Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Lattice homomorphisms between Sobolev spaces

We show in Theorem 4.4 that every vector lattice homomorphism T from \({\mathsf{W}^{1,p}_0(\Omega_1)}\) into \({\mathsf{W}^{1,q}(\Omega_2)}\) for \({p,q\in (1,\infty)}\) and open sets \({\Omega_1,\Omega_2\subset\mathbb{R}^N}\) has a representation of the form \({T\mathsf{u}=(\mathsf{u}\circ\xi)g}\) (Cap _q -quasi everywhere on Ω₂) with mappings ξ : Ω₂ → Ω₁ and g : Ω₂ → [0, ∞). This representation follows as an application of an abstract and more general representation theorem (Theorem 3.5). Other applications are also given.     

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

In this paper the authors study the boundedness for a large class of sublinear operators \({T_{\alpha}, \alpha \in [0,n)}\) generated by Calderón–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces \({M_{p,\varphi}}\) . As an application of the above result, the boundeness of the commutator of sublinear operators \({T_{b,\alpha}, \alpha \in [0,n)}\) on generalized Morrey spaces is also obtained. In the case \({b \in BMO}\) and T _b,α is a sublinear operator, we find the sufficient conditions on the pair \({(\varphi_1,\varphi_2)}\) which ensures the boundedness of the operators \({T_{b,\alpha}, \alpha \in [0,n)}\) from one generalized Morrey space \({M_{p,\varphi_1}}\) to another \({M_{q,\varphi_2}}\) with 1/p ? 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({(\varphi_1,\varphi_2)}\) , which do not assume any assumption on monotonicity of \({\varphi_1, \, \varphi_2}\) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.     

On a two-phase free boundary condition for <Emphasis Type="Italic">p</Emphasis>-harmonic measures

Let \({\Omega^i\subset {\bf R}^n, i\in\{1,2\}}\) , be two (δ, r ₀)-Reifenberg flat domains, for some \({0 < \delta < \hat \delta}\) and r ₀ > 0, assume \({\Omega^1\cap\Omega^2=\emptyset}\) and that, for some \({w\in {\bf R}^n}\) and some 0 < r, \({w\in\partial\Omega^1\cap\partial\Omega^2, \partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . Let p, 1 < p < ∞, be given and let u ⁱ , \({i\in\{1,2\}}\) , denote a non-negative p-harmonic function in Ω ⁱ , assume that u ⁱ , \({i\in\{1,2\}}\), is continuous in \({\bar\Omega^i\cap B(w,2r) }\) and that u ⁱ  = 0 on \({\partial\Omega^i\cap B(w,2r)}\) . Extend u ⁱ to B(w, 2r) by defining \({u^i\equiv 0}\) on \({B(w,2r) {\setminus} \Omega^i}\). Then there exists a unique finite positive Borel measure μ ⁱ , \({i\in\{1,2\}}\) , on R ⁿ , with support in \({\partial\Omega^i\cap B(w,2r)}\) , such that if \({\phi \in C_0^\infty (B(w,2r))}\) , then

$\int\limits_{\mathbf R^n} \,|\nabla u^i|^{ p-2} \,\langle \nabla u^i, \,\nabla \phi \rangle \,dx =- \int\limits_{\mathbf R^n} \,\phi \,d \mu^i.$

Let \({\Delta(w,2r)=\partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . The main result proved in this paper is the following. Assume that μ ² is absolutely continuous with respect to μ ¹ on Δ(w, 2r), d μ ² = kd μ ¹ for μ ¹-almost every point in Δ(w, 2r) and that \({\log k\in VMO(\Delta(w,r),\mu^1)}\) . Then there exists \({\tilde \delta = \tilde \delta(p,n) > 0}\) , \({\tilde \delta < \hat \delta}\) , such that if \({\delta\leq\tilde\delta}\) , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

     

