A combination of Bestvina–Brady Morse theory and an acyclic reflection group trick produces a torsion-free finitely presented $\mathbb{Q}$-Poincaré duality group which is not the fundamental group of an aspherical closed ANR $\mathbb{Q}$-homology manifold.

The acyclic construction suggests asking which $\mathbb{Q}$-Poincaré duality groups act freely on $\mathbb{Q}$-acyclic spaces (i.e., which groups are $\operatorname{FH} (\mathbb{Q})$). For example, the orbifold fundamental group $\Gamma$ of a good orbifold satisfies $\mathbb{Q}$-Poincaré duality, and we show $\Gamma$ is $\operatorname{FH} (\mathbb{Q})$ if the Euler characteristics of certain fixed sets vanish.

We prove two splitting theorems, one topological, the other metric, for open manifolds with nonnegative sectional curvature.

We investigate the boundedness of the evolution operators $e^{itH}$ and $e^{it\mathcal{H}}$ in the sense of $L^{1}\to L^{\infty}$ for both the scalar Schrödinger operator $H=-\Delta+V$ and the non-selfadjoint matrix Schrödinger operator

\begin{eqnarray*}\mathcal{H}=\left [\begin{array}{c@{\quad}c}-\Delta+\mu-V_{1}&-V_{2}\\V_{2}&\Delta-\mu+V_{1}\end{array}\right ]\end{eqnarray*}

in dimension five. Here $\mu>0$ and $V_{1}$, $V_{2}$ are real-valued decaying potentials. The matrix operator arises when linearizing about a standing wave in certain nonlinear partial differential equations. We apply some natural spectral assumptions on $\mathcal{H}$, including regularity of the edges of the spectrum $\pm\mu$.

We establish an upper bound for the sectional genus of varieties which are invariant under Pfaff fields on projective spaces.

We consider continuously differentiable curves embedded in graded groups, along with the points of these curves that have the maximal transversality with respect to the grading of the group. We prove a blow-up theorem at these points and we show that the remaining points are negligible with respect to the Hausdorff measure whose dimension equals the Hausdorff dimension of the curve. This leads us to an area-type formula for the intrinsic spherical Hausdorff measure of any of these curves embedded in an arbitrary graded group.

In this paper, we will study the boundedness of intrinsic square functions on the weighted Hardy spaces $H^{p}(w)$ for $0<p<1$, where $w$ is a Muckenhoupt’s weight function. We will also give some intrinsic square function characterizations of weighted Hardy spaces $H^{p}(w)$ for $0<p<1$.

Given a parallel calibration $\varphi\in\Omega^{p}(M)$ on a Riemannian manifold $M$, I prove that the $\varphi$-critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $\varphi$-critical submanifolds are precisely the integral manifolds of a $\mathscr{C}^{\infty}(M)$-linear subspace $\mathscr{P}\subset\Omega^{p}(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi–Yau manifolds, (co)associative calibrations on $G_{2}$-manifolds, and the Cayley calibration on $\operatorname{Spin}(7)$-manifolds.)

We classify the squarefree ideals which are Gotzmann in a polynomial ring.

We are interested in a new kind of bi-dimensional bilinear paraproducts (appearing in (Amer. J. Math. 132 (2010) 201–256)), which do not fit into the setting of bilinear Calderón–Zygmund operators. In this paper, we propose a fiber-wise Calderón–Zygmund decomposition, which is specially adapted to this kind of bi-dimensional paraproduct.

We show that every central simple algebra $A$ over a field $k$ is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field. This shows that the natural generalization of the Schur group for Hopf algebras (which we call the Hopf Schur group) is in fact the entire Brauer group of $k$. If the characteristic of the field is zero, or if the algebra has a Galois splitting field with certain properties, we can take this Hopf algebra to be semisimple. We also show that if $F$ is any finite separable extension of $k$, then $F$ is a quotient of a finite dimensional commutative semisimple and cosemisimple Hopf algebra over $k$.

We study several Laguerre semigroups appearing in the literature and find sharp ranges of type parameters for which these semigroups are contractive on all $L^{p}$ spaces, $1\le p\le\infty$. We also answer a similar question for Bessel semigroups, which in a sense are closely related to the Laguerre semigroups. A bit surprisingly, in some cases the ranges turn out to be disconnected sets.

