Our aim in this paper is to show the vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, as an improvement of continuity results of Sobolev functions. We also show the vanishing double exponential integrability.

The local behavior of a harmonic function on the Sierpinski gasket in the neighborhood of a periodic point is governed by the eigenvalues of the $3 \times 3$ matrix that corresponds to zooming in to that point. We study the case when the matrix has complex conjugate eigenvalues. We develop a theory of local derivatives in this case. We give numerical evidence for the decay in relative frequency of this case, but we show how to construct infinitely many distinct points that fall into this case.

A singular masa $A$ in a ${\mathrm{II}}_1$ factor $N$ is defined by the property that any unitary $w\in N$ for which $A=wAw^*$ must lie in $A$. A strongly singular masa $A$ is one that satisfies the inequality

\[ \|\bb E_A-\bb E_{wAw^*}\|_{\infty,2}\geq\|w-\bb E_A(w)\|_2 \]

for all unitaries $w\in N$, where $\bb E_A$ is the conditional expectation of $N$ onto $A$, and $\|\cdot\|_{\infty,2}$ is defined for bounded maps $\phi :N\to N$ by $\sup\{\|\phi(x)\|_2:x\in N,\ \|x\|\leq 1\}$. Strong singularity easily implies singularity, and the main result of this paper shows the reverse implication.

Let $I$ be a monomial ideal in a polynomial ring $A= K[x_1,\ldots,x_n]$. We call a monomial ideal $J$ a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of~$I$. We prove that there exists a unique minimal monomial reduction ideal $J$ of $I$ and we show that the maximum degree of a monomial generator of $J$ determines the slope $p$ of the linear function $\reg(I^t)=pt+c$ for $t\gg 0$. We determine the structure of the reduced fiber ring $\mathcal{F}(J)_{\red}$ of $J$ and show that $\mathcal{F}(J)_{\red}$ is isomorphic to the inverse limit of an inverse system of semigroup rings determined by convex geometric properties of $J$.

In this paper, we study two `dual' problems in the operator space theory. We first show that if $L$ is a finite-dimensional operator space, then $L$ has the OLLP if and only if for any indexed family of operator spaces $(W_{i})_{i\in I}$ and a free ultrafilter $\mathcal{U}$ on $I$, we have a complete isometry \[ \prod(L\ha W_{i})/\mathcal{U}=L\ha\prod W_{i}/\mathcal{U}. \] Next, we show that if $W$ is an operator space, then $(T_{n}\ck W )^{**}=T_{n}\ck W^{**}$ holds if and only if $W$ is $\mathcal{T}$-locally reflexive, if and only if for any finitely representable operator spaces $V$, we have an isometry $\mathcal{I}(V, W^{*})=(V\ck W)^{*}$.

We prove that the ergodic bilinear Hilbert transform converges almost everywhere for pairs of bounded functions. We also give a proof along the same lines of Bourgain's analog result for averages.

Let $X$ be an infinite compact metric space with finite covering dimension. Let $\af,\bt: X\to X$ be two minimal homeomorphisms. Suppose that the range of $K_0$-groups of both crossed products are dense in the space of real affine continuous functions on the tracial state space. We show that $\af$ and $\bt$ are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.

In this paper we study Q-degrees of $n$-computably enumerable ($n$-c.e.) sets. It is proved that $n$-c.e. sets form a true hierarchy in terms of Q-degrees, and that for any $n\ge 1$ there exists a $2n$-c.e. Q-degree which bounds no noncomputable c.e. Q-degree, but any $(2n+1)$-c.e. non $2n$-c.e. Q-degree bounds a c.e. noncomputable Q-degree. Studying weak density properties of $n$-c.e. Q-degrees, we prove that for any $n\ge 1$, properly $n$-c.e. Q-degrees are dense in the ordering of c.e. Q-degrees, but there exist c.e. sets $A$ and $B$ such that $A-B<_QA\equiv_Q\emptyset'$, and there are no c.e. sets for which the Q-degrees are strongly between $A-B$ and $A$.

Let $E$ be a separable r.i. space which is an interpolation space between $L^{r}(\mathbb{R}^{+})$ and $L^{q}(\mathbb{R}^{+}),1<r<q<\infty$. We give sufficient conditions on $E$ implying that the symmetric operator space $E(\mathcal{M},\tau )$ has the $p$-Banach-Saks property for a suitable $p$, for an arbitrary semifinite von Neumann algebra $\mathcal{M}$.

We study and characterize laws of measurable processes and their convergence with general state space and parameter set. Using those results, it is shown that convergence of the prediction processes implies that of the given processes. We also give a simple condition for convergence of the prediction processes when the given processes are Markovian.

We answer a question of Cui and Zinsmeister by constructing a quasiconformal map $f$ of the disk to itself with dilatation $\mu$ such that map corresponding to dilatation $\sfrac 12 \mu$, maps the circle to a curve of dimension $> 1$.

In this note we present a new proof of the Carleson Embedding Theorem on the unit disc and unit ball in $\C^n$. The only technical tool used in the proof of this fact is Green's formula. The starting point is that every Carleson measure gives rise to a bounded subharmonic function. Using this function we construct a new related Carleson measure that allows for a simple embedding. In the case of the disc $\D$ this gives the best known constant, with the previous best given by N.~Nikolskii.

