Papers

We prove that the classical Cesàro operator belongs to the Toeplitz algebra, providing an independent solution to a question raised by Barría and Halmos. Our approach is based on a discrete Mellin calculus for the sampled-ratio matrices \[ W(\kappa)_{jk} = \frac{1}{j+1}\, \kappa\!\left(\frac{k+1}{j+1}\right). \] For a natural algebra of kernels $A$, we prove that this quantization is multiplicative modulo Hilbert–Schmidt operators, \[ W(\kappa)W(\eta) - W(\kappa \star \eta) \in S_2, \qquad \kappa,\eta \in A. \] We further show that every operator $W(\kappa)$, $\kappa \in A$, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel $\kappa = \mathbf 1_{(0,1]}$, this resolves the Barría–Halmos question as a special case of the general framework.

We extend the recent results of Arnold on estimates for Kreiss positive semigroups on $L_p$-spaces to the non commutative setting, $1<p<\infty$. We also provide a sharp example. All the results are achieved thanks to the use of the Schur test, including a new version for von Neumann algebras.

We prove the Trudinger inequality with exponent $\frac{s}{s-1}$ for $M^{1,s}$ Sobolev spaces on metric measure spaces under the sole measure growth assumption $\mu(B(x,r))\ge br^s$. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as $p\uparrow s$. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a $(1,p)$-Poincaré inequality when $p<s$. An Ahlfors regular counterexample shows that this extension fails at the critical exponent $p=s$.

Let $\Omega_{n}=\{-1,1\}^n$ be the $n$-dimensional hypercube equipped with the normalized uniform measure, let $\nabla$ be the Walsh gradient and let $\Delta$ be the Walsh Laplacian. For every $1<p\leq 2$ we prove the following estimate \[ \|\nabla f\|_{L_p(\Omega_n;\ell_2^n)} \leq c_{{\rm abs}}(p-1)^{-2}\|\Delta^{1/p}f\|_{L_p(\Omega_n)}. \] The exponent $\frac1p$ is optimal, thus this settles the open problem on the sharp fractional Riesz estimate by Efraim and Lust-Piquard \cite{E-LP2008} which was subsequently highlighted by Ivanisvili and Volberg \cite{I-V2022}. We also establish the higher-order counterpart. As applications of our results, we obtain simpler proofs of the optimal short-time estimate for $\nabla e^{-t\Delta}$, and the Bernstein-Markov type inequality for $d$-bounded degree functions.

We establish a necessary and sufficient weight condition so that a modular inequality holds for a generalized integral operator $T_\phi$ on non-negative non-increasing functions in weighted variable Lebesgue spaces $L_w^{p(x)}.$ We give a condition under which the mentioned modular inequality leads to a norm inequality. Lastly, examples are given to demonstrate our main result, to obtain modular inequalities for some well known operators.

Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\subseteq\mathbb{R}^n$, let $\mu_{K,t} (\mathrm{d} x) \propto e^{-t\| x \| ^2} \mathbb{1}_K(x)\,\mathrm{d} x$, and let $Q_n$ be the supremum of $\|\operatorname{Cov}(\mu_{K,t})\|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=\Theta(n^{2/5})$, closing the gap between the known $\Omega(n^{1/3})$ lower bound and the $O(\sqrt{n\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $\Omega(n^{2/5})$.

For $d\geqslant3$, let $p^*=\frac{2d}{d-2}$. We prove that the Born-Jordan distribution maps $L^{2,p^*}(\mathbb R^d)\times L^{2,p^*}(\mathbb R^d)$ boundedly into $L^{p^*}(\mathbb R^{2d})$. The Lorentz index $p^*$ cannot be increased in either variable. In particular, this gives the endpoint $L^2\times L^2\to L^{p^*}$ estimate, settling the critical boundedness problem raised by Stra, Svela, and Trapasso \cite{SST26}.

We give a negative solution to the complemented subspace problem for Banach spaces with unconditional bases over both the real and complex fields. For every $\rho>0$, we construct a projection $P_\rho$ of norm less than $1+\rho$ on a separable superreflexive space \begin{equation*} X_\rho=\left(\bigoplus_{j=1}^{\infty}\ell_{p_j}^{N_j}\right)_2, \qquad p_j\downarrow2, \end{equation*} such that $Z_\rho=P_\rho(X_\rho)$ and its dual $Z_\rho^*$ have Schauder bases but admit no unconditional bases. Over the real field, both spaces have Gordon–Lewis local unconditional structure (GL-lust) but fail Dubinsky–Pełczyński–Rosenthal local unconditional structure (DPR-lust), disproving a conjecture of Figiel, Johnson and Tzafriri. In particular, neither is isomorphic to a Banach lattice, giving a negative solution to the separable Banach-lattice complemented subspace problem. A modification of the construction also shows that the class of separable real Banach lattices is not primary. A Lean 4 formalisation of the main results accompanies the paper.

