Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

Quasi-Banach spaces of almost universal disposition

We show that for each p∈(0,1]$p\in (0,1]$ there exists a separable p  -Banach space _Gp${\mathbb{G}}_p$ of almost universal disposition, that is, having the following extension property: for each ε>0$\epsilon >0$ and each isometric embedding g:X→Y$g:X\to Y$, where Y is a finite-dimensional p-Banach space and X   is a subspace of _Gp${\mathbb{G}}_p$, there is an ε  -isometry f:Y→_Gp$f:Y\to {\mathbb{G}}_p$ such that x=f(g(x))$x=f(g(x))$ for all x∈X$x\in X$.     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

Weakly compact operators and strict topologies

Let X   be a completely regular Hausdorff space and _Cb(X)$C_b(X)$ be the Banach space of all real-valued bounded continuous functions on X, endowed with the uniform norm. It is shown that every weakly compact operator T   from _Cb(X)$C_b(X)$ to a quasicomplete locally convex Hausdorff space E   can be uniquely decomposed as T=_T1+_T2+_T3+_T4$T=T_1+T_2+T_3+T_4$, where _Tk:_Cb(X)→E$T_k:C_b(X)\to E$(k=1,2,3,4)$(k=1,2,3,4)$ are weakly compact operators, and _T1$T_1$ is tight, _T2$T_2$ is purely τ  -additive, _T3$T_3$ is purely σ  -additive and _T4$T_4$ is purely finitely additive. Moreover, we derive a generalized Yosida–Hewitt decomposition for E-valued strongly bounded regular Baire measures.     

Normal subgroups of profinite groups of non-negative deficiency

The principal goal of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group G of non-negative deficiency gives rather strong consequences for the structure of G. To make this precise we introduce the notion of p-deficiency (p a prime) for a profinite group G. We prove that if the p-deficiency of G is positive and N is a finitely generated normal subgroup such that the p  -Sylow subgroup of G/N$G/N$ is infinite and p divides the order of N   then we have _cdp(G)=2${\mathrm{cd}}_p(G)=2$, _cdp(N)=1${\mathrm{cd}}_p(N)=1$ and _vcdp(G/N)=1${\mathrm{vcd}}_p(G/N)=1$ for the cohomological p-dimensions; moreover either the p  -Sylow subgroup of G/N$G/N$ is virtually cyclic or the p-Sylow subgroup of N is cyclic. If G is a profinite Poincaré duality group of dimension 3 at a prime p   (PD³${\mathit{PD}}^3$-group at p) we show that for N and p as above either N   is PD¹${\mathit{PD}}^1$ at p   and G/N$G/N$ is virtually PD²${\mathit{PD}}^2$ at p or N   is PD²${\mathit{PD}}^2$ at p   and G/N$G/N$ is virtually PD¹${\mathit{PD}}^1$ at p.     

The Zassenhaus filtration,Massey products,and representations of profinite groups

We consider the p  -Zassenhaus filtration (_Gn)$(G_n)$ of a profinite group G  . Suppose that G=S/N$G=S/N$ for a free profinite group S and a normal subgroup N of S   contained in _Sn$S_n$. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p  -cohomological dimension ≤ 1), we prove that _Gn+1$G_{n+1}$ is the intersection of all kernels of upper-triangular unipotent (n+1)$(n+1)$-dimensional representations of G   over _Fp${\mathbb{F}}_p$. This extends earlier results by Miná?, Spira, and the author on the structure of absolute Galois groups of fields.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Best constants in a borderline case of second-order Moser type inequalities

We study optimal embeddings for the space of functions whose Laplacian Δu   belongs to L¹(Ω)$L^1(\Omega )$, where Ω⊂RN$\Omega \subset {\mathbb{R}}^N$ is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω)$W^{2,1}(\Omega )$ in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2$N=2$, we establish a sharp embedding inequality into the Zygmund space Lexp(Ω)$L_{\mathit{exp}}(\Omega )$. On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)∈L¹(Ω)$\mathrm{\Delta }u=f(x)\in L^1(\Omega )$, u=0$u=0$ on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N?3$N?3$ are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.