Lambda_{\boldsymbol{\omega}}\subset\mathbb{Z}$ with $t\Lambda_{\boldsymbol{\omega}}$ also a spectrum of $\mu^{\mathbf{n}}(\boldsymbol{\omega})$. Finally, for the important case where each digit set $B_k$ is a consecutive set $\{0,1,\dots,b_k-1\}$, we completely characterise the positive integer spectral eigenvalues of $\mu^{\mathbf{n}}(\boldsymbol{\omega})$, proving that they are exactly the integers coprime to every $b_k$.

A result of Cembranos states that a non-trivial injective tensor product of a $C$-space contains a complemented copy of $c_{0}$, and in particular fails the Grothendieck property. We establish a positive analogue of the Cembranos theorem for tensor products of Banach lattices. If $E$ and $F$ are infinite dimensional Banach lattices, with $E$ containing $c_{0}$ and $E^{\ast}$ or $F^{\ast}$ having the bounded positive approximation property, then the Wittstock tensor product $E\widetilde{\otimes}_{\vert{\varepsilon}\vert}F$ contains a complemented copy of $c_{0}$. If $F$ is in addition reflexive, then $E\widetilde{\otimes}_{\vert{\varepsilon}\vert}F$ fails the positive Grothendieck property.

We prove that sparse domination of a bilinear form implies convex body domination. More precisely, if a bilinear form admits an $(r,s)$-sparse bound, then its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound. The proof relies on a randomization argument. We establish the result both for sparse families in a fixed dyadic lattice and for sparse families of arbitrary cubes. As an application, we deduce sparse domination for iterated commutators, with the local oscillations of the symbol appearing in the sparse form.

We show that every smooth manifold $M$ of positive dimension admits fully supported distributions $u,v\in \mathcal D'(M)$ with $\mathrm{WF}(u)\cap({-}\mathrm{WF}(v))=\emptyset$ and $uv=0$. Neither factor can be continuous on any non-empty open set, but $u$ may be chosen of optimal Sobolev order $\dim M/2$, with wavefront set contained in any prescribed closed conical set $\Gamma\subset T^\ast M\setminus 0$ with non-empty and non-maximal fibers. Simultaneously, the wavefront set of $v$ may be confined to any compatible open conical set or, in geometric situations, to closed conical sets such as conormal bundles of foliations or ray bundles generated by locally conformally closed one-forms. We give positivity criteria and show that full-support zero divisor pairs accumulate at $(1,0)$ in Sobolev-microlocal topologies. The one-dimensional case uses a Fourier null series of Kozma-Olevskiĭ.

A free spectrahedron is the matricial solution set of a free linear matrix inequality $L_A(X) = I-A_1 \otimes X_1 - \dots - A_g \otimes X_g \succeq 0$. In this dimension-free setting, free extreme points play the role of classical extreme points. In particular, every bounded real free spectrahedron is the matrix convex hull of its free extreme points. In this qualitative sense, free extreme points of free spectrahedra are abundant. However, quantifications of this abundance have remained elusive. In particular, outside simplices, it is not known whether there exist bounded real free spectrahedra that have finitely many free extreme points. A necessary condition for having finitely many free extreme points is that the classical spectrahedron defined by $L_A(x) \succeq 0$ is a polytope. We strengthen this necessary condition through two constructions. First, we provide a geometric construction of an infinite family of free extreme points at the second level of the maximal matrix convex set over a polygon with at least four sides. Second, we develop a graded face lifting technique for general bounded real free spectrahedra, which allows us to construct free extreme points of the full spectrahedron from free extreme points of the graded face lift. As corollaries, we obtain obstructions to minimal matrix convex sets over polytopes being free spectrahedra and show that much of the scalar extreme-point geometry of maximal matrix convex sets over polytopes can be captured by higher-level free extreme points.

We prove dimension-free higher-order Sobolev norm estimates on open convex subsets of $\mathbb{R}^n$ with respect to Gaussian measure and use them to obtain norm equivalence on nonempty open convex subsets of $\ell^2$ endowed with a nondegenerate Gaussian measure. To the best of our knowledge, this is the first such equivalence theorem on a proper open subset of an infinite-dimensional Hilbert space, beyond the earlier whole-space results. We also prove the Malliavin–Sobolev norm equivalence for all $p\in[1,\infty)$ and $k\ge2$, including the case $p=1$, $k\ge3$ left open by Addona–Muratori–Rossi in \cite{AddonaMuratoriRossi}.