A note on standard modules and Vogan L-packets

Let F be a non-Archimedean local field of characteristic 0, let G be the group of F-rational points of a connected reductive group defined over F and let \({G\prime}\) be the group of F-rational points of its quasi-split inner form. Given standard modules \({I(\tau, \nu )}\) and \({I(\tau\prime, \nu\prime)}\) for G and \({G\prime}\) respectively with \({\tau\prime}\) a generic tempered representation, such that the Harish-Chandra \({\mu}\)-function of a representation in the supercuspidal support of \({\tau}\) agrees with the one of a generic essentially square-integral representation in some Jacquet module of \({\tau\prime}\) (after a suitable identification of the underlying spaces under which \({\nu = \nu\prime}\)), we show that \({I(\tau, \nu)}\) is irreducible whenever \({I(\tau\prime, \nu\prime)}\) is. The conditions are satisfied if the Langlands quotients \({J(\tau, \nu})\) and \({J(\tau\prime, \nu\prime)}\) of respectively \({I(\tau, \nu)}\) and \({I(\tau\prime, \nu\prime)}\) lie in the same Vogan L-packet (whenever this Vogan L-packet is defined), proving that, for any Vogan L-packet, all the standard modules with Langlands quotient in a given Vogan L-packet are irreducible, if and only if this Vogan L-packet contains a generic representation. This result for generic Vogan L-packets was proven for quasi-split orthogonal and symplectic groups by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these groups.     

Isoperimetric inequality for the polydisk

Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E ^r (Ω ⁿ ) be the holomorphic Smirnov class on Ω ⁿ with index r. We show that the generalized isoperimetric inequality

$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $

holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.

     

On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation

$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$

where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form

$a_\vartheta = a+b_\vartheta-b,$

where a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods

$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$

as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ _T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form

$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$

where \({H\in[1/2,1]}\) and B ^H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ _T  = T ^?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.

     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

Perturbation from symmetry for indefinite semilinear elliptic equations

We prove the existence of an unbounded sequence of solutions for an elliptic equation of the form \({-\Delta u=\lambda u + a(x)g(u)+f(x), u\in H^1_0(\Omega)}\), where \({\lambda \in \mathbb{R}, g(\cdot)}\) is subcritical and superlinear at infinity, and a(x) changes sign in Ω; moreover, g( ? s) =  ? g(s) \({\forall s}\). The proof uses Rabinowitz’s perturbation method applied to a suitably truncated problem; subsequent energy and Morse index estimates allow us to recover the original problem. We consider the case of \({\Omega\subset \mathbb{R}^N}\) bounded as well as \({\Omega=\mathbb{R}^N, \, N\geqslant 3}\).     

Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent {p > 2}

Let \({L(n)}\) be the language of group theory with n additional new constant symbols \({c_1,\ldots,c_n}\). In \({L(n)}\) we consider the class \({{\mathbb{K}}(n)}\) of all finite groups G of exponent \({p > 2}\), where \({G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z(G)}\) and \({c_1^G,\ldots,c_n^G}\) are linearly independent. Using amalgamation we show the existence of Fraïssé limits \({D(n)}\) of \({{\mathbb{K}}(n)}\). \({D(1)}\) is Felgner’s extra special p-group. The elementary theories of the \({D(n)}\) are supersimple of SU-rank 1. They have the independence property.     

Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

Set \({T=N^{\frac{1}{3}-\epsilon}}\). It is proved that for all but \({\ll TL^{-H},\,H > 0}\), exceptional prime numbers \({k\leq T}\) and almost all integers b ₁, b ₂ co-prime to k, almost all integers \({n\sim N}\) satisfying \({n\equiv b_{1}+b_{2}(mod\,k)}\) can be written as the sum of two primes p ₁ and p ₂ satisfying \({p_{i}\equiv b_{i}(mod\,k),\,i=1,2}\). For prime numbers \({k\leq N^{\frac{5}{24}-\epsilon}}\), this result is even true for all but \({\ll (\log\,N)^{D}}\) primes k and all integers b ₁, b ₂ co-prime to k.     

Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.     

Symmetry for a general class of overdetermined elliptic problems

Let \({\Omega}\) a bounded domain in \({\mathbb{R} ^N }\), and let \({u\in C^1 (\overline{\Omega})}\) a weak solution of the following overdetermined BVP: \({-\nabla (g(|\nabla u|)|\nabla u|^{-1}\nabla u)=f(|x|,u)}\), \({ u > 0 }\) in \({\Omega }\) and \({u=0, \ |\nabla u(x)|=\lambda (|x|)}\) on \({\partial \Omega }\), where \({g\in C([0,+\infty)\cap C^1 ((0,+\infty ) ) }\) with \({g(0)=0}\), \({g'(t) > 0}\) for \({t > 0}\), \({f\in C([0,+\infty ) \times [0, +\infty ) )}\), f is nonincreasing in \({|x|}\), \({\lambda \in C([0, +\infty )) }\) and \({\lambda }\) is positive and nondecreasing. We show that \({\Omega }\) is a ball and u satisfies some “local” kind of symmetry. The proof is based on the method of continuous Steiner symmetrization.     

Limits of Besov norms

Besov spaces \({{\mathbf B}^s_{p,q} ({\mathbb R}^n)}\) with s > 0 can be normed in terms of the differences \({\Delta^m_h f}\) and related moduli of smoothness ω _m (f, t) _p , where \({0 < s < m \in {\mathbb N}}\). The paper deals with the question what happens if \({s {\uparrow} m}\) and how the outcome is related to the Sobolev spaces \({{\mathbf W}^m_p ({\mathbb R}^n)}\).     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

Evolution of curvatures on a surface with boundary to prescribed functions

Let (M, g ₀) be a compact Riemann surface with boundary and with negative Euler characteristic. Let f(x) be a strictly negative smooth function on \({\bar{M}}\) and denote by \({\sigma(x)}\) the value of f in the interior and \({\zeta(x)}\) the value of f on the boundary. By studying the evolution of curvatures on M, we prove that there exist a constant \({\lambda_\infty}\) and a conformal metric \({g_\infty}\) such that \({\lambda_\infty\sigma(x)}\) and \({\lambda_\infty\zeta(x)}\) can be realized as the Gaussian curvature and boundary geodesic curvature of \({g_\infty}\) respectively.     

Lifting Fixed Points of Completely Positive Semigroups

Let \({\{\phi_s\}_{s\in S}}\) be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra N. Assume there exists a semigroup \({\{\alpha_s\}_{s\in S}}\) of weak*-continuous *-endomorphisms of some larger von Neumann algebra \({M\supset N}\) and a projection \({p\in M}\) with N = pMp such that α _s (1 ? p) ≤ 1 ? p for every \({s\in S}\) and \({\phi_s(y)=p\alpha_s(y)p}\) for all \({y\in N}\). If \({\inf_{s \in S}\alpha_s(1-p)=0}\) then we show that the map \({E:M\to N}\) defined by E(x) = pxp for \({x\in M}\) induces a complete isometry between the fixed point spaces of \({\{\alpha_s\}_{s\in S}}\) and \({\{\phi_s\}_{s\in S}}\).     

Strichartz estimates via the Schrödinger maximal operator

We consider the Schrödinger operator \({e^{it\Delta}}\) acting on initial data f in \({\dot{H}^s}\). We show that an affirmative answer to a question of Carleson, concerning the sharp range of s for which \({\lim_{t\to 0}e^{it\Delta}f(x)=f(x)}\) a.e. \({x\in \mathbb {R}^n}\), would imply an affirmative answer to a question of Planchon, concerning the sharp range of q and r for which \({e^{it\Delta}}\) is bounded in \({L_x^q(\mathbb {R}^n,L^r_t(\mathbb {R}))}\). When n  =  2, we unconditionally improve the range for which the mixed norm estimates hold.