We show that for any length-compact metric space $Y$ and any $1 \lt q\leq n$, there is a continuous surjection in a suitably defined Sobolev–Lorentz space space $W^{1,n,q}([0,1]^n, Y)$. On the other hand, we show that mappings in the space $W^{1,n,1}([0,1]^n, Y)$ satisfy condition (N). This implies that the target $Y$ can be at most $n$-dimensional.

Let $k \geq2$ and let $f$ be meromorphic in the unit disc $\Delta$, such that $f(z) f^{(k)}(z) \neq0$ for all $z \in\Delta$ and the poles of $f$ in $\Delta$ have bounded multiplicities. Then $f $ has asymptotic values on a dense subset of $\partial\Delta$.

If $T$ is the operator given by convolution with surface measure on the sphere, $(E,F)$ is a quasi-extremal pair of sets for $T$ if $\langle T\chi_E, \chi_F \rangle\gtrsim|E|^{d/(d+1)}|F|^{d/(d+1)}$. In this article, we explicitly define a family $\mathcal{F}$ of quasi-extremal pairs of sets for $T$. We prove that $\mathcal{F}$ is fundamental in the sense that every quasi-extremal pair $(E,F)$ is comparable (in a rather strong sense) to a pair from $\mathcal{F}$. This extends work carried out by Christ for convolution with surface measure on the paraboloid.

In this paper, we prove that the Stanley–Reisner ideal of any connected simplicial complex of dimension $\ge2$ that is locally complete intersection is a complete intersection ideal.

As an application, we show that the Stanley–Reisner ideal whose powers are Buchsbaum is a complete intersection ideal.

We prove existence and uniqueness of the solution of the Björling problem for minimal surfaces in a three-dimensional Lie group.

Let $0\rightarrow I\rightarrow A\rightarrow A/I \rightarrow 0$ be a short exact sequence of $C^*$-algebras with $A$ unital. Suppose that $I$ has tracial topological rank no more than one and $A/I$ is TAI (in particular, if $A/I$ is simple and has tracial topological rank no more than one). It will be proved that $A$ has tracial topological rank no more than one if the extension is quasidiagonal, and $A$ has the property ($P_1$) if the extension is tracially quasidiagonal.

In this paper, we prove some noncommutative analogues of Yano’s classical extrapolation theorem. Applying one of them to noncommutative martingales, we obtain a maximal inequality for noncommutative martingales from $L\log^2L$ to $L_1$. Moreover, the exponent $2$ is optimal. We also obtain the noncommutative analogue of the classical theorem of Burkholder and Chow on the iterations of two conditional expectations.

We prove that seesaw words exist in Thompson's group $F(N)$ for $N=2,3,4,\ldots$ with respect to the standard finite generating set $X$. A seesaw word $w$ with swing $k$ has only geodesic representatives ending in $g^k$ or $g^{-k}$ (for given $g\in X$) and at least one geodesic representative of each type. The existence of seesaw words with arbitrarily large swing guarantees that $F(N)$ is neither synchronously combable nor has a regular language of geodesics. Additionally, we prove that dead ends (or $k$-pockets) exist in $F(N)$ with respect to $X$ and all have depth 2. A dead end $w$ is a word for which no geodesic path in the Cayley graph $\Gamma$ which passes through $w$ can continue past $w$, and the depth of $w$ is the minimal $m\in\mathbb{N}$ such that a path of length $m+1$ exists beginning at $w$ and leaving $B_{|w|}$. We represent elements of $F(N)$ by tree-pair diagrams so that we can use Fordham's method for computing word length. This paper generalizes results by Cleary and Taback, who proved the case $N=2$.

We describe the growth of the naturally defined argument of a bounded analytic function in the unit disk in terms of the complete measure introduced by A. Grishin. As a consequence, we characterize the local behavior of a logarithm of an analytic function. We also find necessary and sufficient conditions for closeness of $\log f(z)$, $f\in H^\infty$, and the local concentration of the zeros of $f$.

Let $E$ be a subset in $(n+1)$-dimensional Euclidian space with parabolic homogeneity, codimension $1$, and with an appropriate surface measure $\sigma$ associated to it. We define a parabolic version of Corona decomposition of $E$ and establish two results on sufficient conditions for the existence of parabolic Corona decomposition for $E$. Both results are parabolic versions of well-known results due to G. David and S. Semmes.

The connected groups acting by isometries on either the real or the complex hyperbolic spaces are determined. A Lie-theoretic description of the homogeneous Riemannian, respectively Kähler, structures of linear type on these spaces is then found. On both spaces, examples that are not of linear type are given.