In this paper, we prove pathwise uniform convergence at rates up to $1/2$ for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive $C_0$-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form \begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \|U(t_j)-U^j\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*} on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $U^j$ is the tamed exponential Euler approximation at time $t_j=jk$ with step size $k>0$. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.

We determine the optimal order of the scaling parameter in the metric cotype $2$ inequality for complete Alexandrov spaces of nonnegative curvature. It grows linearly with the dimension $n$ of the discrete torus. This answers Question 17 of Eskenazis, Mendel, and Naor. In their sign-vector normalization on $\bigl(\mathbb{Z}/(2m\mathbb{Z})\bigr)^n$, the square of the optimal metric cotype $2$ constant for this class lies between $\max\{1,n/m\}$ and $e\lceil n/m\rceil$ for every dyadic $m\geq 2$. The upper bound follows from a dyadic Bernoulli-thinning argument using only the Lang–Schroeder–Sturm inequality. Snowflake universality of the fixed Wasserstein space $\mathcal{P}_2(\mathbb{R}^3)$ gives the $n/m$ lower bound.

For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metric cotype inequality holds at the sharp scale $m=O(n^{1/q})$. In particular, the quadratic inequality holds with $m=O(\sqrt{n})$, answering the $L_1$ sharp metric cotype question of Mendel and Naor. The proof combines finite cut representations with a nonlinear smoothing estimate and an exact ternary rounding identity.

We consider the space $\mathcal{D}'^E_L(U)$ of all distributions on the open set $U$ whose wave front set measured with respect to a Banach space $E$ lies in a closed conic subset $L$ of $U\times(\mathbb{R}^n\backslash\{0\})$. The Banach space $E$ only satisfies mild technical assumptions. We show that the pullback by a diffeomorphism $f:O\rightarrow U$ is a topological isomorphism $f^*:\mathcal{D}'^E_L(U)\rightarrow\mathcal{D}'^E_{f^*L}(O)$. In the case when $E$ is the $L^p$-Sobolev space $W^{r,p}(\mathbb{R}^n)$ of order $r\in\mathbb{R}$, we study the pullback by a smooth map $f:O\rightarrow U$ of constant rank.

We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator on $\mathrm{L}^1_{\sigma,n}(\Omega)$ does not generate a $\mathrm{C}_0$-semigroup, even though the resolvent problem is uniquely solvable. In stark contrast, its realization on ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$ generates a compact, analytic $\mathrm{C}_0$-semigroup, which leaves $\mathrm{L}^1_{\sigma,n}(\Omega)$ invariant. The key point is that these two realizations, which are canonically identified for $1<p<\infty$ through the Helmholtz decomposition, cease to be equivalent at the endpoint $p=1$. This leads to genuinely different functional analytic properties. Our result provides the first positive generation theorem for the Stokes operator with no-slip boundary conditions in a pure $\mathrm{L}^1$-setting on a bounded domain and settles a problem that had remained open for nearly fifty years; see, e.g., \cite{Koz:01,DHP:01}. In this sense, it completes the theory of the Stokes semigroup across the full scale of solenoidal Lebesgue spaces on (smooth) bounded domains. As an intermediate step, some results on the space of Radon measures are obtained. The proof combines the sun-dual construction with a precise analysis of the failure of the Helmholtz decomposition in $\mathrm{L}^1$, the celebrated result of Abe and Giga \cite{AG:12} on the Stokes semigroup on $\mathrm{C}_{\sigma,0}(\Omega)$ and the regularity theory refinements for the Stokes operator recently developed by Breit and the second author \cite{BG:25}.

This note presents a proof that the Riesz projection is contractive from $L^q$ to $L^{4(1-1/q)}$ for the case $1<q<\infty$. This fills in the previously unresolved parameter range of a conjecture of Brevig, Ortega-Cerdà, Seip, and Zhao.

