A class of \ell ^p saturated Banach spaces

We present a new class of reflexive \(\ell ^p\) saturated Banach spaces \(\mathfrak{X }_p\) for \(1<p<\infty \) with rather tight structure. The norms of these spaces are defined with the use of a modification of the standard method yielding hereditarily indecomposable Banach spaces. The space \(\mathfrak{X }_p\) does not embed into a space with an unconditional basis and for any analytic decomposition into two subspaces, it is proved that one of them embeds isomorphically into the \(\ell ^p\) -sum of a sequence of finite dimensional normed spaces. We also study the space of operators of \(\mathfrak{X }_p\) .     

Pointwise convergence in Pringsheim’s sense of the summability of Fourier transforms on Wiener amalgam spaces

New multi-dimensional Wiener amalgam spaces \(W_c(L_p,\ell _\infty )(\mathbb{R }^d)\) are introduced by taking the usual one-dimensional spaces coordinatewise in each dimension. The strong Hardy-Littlewood maximal function is investigated on these spaces. The pointwise convergence in Pringsheim’s sense of the \(\theta \) -summability of multi-dimensional Fourier transforms is studied. It is proved that if the Fourier transform of \(\theta \) is in a suitable Herz space, then the \(\theta \) -means \(\sigma _T^\theta f\) converge to \(f\) a.e. for all \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) . Note that \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset W_c(L_r,\ell _\infty )(\mathbb{R }^d) \supset L_r(\mathbb{R }^d)\) and \(W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d) \supset L_1(\log L)^{d-1}(\mathbb{R }^d)\) , where \(1 . Moreover, \(\sigma _T^\theta f(x)\) converges to \(f(x)\) at each Lebesgue point of \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) .     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Composition operators acting on Besov spaces on the real line

We study the composition operator \(T_f(g):= f\circ g\) on Besov spaces \(B_{{p},{q}}^{s}(\mathbb{R })\) . In case \(1 < p< +\infty ,\, 0< q \le +\infty \) and \(s>1+ (1/p)\) , we will prove that the operator \(T_f\) maps \(B_{{p},{q}}^{s}(\mathbb{R })\) to itself if, and only if, \(f(0)=0\) and \(f\) belongs locally to \(B_{{p},{q}}^{s}(\mathbb{R })\) . For the case \(p=q\) , i.e., in case of Slobodeckij spaces, we can extend our results from the real line to \(\mathbb{R }^n\) .     

Stationary and invariant densities and disintegration kernels

Let \(G\) be a locally compact topological group, acting measurably on some Borel spaces \(S\) and \(T\) , and consider some jointly stationary random measures \(\xi \) on \(S\times T\) and \(\eta \) on \(S\) such that \(\xi (\cdot \times T)\ll \eta \) a.s. Then there exists a stationary random kernel \(\zeta \) from \(S\) to \(T\) such that \(\xi =\eta \otimes \zeta \) a.s. This follows from the existence of an invariant kernel \(\varphi \) from \(S\times {\mathcal {M}}_{S\times T}\times {\mathcal {M}}_S\) to \(T\) such that \(\mu =\nu \otimes \varphi (\cdot ,\mu ,\nu )\) whenever \(\mu (\cdot \times T)\ll \nu \) . Also included are some related results on stationary integration, absolute continuity, and ergodic decomposition.     

The Cauchy–Riemann equations in the unit ball of l^{2}

We study the local exactness of the \(\overline{\partial }\) operator in the Hilbert space \(l^2\) for a particular class of \((0,1)\) -forms \(\omega \) of the type \(\omega (z) = \sum _i z_i\omega ^i(z) d\overline{z}_i\) , \(z = (z_i)\) in \(l^2\) . We suppose each function \(\omega ^i\) of class \(C^\infty \) in the closed unit ball of \(l^2\) , of the form \(\omega ^i(z) = \sum _k \omega ^i_k\left( z^k\right) \) , where \(\mathbf N = \bigcup I_k\) is a partition of \(\mathbf N\) , \((\) card \(I_k < +\infty )\) and \(z^k\) is the projection of \(z\) on \(\mathbf C^{I_k}\) . We establish sufficient conditions for exactness of \(\omega \) related to the expansion in Fourier series of the functions \(\omega ^i_k\) .     