We give estimates of the sizes of certain unions of hyperplanes or of spheres.

We extend the well-known Moran-Hutchinson Theorem, which says that under the open set condition the Hausdorff dimension of self-similar sets coincides with the similarity dimension, to the more general setting of doubling metric spaces and prove its reverse implication. We also provide examples of iterated function systems in doubling metric spaces where our results apply.

Let $K$ be an algebraically closed field of characteristic zero and let $I=(f_1 \komdots f_n)$ be a homogeneous $R_+$-primary ideal in $R:=K[X,Y,Z]$. If the corresponding syzygy bundle $\Syz(f_1 \komdots f_n)$ on the projective plane is semistable, we show that the Artinian algebra $R/I$ has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection ($n=3$) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection ($n=4$) does not necessarily have the Weak Lefschetz property, answering negatively a question of Migliore and Miró-Roig. We prove that an almost complete intersection has the Weak Lefschetz property if the corresponding syzygy bundle is not semistable.

Braverman and Kappeler introduced a refinement of the Ray-Singer analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold. We study this notion and improve the Braverman-Kappeler theorem comparing the refined analytic torsion with the Farber-Turaev refinement of the combinatorial torsion. Using this result we establish, modulo sign, the Burghelea-Haller conjecture, comparing their complex analytic torsion with the Farber-Turaev torsion in the case when the flat connection can be deformed in the space of flat connections to a Hermitian connection. We then compute the refined analytic torsion of lens spaces and answer some of the questions posed in [BK1, Remark 14.9].

We compute lower bounds for the Morse index and nullity of constant mean curvature tori of revolution in the three-dimensional unit sphere. In particular, all such tori have index at least five, with index growing at least linearly with respect to the number of the surfaces' bulges, and the index of such tori can be arbitrarily large.

We investigate some extremal cases of exactness constants and completely bounded projection constants. More precisely, for an $n$-dimensional operator space $E$ we prove that $\lambda_{cb}(E) = \sqrt{n}$ if and only if $\ex(E) = \sqrt{n}$.

It is known that, in a complete Riemannian manifold $(M, g)$, if the Hessian of a real valued function satisfies some suitable conditions, then it restricts the geometry of $(M, g)$. In this paper we give a characterization of a certain class of Kähler manifolds admitting a real valued function $u$ such that the Hessian has two eigenvalues $u$ and $\frac{1+u}{2}$.

The Szegö kernel $S(z,\zeta)$ on the boundary of strictly pseudoconvex domains has been studied extensively. We can consider model domains $\Omega = \{ (z_1,z_2) \in \mathbb{C}^2 \mid -\Im z_2 > b(\Re z_1)\}$. If $b$ is convex, one has $|S(z,\zeta)| \le c|B(z,\delta)|^{-1}$, where $B(z,\delta)$ is the nonisotropic ball with center $z$ and radius $\delta$, and $\delta $ is the nonisotropic distance from $z$ to $\zeta$. The only singularities are on the diagonal $z=\zeta$. In this paper, we obtain estimates for $|S|$ when the function $b$ is a certain non-convex function. We show that near certain points, there are singularities off the diagonal.

We determine a term order on the monomials in the variables $\varx{i}{j}$, $1 \leq i < j \leq n$, such that corresponding initial ideal of the ideal of Pfaffians of degree $r$ of a generic $n$ by $n$ skew-symmetric matrix is the Stanley-Reisner ideal of a join of a simplicial sphere and a simplex. Moreover, we demonstrate that the Pfaffians of the $2r$ by $2r$ skew-symmetric submatrices form a Gröobner basis for the given term order. The same methods and similar term orders as for the Pfaffians also yield squarefree initial ideals for certain determinantal ideals. Yet, in contrast to the case of Pfaffians, the corresponding simplicial complexes are balls that do not decompose into a join as above.

Suppose that $Y_t$ is a $d$-dimensional symmetric Lévy process such that its Lévy measure differs from the Lévy measure of the isotropic $\alpha$-stable process ($0 < \alpha < 2$) by a finite signed measure. For a bounded Lipschitz open set $D$ we compare the Green functions of the process $Y$ with those of its stable counterpart, and we prove several comparability results, both one-sided and two-sided. In particular, assuming an additional condition about the difference between the densities of the Lévy measures, namely that it is of the order of $|x|^{-d+\varrho}$ as $|x|\to 0$, where $\varrho > 0$, we prove that the Green functions are comparable, provided $D$ is connected. These results apply, for example, to the relativistic $\alpha$-stable process. The bounds for its Green functions were previously known for $d > \alpha$ and smooth sets. Here we consider also the one-dimensional case for $\alpha \ge 1$, and we prove that the Green functions for a bounded open interval are comparable, a case that, to the best of our knowledge, had not been treated in the literature.

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring $(R,\fm)$ with $\fm^3=0$. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes, and new sufficient conditions are given for total acyclicity. Results are also obtained on the structure of rings that are not Gorenstein and admit acyclic complexes; part of this structure is exhibited by every ring $R$ that admits a non-free finitely generated module $M$ with $\Ext{n}{R}{M}{R}=0$ for a few $n>0$.