We study the critical fractional $p$-Hardy-Sobolev equation \begin{equation}\tag{$\mathcal{P}$}\label{a-main} (-\Delta_p)^s u -\mu\dfrac{|u|^{p-2}u}{|x|^{sp}}=\dfrac{|u|^{p^*_s(\alpha)-2}u}{|x|^{\alpha}}+f \;\mbox{ in }\,\mathbb{R}^d, \quad u\in \mathcal{D}^{s,p}(\mathbb{R}^d), \end{equation} where $1<p<\infty$, $0<s<1$, $0\leq\alpha<sp<d$, $\mu>0$, $p^*_s(\alpha):= p(d-\alpha)/(d-sp)$ is the critical Hardy-Sobolev exponent, and $f$ is a nontrivial nonnegative functional in $(\mathcal{D}^{s,p}(\mathbb{R}^d))^*$. We first establish global compactness results for Palais-Smale sequences associated with the corresponding energy functional. When $\alpha>0$, the loss of compactness is described by dilations of solutions of the Hardy-Sobolev limit problem. The case $\alpha=0$ has a different structure: in addition to Hardy profiles, pure Sobolev profiles may occur when the centre of concentration escapes from the Hardy singularity relative to its scale. We give a direct centre-scale analysis of these two concentration regimes and obtain the corresponding energy decomposition and profile separation. As an application, under an explicit smallness assumption on $f$, we first obtain a positive solution for \eqref{a-main} with negative energy. We then construct a nonlinear path based on hidden convexity whose energy remains strictly below the first bubbling threshold. A minimax argument, combined with the global compactness theorem, then yields a second distinct positive solution for \eqref{a-main}.

Let $R\subset\mathbb{R}^d$ be a reduced root system, $G$ the associated finite reflection group, and $k\ge0$ a $G$-invariant multiplicity function. We develop a Dunkl analogue of the obstacle/partial-balayage method of Ouyang, Spector, and Stockdale (https://arxiv.org/abs/2608.18068) for the Euclidean fractional Laplacian. For $0<s<2$, nonnegative $f\in L^1(\nu_k)\cap L^2(\nu_k)$, and $\lambda>0$, we obtain a decomposition \[ f=\mu+(-\Delta_k)^{s/2}u, \qquad 0\le\mu\le\lambda, \] with $\mu=\lambda$ on $\Omega=\{u>0\}$ and \[ \lambda\nu_k(\Omega)\le\|f\|_{L^1(\nu_k)}. \] As an application, we prove that the vector Dunkl–Riesz transform $\mathcal R_k=\nabla_k(-\Delta_k)^{-1/2}$ is of weak type $(1,1)$ with constant at most $(M_k+2)$, where \[ M_k=\#\{\alpha\in R_+:k(\alpha)>0\}. \] For $G$-invariant functions, the reflection terms vanish and the same argument gives the universal constant $2$. We further establish a dimension-free weak-type $(1,1)$ estimate for the Dunkl–Schrödinger Riesz transform.

In the paper J. Phys. A: Math. Theor. 53 (2020), 345303, the limiting mutual information between two intervals separated by one lattice site in a free-fermion chain was evaluated. The calculation replaces the entropy function by a regularized version depending on $\varepsilon>0$, takes the block lengths to infinity, and only then lets $\varepsilon$ tend to zero. We prove that the opposite order gives the same constant $2\log2-1$. Moreover, we show that the same limit is obtained whenever both block lengths tend to infinity and $\varepsilon$ tends to zero simultaneously, with no restriction on their relative rates. In fact, the error caused by the regularization is bounded by $(1+\varepsilon)\log(1+\varepsilon)-\varepsilon\log\varepsilon$, uniformly in both block lengths.

A separable real or complex Banach space fails the Schur property if and only if it admits an equivalent Gâteaux smooth, weakly locally uniformly rotund norm which is not midpoint locally uniformly rotund. The construction enlarges an LUR unit ball by a weakly compact set and adds a weighted Hilbert-space term. The resulting norms can be chosen arbitrarily close to any prescribed equivalent LUR norm, and are Gâteaux smooth whenever the prescribed norm is Gâteaux smooth; Fréchet smoothness is preserved as well. Weakly locally uniformly rotund norms which are not midpoint locally uniformly rotund are dense among all equivalent norms on each separable non-Schur space. For arbitrary real or complex Banach spaces, such a renorming exists exactly when the space is LUR-renormable and fails the Schur property. We give a direct construction for this last assertion.

