We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $\delta>1/2$ there is a constant $C_M(\delta)$, independent of the group $G$, such that $\|\mu\|_{M(G)}\le1$ and $\inf_{\gamma\in\widehat G}|\widehat\mu(\gamma)|\ge\delta$ imply that $\mu$ is invertible in $M(G)$ and $\|\mu^{-1}\|_{M(G)}\le C_M(\delta)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $\delta=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.
We prove that every infinite compact abelian group $G$ admits a countably infinite proper set $E\subset\widehat G$ that is strongly small-2 but not Riesz. More precisely, $|\alpha|*|\beta|$ is absolutely continuous for all $\alpha,\beta\in M_E(G)$, whereas $M_E(G)$ contains a singular complex measure of total mass one...
For a partition $\lambda$, let $G_\lambda^{(\beta)}$ be the stable $\beta$-Grothendieck polynomial attached to $\lambda$. Each homogeneous component of the $\beta = 1$ specialization $G_\lambda^{(1)}$ is Schur-positive and hence positive in the fundamental basis of quasisymmetric functions. For $m\ge|\lambda|$, let $G_...
Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=\lambda_0\leq\lambda_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{\phi_j\}_{j\geq0}$. Let $\{\omega_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured...
Let $0\leq\alpha<n$, and let $\sigma$ and $\omega$ be locally finite positive Borel measures on $\R^n$, possibly with common point masses. Suppose that one measure is supported on a compact regular $C^{1,\mathrm{Dini}}$ curve. We prove that the $\alpha$-fractional Riesz vector maps $L^2(\sigma)$ to $L^2(\omega)$ if and...
Ji Li, Chun-Yen Shen, Eric T. Sawyer et al.· 0 citations
The eigenvalue problem for a probability measure $\mu$ with compact support in $\R$ is whether there exist a countable set $\Lambda$ and a nonzero real $t\ne 1$ such that both $\Lambda$ and $t\Lambda$ are spectra of $\mu$, that is, the family $$E_{a\Lambda}=\{e^{-2\pi i a\lambda x}:\lambda\in\Lambda\}$$ is an orthonorm...
Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a H\"ormander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood--Paley square functions. Let...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
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