In this paper, we study the extension of isometric operator between unit spheres of normed spaces, and give an equivalent statement of Tingley problem. We also give another statement of Mazur–Ulam theorem: Let $V : E\rightarrow F$ be an isometric operator, and $V|_{S(E)}$ denotes the operator $V$ restricted to the set $S(E)$. If $V|_{S(E)}$ is an onto isometric operator from $S(E)$ to $S(F)$, then $V$ must be linear.

A hypersurface $M_1^3$ in the four-dimensional pseudo-Euclidean space $E_1^4$ is called a Lorentz hypersurface if its normal vector is space-like. We show that if the mean curvature vector field of $M_1^3$ satisfies the equation $\Delta\vec H=\alpha\vec H$ ($\alpha$ a constant), then $M_1^3$ has constant mean curvature. This equation is a natural generalization of the biharmonic submanifold equation $\Delta\vec H=\vec0$.

We give a characterization of the Lefschetz elements in Artinian Gorenstein rings over a field of characteristic zero in terms of the higher Hessians. As an application, we give new examples of Artinian Gorenstein rings which do not have the strong Lefschetz property.

Let $f$ be a transcendental meromorphic function such that all but finitely many of the poles of $f$ and zeroes of $f'$ are real. Generalising a result of Hinkkanen and Rossi (Proc. Amer. Math. Soc. 92 (1984) 72–74), we characterize those $f$ such that $f'$ takes some nonzero value only finitely often, and show that all but finitely many of the zeroes of $f''$ are real in this case. We also prove a related asymptotic result about real meromorphic functions with a nonzero deficient value $\alpha$ and only finitely many nonreal zeroes, poles and $\alpha$-points.

We study solid sequence $F$-spaces $\lambda_{0}(\eta)$, nonseparable in general, and their closed separable subspaces $\lambda_{00}(\eta)$. The space $\lambda_{0}(\eta)$ is associated with a strictly positive submeasure $\eta$ on $\mathbb{N}$ and equipped with the topology of convergence in submeasure. While $\lambda_{0}(\eta)$'s may be viewed as analogs of usual $L_0$-spaces, the relation between $\lambda_{00}(\eta)$ and $\lambda_{0}(\eta )$ often resembles that between $c_0$ and $l_\infty$. For many $\eta$'s, the weak topology of these spaces coincides with that of coordinate-wise convergence, they are not locally pseudoconvex and yet have the Bounded Multiplier Property. Further, in agreement with the analogy to $L_0=L_0[0,1]$, they possess copies of $l_p$ for $0 \lt p \leq 2$, and yet in contrast to $L_0$ they contain a lot of well-located copies of $c_0$ and $l_\infty$; also, the quotient $\lambda_{0}(\eta)/\lambda_{00}(\eta)$ contains a copy of $L_0$. All of this happens already for the spaces $\lambda_{0}=\lambda _{0}(\bar d)$ and $\lambda_{00}=\lambda_{00}(\bar d)$ with $\bar d$ being a submeasure closely related to the standard density $d$, in which case, moreover: (1) There is a series in $\lambda_{00}$ all of whose subseries of density zero are convergent, and yet its partial sums are unbounded. (2) The Orlicz–Pettis theorem fails in $\lambda_{0}$. (3) $\lambda_{00}$ can be used to show that some earlier constructed normed barrelled spaces are not ultrabarrelled.

For a relatively compact domain $M$ in a complex manifold, we completely characterize in terms of Jensen measures the annihilating measures of the algebra $A(\overline M)$ of holomorphic functions and the space $h(\overline M)$ of pluriharmonic functions continuous on $\overline M$. We also establish the equivalence of Mergelyan type approximation properties of a domain for different function spaces.

A metric space $\mathcal{S}$ is called a quasisphere if there is a quasisymmetric homeomorphism $f : S^2\to\mathcal{S}$. We consider the elliptic harmonic measure, i.e., the push forward of $2$-dimensional Lebesgue measure by $f$. It is shown that for certain self similar quasispheres $\mathcal{S}$ (snowspheres) the dimension of the elliptic harmonic measure is strictly less than the Hausdorff dimension of $\mathcal{S}$. This result is obtained by representing the self similarity of a snowsphere by a postcritically finite rational map, and showing a corresponding result for such maps. As a corollary a metric characterization of Lattès maps is obtained. Furthermore, a method to compute the dimension of elliptic harmonic measure numerically is presented, along with the (numerically computed) values for certain examples.