The universal property of the Lipschitz-free spaces allows for the linearisation of a base point preserving Lipschitz map $f\colon M\to M$ to a bounded linear operator $T_f\colon \mathcal{F}(M) \to \mathcal{F}(M)$. While it is known that the operator $T_f$ is weakly mixing whenever $f$ has this property, it is open whether a similar inheritance result for topological transitivity holds. Motivated by this question we investigate properties of $T_f$ under the assumption that $f$ is topologically transitive. In particular we prove Kitai's theorem for Lipschitz operators, i.e. we show that every connected component of the complex variant of $T_f$ intersects the unit circle if $f$ is topologically transitive.

We study Sobolev and $BV$-extension properties of planar subsets. In particular, we prove that fat Sierpiński carpets that support a weak $(1,1)$-Poincaré inequality are $W^{1,1}$-extension sets, and we provide an explicit linear extension operator for them. We also show that for planar domains satisfying a weak $(1,1)$-Poincaré inequality the $W^{1,1}$-extension property and the $BV$-extension property are equivalent. Finally, as a consequence of the previous result, we show that if a simply-connected planar domain is Ahlfors regular and supports a weak $(1,1)$-Poincaré inequality, then it is a (linear) $W^{1,1}$-extension domain.

We establish that a JB$^*$-triple $E$ is commutative if and only if its numerical index $n(E)$ is one. As a consequence, $n(E) = 1$ implies $n(E^*) = n(E^{**}) = 1$. This settles a question left open in the literature since 2008.

We study two means on the five-dimensional Vinberg cone $\Omega$, viewed as a sparse cone of positive definite $3\times3$ matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group $H$ associated with $\Omega$. It is $H$-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining $\Omega$.

We show the following equivalence of solutions to general fractional Helmholtz equations: $u$ is a solution to $$A^su=\lambda^su$$ for some fixed $0<s<2$ and some $\lambda>0$ if and only if $u$ is a solution to the same equation for all $s\neq0$. Furthermore, we show as well that $u$ equivalently solves the logarithmic Helmholtz equation $$\log(A)u=(\log\lambda)u.$$ Here, $A$ is a nonnegative, linear operator on a Banach space $X$. In particular, our results hold whenever $A=-L$, where $L$ is the infinitesimal generator of a $C_0$-semigroup in a Banach space $X$ satisfying mild assumptions that are typical in applications. As a particular case, we recover known results for the fractional Laplacian $(-\Delta)^s$ in $\mathbb{R}^n$. More importantly, we provide a list of examples of other fractional power operators in settings where the Fourier transform is not available and for which our theorems apply. These include fractional powers and logarithms of second order elliptic operators in bounded domains, Laplace–Beltrami operators on Riemannian manifolds, space-time master equations, the discrete Laplacian, and fractional derivatives.

Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic–harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We consider the more general problem of when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection determine three midpoint maps associated with the connection, its metric dual, and the Levi–Civita connection. We show that the Gauss composition of the first two equals the Levi–Civita midpoint if and only if the latter is invariant under one step of the iteration. We give a sufficient condition for this invariance in terms of an isometry which acts as the point reflection about the Levi–Civita midpoint and exchanges the connection with its dual. Under this invariance, the iterations converge quadratically for sufficiently close initial pairs. Nakamura's iteration is an example of this criterion. The criterion also applies to dual pairs of matrix power means whose Gauss composition is the geometric matrix mean. We give a dually flat Hessian example which shows that duality alone does not suffice. We also give non-flat examples with a Euclidean metric and a parallel cubic form in every dimension larger than one. In dimension one, we characterize the invariance completely by using the Matkowski–Sutô equation. For Euclidean dual pairs, we conjecture that the invariance implies that the cubic form is constant. We prove this conjecture in dimension one and for scalar multiples of a constant cubic form.