Let $\cH$ be a separable complex Hilbert space with $\dim\cH\ge3$. We characterize the bounded bijective complex-linear maps $T$ on $\BH$ for which there exists a bijection $\psi$ of the set of conjugations,preserving commutativity in both directions, such that $T$ maps the space of $C$-symmetric operators onto the space of $\psi(C)$-symmetric operators for every conjugation $C$. We show that this condition is equivalent to the existence of a bijection $\varphi$ of the set of all orthonormal bases such that $T$ maps the set of all $(e_n)$-diagonal operators onto the set of all $\varphi((e_n))$-diagonal operators. We also characterize the corresponding preservers in dimension two using Pauli coordinates, without assuming the commutativity condition on $\psi$.

Let $(\Omega,\mu)$ and $(\Lambda,\lambda)$ be complete atomless localizable semifinite measure spaces. Suppose that $E(\Omega,\mu)$ and $F(\Lambda,\lambda)$ are real rearrangement-invariant Banach function spaces with order-continuous norms, in the Banach-lattice sense, and that neither norm is proportional to the $L_2$-norm. Every surjective real-linear isometry $U:E(\Omega,\mu)\longrightarrow F(\Lambda,\lambda)$ has the form $Uf=w\Phi(f)$, where $w$ has full support and $\Phi$ is induced by a complete measure-class Boolean isomorphism. Both factors are uniquely determined by $U$. Let $(\mathcal{M},\tau)$ and $(\mathcal{N},\nu)$ be atomless semifinite von Neumann algebras, and let $E(\mathcal{M},\tau)$ and $F(\mathcal{N},\nu)$ be symmetric operator spaces satisfying the same assumptions on their norms. Every surjective real-linear isometry $V:E(\mathcal{M},\tau)_{\mathrm{sa}}\longrightarrow F(\mathcal{N},\nu)_{\mathrm{sa}}$ has the form $V(x)=hJ(x)$, where $J:\mathcal{M}\longrightarrow\mathcal{N}$ is a normal surjective Jordan $*$-isomorphism and $h\in LS(\mathcal{Z}(\mathcal{N}))_{\mathrm{sa}}$ is central with full support; again, the two factors are unique. We also identify the bounded skew-Hermitian operators on the real self-adjoint part and derive commutative and noncommutative isometric forms of Mityagin's question.

We determine the exact finite-dimensional threshold in the zero-subspace problem of Aron and Rueda for complex homogeneous polynomials. More precisely, for every $d$ and $k$ we determine the least $m$ such that every $d$-homogeneous polynomial on $\mathbb{C}^m$ vanishes on a $k$-dimensional linear subspace. We also determine the exact threshold for arbitrary polynomials of degree at most $d$ to be constant on a $k$-dimensional linear subspace. The two thresholds are different. In the homogeneous case the exact threshold follows from Tevelev's theorem on isotropic subspaces and closedness of the incidence locus. In the bounded-degree case we first eliminate the linear homogeneous component by passing to its kernel; the remaining components, of degrees $2,\ldots,d$, form the system to which the Debarre–Manivel theorem is applied. For $k=2$ we give a separate proof using top Chern classes and Newton's inequalities.

Motivated by both the research on Lipschitz-free spaces and on Lipschitz harmonic functions on graphs, we study invariant pointwise closed subspaces of spaces of Lipschitz functions over graphs, with special emphasis on finitely generated groups as graphs. Such spaces form a natural class of $\text{weak}^*$-closed subspaces, that can be fully described in some cases, and hence have canonical quotient preduals of the corresponding Lipschitz-free spaces. We show that these spaces are described by finite local constraints; in the group case as Lipschitz solutions of systems of convolution equations. We describe and characterize their preduals via a universal property and show that whenever they contain a non-zero element with a $c_0$-gradient, then they contain $\ell_\infty$, and consequently their preduals contain a complemented copy of $\ell_1$. A guiding question is whether this class of Lipschitz spaces and their preduals contains an infinite-dimensional reflexive Banach space. In this regard, the main result of the paper is the following dichotomy proved using abstract harmonic analysis. Every translation-invariant pointwise closed subspace of the Lipschitz space over $\mathbb{Z}^d$ is either finite-dimensional or it is non-separable –in particular, the corresponding predual is either finite-dimensional or non-reflexive.

Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincaré inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincaré spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $\gamma\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(\mu)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(\mu)} \sim \|\rho^{-1}\phi^{-\gamma}F\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}, \qquad F\in\{\Delta f,m_f\},\quad \phi\in\{\rho,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(\mu)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu(B(x,\rho(x,y)))$ and $\Delta f(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(\mu)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma\in(-\infty,-1)\cup(0,\infty)$ for $\phi=\rho$ or $\gamma\in (-\infty,-\frac1d)\cup(0,\infty)$ for $\phi=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.