In this paper, we study weighted local Hardy spaces $h^{p}_{\omega}({\mathbb{R}}^{n})$ associated with local weights which include the classical Muckenhoupt weights. This setting includes the classical local Hardy space theory of Goldberg, and the weighted Hardy spaces of Bui.

In this short note, the norm of Berezin transform, acting on $L^{p}(\mathbb{B}_{n})$, is determined to be

\[\bigl\|B_{\mathbb{B} _{n}}:L^{p}(\mathbb{B}_{n})\to L^{p}(\mathbb{B}_{n})\bigr\|=\frac{1}{p}\prod_{k=1}^{n}\biggl(1+\frac{1}{kp}\biggr)\frac{\pi}{\sin(\pi/p)}.\]

This extends a result of Dostanić (J. Anal. Math. 104 (2008) 13–23) to several complex variables.

In this paper, we are concerned with exponential integrability for Riesz potentials of functions in Orlicz spaces $\Phi_{p,\varphi}(L,\textbf{R} ^{n})$. As an application, we study exponential integrability for BLD (Beppo Levi and Deny) functions.

We consider the maximal function of oscillatory integrals $S^{a}f$ where $(S^{a}f)(t)\widehat{\ }(\xi )=\exp(it|\xi|^{a})\widehat{f}(\xi)$ and $a\in \ ]0,1[$. For a fixed $n\geq2$ we prove the global estimate

\[\bigl\|S^{a}f\bigr\|_{L^{2}(\mathbf{R}^{n},L^{\infty}(-1,1))}\leq C\|f\|_{H^{s}(\mathbf{R}^{n})},\quad s>a/4\]

with $C$ independent of the radial function $f$. We also prove that this result is almost sharp with respect to the Sobolev regularity $s$. This extends work of Sjölin who proved these result for $a>1$.

All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of lexsegment ideals, the corresponding Rees algebra has a quadratic Gröbner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

We prove a nontrivial upper bound for the quantity (with $\mathbf{e}(z)=e^{2\pi iz}$),

\[\biggl\vert \sum_{X\le n\le 2X}\lambda (n)\mathbf{e}(\alpha{\sqrt{n}} )\biggr\vert ,\]

where $\alpha$ is any nonzero real number. This upper bound is an improvement of the earlier known results.

For an infinite compact Abelian group $G$ and $1<p<2$, it was shown in [9] that there exists a $L^{p}(G)$ multiplier which is not completely bounded. In this note, we show that in infinite every locally compact Abelian group $G$ there is a $L^{p}(G)$ multiplier which is not completely bounded.

This paper studies Newtonian Sobolev–Lorentz spaces. We prove that these spaces are Banach. We also study the global $p,q$-capacity and the $p,q$-modulus of families of rectifiable curves. Under some additional assumptions (that is, $X$ carries a doubling measure and a weak Poincaré inequality), we show that when $1\le q<p$ the Lipschitz functions are dense in those spaces; moreover, in the same setting we show that the $p,q$-capacity is Choquet provided that $q>1$. We also provide a counterexample to the density result in the Euclidean setting when $1<p\le n$ and $q=\infty$.

Let $A=\sum_{k\geq1}T_{\varphi_{k}}T_{\varphi_{k}}^{*}$ be a bounded linear operator on Bergman space $L_{a}^{2}(B_{d})$ or Hardy space $H^{2}(B_{d})$, where $\varphi_{k}$ is a multiplier for each $k$. We will show by trace estimation that for such an operator, $[A,T_{z_{i}}]$ belongs to Schatten class $\mathcal{L}_{2p}$ for $p>d$, and satisfies $\|[A,T_{z_{i}}]\|_{2p}\leq C\|A\|$ for some constant $C$ depending only on $p$ and $d$.

We study the abstract complete geometric notion of unitaries in an operator space characterized in terms of the matrix numerical index, which is a quantity determined by a norm-one element and the matrix numerical radius.

In the paper, we consider the question whether a unital full amalgamated free product of RFD (residually finite dimensional) C^∗-algebras is RFD again. One example shows that the answer to the general case is no. We give a necessary and sufficient condition such that a unital full amalgamated free product of RFD C^∗-algebras with amalgamation over a finite dimensional C^∗-algebra is RFD. Applying this result, we conclude that a unital full free product of two same RFD C^∗-algebras with amalgamation over a finite-dimensional C^∗-algebra is always RFD.