For a prime number $p > 2$, let $\bm{0} \in D \subset \mathbb{Z}^n$ be a $p$-element digit set satisfying $ \mathcal{Z}(\widehat{\delta}_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some \( \bm{a} \in \{ (i_1, \dots, i_n)^t : i_k \in [1, p-1] \cap \mathbb{Z}, 1\leq k\leq n \} \), where $\mathcal{Z}(\widehat{\delta}_D)$ is the zero set of the Fourier transform of $\delta_D$. Let $Q$ be an integer expansive diagonal matrix in $\mathbb{R}^{n}$, the self-affine measure $\mu_{Q,D}$ is defined by \[ \mu_{Q,D}(\cdot) = \frac{1}{\#D} \sum_{d \in D} \mu_{Q,D}(Q(\cdot) - d). \] In this paper, we first provide sufficient condition for a maximal orthogonal family $E_{\Lambda}=\{ e^{-2\pi i \langle \lambda, x \rangle} : \lambda\in \Lambda \subset\mathbb{R}^{n}\}$ to be an orthogonal basis of $L^2(\mu_{Q,D})$ when $Q = p\operatorname{diag}[q, \dots, q]$ with $q\geq 1$. Then we obtain necessary and sufficient conditions for the integer matrix $R$ such that $E_{\Lambda}$ and $E_{R\Lambda}$ are both orthogonal basis of $L^{2}(\mu_{Q,D})$. Furthermore, for $Q = p\operatorname{diag}[l_1, \dots, l_n]$ with $|l_1 \dots l_n| > 1$, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of $\mu_{Q,D}$.

We study disorder-free localization in a spin-coupled quantum walk on the hypercube within the quantum Bernoulli noises framework. We derive a general criterion for disorder-free localization: it occurs if and only if some spectral subspace of a reduced unitary evolution contains a vector whose position marginal probability is non-uniform. For the Grover walk with distance-dependent coupling \(\phi_\sigma=|\sigma|\pi/(n+1)\) in dimension \(d=4\), we explicitly construct a non-uniform fixed point in the all \(+1\) spin configuration. This implies that disorder-free localization occurs in our model.

Given $\alpha\in \mathbb R$ and $ B_1,\ldots,B_q\in M_{m,n}$, consider the map \[\Phi^{\pm}_{\alpha,\mathfrak{B}}(X)=\alpha \mbox{tr}(X) I_m\pm \sum_{s=1}^qB_sXB_s^*\ \mbox{for all}\ X\in M_n.\] In this study, necessary and sufficient conditions on $\alpha $ and $B_1,\ldots,B_q$ are given that guarantee $\Phi_{\alpha,\mathfrak{B}}^\pm$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $\alpha \mbox{tr}(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.

Here we investigate the stability of the Kolmogorov–Arnold representation theorem (KART) under adversarial reparameterisations of the hidden layer for multivariate discontinuous and unbounded functions. Our results provide a rigorous mathematical foundation for the structural robustness of modern deep learning architectures, such as Kolmogorov–Arnold Networks (KANs), under adversarial configurations.

In 1940, Levin\footnotemark[1] and Milman proved that a closed linear subspace of $C[0,1]$ whose elements all have bounded Jordan variation must be finite-dimensional. We prove its analogue for variation in the sense of Young and, more generally, for every finite-valued nondecreasing gauge $\varphi:[0,\infty)\to[0,\infty)$ with $\varphi(0)=0<\varphi(t)$ for $t>0$. If $E$ is a closed linear subspace of $C[0,1]$ and every $f\in E$ satisfies $\Var_\varphi(\lambda_f f)<\infty$ at some scale $\lambda_f>0$, then $E$ is finite-dimensional. No continuity, convexity, or doubling condition is needed. The proof combines a lower-semicontinuous regularization that preserves the scaled class, Baire uniformization, Helly selection, and a quantitative nested-peaks construction for nonhomogeneous gauges. Consequently, for every infinite-dimensional closed subspace $F\subset C[0,1]$, the set $F\cap\cV_\varphi([0,1])$ is a meagre $F_\sigma$ subset of $F$. For every Young function, both the scaled and raw finite-variation families are maximal dense-lineable but not spaceable. The scaled family is itself a dense vector subspace of Hamel dimension $\mathfrak c$, whereas without local doubling the raw family need not itself be linear.

We study to what extent the majorisation order between a probability measure and its $1$-Lipschitz image is preserved when both measures undergo Gaussian convolution. We show that the majorisation order is fully preserved in dimensions $n\leq 2$, and obtain dimension-dependent partial preservation in higher dimensions. The perspective taken is that majorisation between densities amounts to comparison through internal energies satisfying a certain pressure condition. Accordingly we introduce a notion of iterated nonnegativity of pressure, closely related to iterated pressures arising in optimal transport, placing the majorisation order within a graded hierarchy of internal energy comparisons. Using the observation that the volume of Euclidean neighbourhoods can be detected from internal energy measurements along the heat flow, we show that our results imply several principal known cases of the Kneser–Poulsen conjecture.