CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into $W^{m,p}\left(\Omega \right)$ into $L^S\left(\Omega \right)$ (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding $W^{m,p}\left(\Omega \right)$ into $L^{\phi }\left(\Omega \right)$...............................................................

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s ≥ 3. Also let ${\Omega }_{\pi }$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f:\partial {\Omega }_{\pi }\to ℝ$ that can be extended to a real-valued pluriharmonic function in ${\Omega }_{\pi }$.

Let ${\left(F_n\right)}_{n\in \mathbb{N}}$ be a sequence of non-decreasing functions from $[0,+\infty )$ into $[0,+\infty )$. Under some suitable hypotheses of ${\left(F_n\right)}_{n\in \mathbb{N}}$, we will prove that if $g\in L^p\left({ℝ}^N\right)$, $1<p<+\infty$, satisfies ${lim\; inf}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy<+\infty$, then $g\in W^{1,p}\left({ℝ}^N\right)$ and moreover ${lim}_{n\to \infty }{\int }_{{ℝ}^N}{\int }_{{ℝ}^N}{F}_n{\left(\right|g\left(x\right)-g\left(y\right)\left|\right)/|x-y|}^{N+p}dxdy=K_{N,p}{\int }_{{ℝ}^N}{\left|\nabla g\left(x\right)\right|}^{p}dx$, where $K_{N,p}$ is a positive constant depending only on $N$ and $p$. This extends some results in J. Bourgain and H-M. Nguyen [A new characterization of Sobolev spaces, C. R. Acad Sci. Paris, Ser. 343 (2006) 75-80] and H-M. Nguyen [Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689-720]. We also present some...

Let ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ be the following algorithmic problem: Given a finite simplicial complex $K$ of dimension at most $k$, does there exist a (piecewise linear) embedding of $K$ into ${ℝ}^d$? Known results easily imply polynomiality of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2}$ ($k=1,2$; the case $k=1,d=2$ is graph planarity) and of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to 2k}$ for all $k\ge 3$. We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{d\to d}$ and ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{(d-1)\to d}$ are undecidable for each ${}_d\ge 5$. Our main result is NP-hardness of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{2\to 4}$ and, more generally, of ${\mathrm{𝙴𝙼𝙱𝙴𝙳}}_{k\to d}$ for all...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

In this note we shall prove that for a continuous function $\varphi :\Delta \to {ℝ}^n$, where $\Delta \subset ℝ$,  the paratingent of $\varphi$ at $a\in \Delta$ is a non-empty and compact set in ${ℝ}^n$ if and only if $\varphi$ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Let ${ℱ}_h^i(k,n)$ be the $i$-th ordered configuration space of all distinct points $H_1,...,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${ℂ}^n$, whose sum is a subspace of dimension $i$. We prove that ${ℱ}_h^i(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\&gt;2$ and equal to the braid group of the sphere $ℂ$ $P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

CONTENTSPREFACE..........................................................................................................................................................................3INTRODUCTION............................................................................................................................................................. 41. Notation. 2. Subject of the paper.Chapter I. DECOMPOSITION OF Σ INTO ${\Sigma }_1$, ${\Sigma }_2$, ${\Sigma }_3$, ${\Sigma }_4$ INESSENTIAL RESTRICTIONOF GENERALITY ...............................................................................................................................................................

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1<p<\infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1\le p<\infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a “common roof”...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...