We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.
In 1965, Chowla conjectured that the mean value of the Liouville function at non-square one-variable polynomials is zero. This conjecture is a generalization of the prime number theorem (PNT). In 2022, Bergelson and Richter established a dynamical generalization of this form of the PNT. For non-square multivariate poly...
The classical P\'olya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized conv...
We prove sharp improved Sobolev inequalities on $\mathbb S^n$, $n\ge3$, for antipodally even functions and for axially symmetric functions whose critical density $|u|^{2n/(n-2)}$ has zero barycenter. In both cases, the Dirichlet coefficient improves by the optimal factor $2^{-2/n}$, while the coefficient of the $L^2$ t...
In his 1957 paper on the Dirichlet problem for higher-order elliptic equations in the plane, Agmon conjectured that there should exist higher-order equations in dimensions greater than two for which a potential-theoretic approach based on multi-layer integral operators can be successfully implemented. We show that this...
Artur Andrade, D. Mitrea, I. Mitrea et al.· 0 citations
We prove that every vector of $\mathrm{II}_{8n+1,1}$, with $n\ge1$, is a sum of two roots if and only if $n=1$ or $2$. This settles the Borisov--Nuer conjecture and implies the existence of Ulrich line bundles for every very ample polarization on a complex unnodal Enriques surface. For vectors of negative norm, we expr...
We establish a polynomial analogue of Lindenstrauss'theorem in full generality. More precisely, we prove that, for every real or complex Banach space X and every positive integer n, the n-homogeneous scalar-valued polynomials whose Aron-Berner extensions to the bidual X** attain their norms form a dense subset of the s...
Javier Falcó· 0 citations
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