Related papers: Towards Hilbert-Kunz density functions in Characte…

Let $E$ be a subset of the unit disc $U$ of the complex plane $\CC$. Recall that $H^p(U)$ is the space of all holomorphic functions $g$ on $U$ for which $\|g\|_{H^p}$ $<$ $\infty$. Put \begin{equation} C_p(\epsilon, R) = \sup \{\sup_{|z|…

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions…

For integers $q\ge p\ge r\ge2$, we say that an $r$-uniform hypergraph $H$ has property $(q,p)$, if for any $q$-vertex subset $Q$ of $V(H)$, there exists a $p$-vertex subset $P$ of $Q$ spanning a clique in $H$. Let $T_{r}(n,q,p)=\min\{ e(H):…

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple $(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$, the limit $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j=…

In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $\mu_r^{(s)}$ be…

We study two families of integral functionals indexed by a real number $p > 0$. One family is defined for 1-dimensional curves in $\R^3$ and the other one is defined for $m$-dimensional manifolds in $\R^n$. These functionals are described…

We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2\epsilon)^{1/2p}$-uniformly Lipschitz) with $p$…

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their…

Recently, Charpentier showed that there exist holomorphic functions $f$ in the unit disk such that, for any proper compact subset $K$ of the unit circle, any continuous function $\phi$ on $K$ and any compact subset $L$ of the unit disk,…

We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We…

In this paper, we show the equivalence between the boundedness of the Riesz transform $d\Delta^{-1/2}$ on $L^p$, $p\in (2,p_0)$, and the equality $H^p=L^p$, $p\in(2,p_0)$, in the class of manifold whose measure is doubling and for which the…

In this paper density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density, $\rho$, and paramagnetic current density, $j^p$. This approach is…

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients and $(p_-(L),\, p_+(L))$ be the maximal interval of exponents $q\in[1,\,\infty]$ such that the semigroup…

For a given algebraically closed field $k$ of characteristic $p>0$ we consider the set ${\mathcal C}_k$, of graded isomorphism classes of {\em standard graded pairs} $(R, I)$, where $R$ is a standard graded ring over the field and $I$ is a…

We initiate the study of synthetic curvature-dimension bounds in sub-Finsler geometry. More specifically, we investigate the measure contraction property $\mathsf{MCP}(K, N)$, and the geodesic dimension on the Heisenberg group equipped with…

We study spectral properties of Schr\"odinger operators on $\RR^d$. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in $\ZZ^d$, with the property that frequencies of finite patterns are…

In this paper we investigate the (Kohn-Sham) density-to-potential map in the case of spinless fermions in one spatial dimension, whose existence has been rigorously established by the first author in [arXiv:2504.05501 (2025)]. Here, we…

In this paper, for general plane curves $\gamma$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,\gamma}$ along the variable…

The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2…

Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$,…