A doubling subset of L_p for p>2 that is inherently infinite dimensional

It is shown that for every \(p\in (2,\infty )\) there exists a doubling subset of \(L_p\) that does not admit a bi-Lipschitz embedding into \(\mathbb R^k\) for any \(k\in \mathbb N\) .     

Square functions in the Hermite setting for functions with values in UMD spaces

In this paper, we characterize the Lebesgue Bochner spaces \(L^p({\mathbb{R }}^{n},B),\, 1 , by using Littlewood–Paley \(g\) -functions in the Hermite setting, provided that \(B\) is a UMD Banach space. We use \(\gamma \) -radonifying operators \(\gamma (H,B)\) where \(H=L^2((0,\infty ),\frac{\mathrm{d}t}{t})\) . We also characterize the UMD Banach spaces in terms of \(L^p({\mathbb{R }}^{n},B)-L^p({\mathbb{R }}^{n},\gamma (H,B))\) boundedness of Hermite Littlewood–Paley \(g\) -functions.     

Hyperreflexivity of bounded N-cocycle spaces of banach algebras

Let \(X\) and \(Y\) be Banach spaces, \(n\in \mathbb {N}\) , and \(B^n(X,Y)\) the space of bounded \(n\) -linear maps from \(X\times \ldots \times X\) ( \(n\) -times) into \(Y\) . The concept of hyperreflexivity has already been defined for subspaces of \(B(X,Y)\) , where \(X\) and \(Y\) are Banach spaces. We extend this concept to the subspaces of \(B^n(X,Y)\) , taking into account its \(n\) -linear structure. We then investigate when \(\mathcal {Z}^n(A,X)\) , the space of all bounded \(n\) -cocycles from a Banach algebra \(A\) into a Banach \(A\) -bimodule \(X\) , is hyperreflexive. Our approach is based on defining two notions related to a Banach algebra, namely the strong property \((\mathbb {B})\) and bounded local units, and then applying them to find uniform criterions under which \(\mathcal {Z}^n(A,X)\) is hyperreflexive. We also demonstrate that these criterions are satisfied in variety of examples including large classes of C \(^*\) -algebras and group algebras and thereby providing various examples of hyperreflexive \(n\) -cocyle spaces. One advantage of our approach is that not only we obtain the hyperreflexivity for bounded \(n\) -cocycle spaces in different cases but also our results generalize the earlier ones on the hyperreflexivity of bounded derivation spaces, i.e. when \(n=1\) , in the literature. Finally, we investigate the hereditary properties of the strong property \((\mathbb {B})\) and b.l.u. This allows us to come with more examples of bounded \(n\) -cocycle spaces which are hyperreflexive.     

Representations of set-valued risk measures defined on the l-tensor product of Banach lattices

We obtain a representation for set-valued risk measures which are defined on the completed \(l\) -tensor product \(E\widetilde{\otimes }_l G\) of Banach lattices \(E\) and \(G\) . This representation extends known representations for set-valued risk measures defined on Bochner spaces \(L^p(\mathbb {P}, \mathbb {R}^d)\) of \(p\) -integrable functions with values in \(\mathbb {R}^d\) .     

Intrinsically p-biharmonic maps

For a compact Riemannian manifold \(N\) , a domain \(\Omega \subset \mathbb {R}^m\) and for \(p\in (1, \infty )\) , we introduce an intrinsic version \(E_p\) of the \(p\) -biharmonic energy functional for maps \(u : \Omega \rightarrow N\) . This requires finding a definition for the intrinsic Hessian of maps \(u : \Omega \rightarrow N\) whose first derivatives are merely \(p\) -integrable. We prove, by means of the direct method, existence of minimizers of \(E_p\) within the corresponding intrinsic Sobolev space, and we derive a monotonicity formula. Finally, we also consider more general functionals defined in terms of polyconvex functions.     

Mean Squared Error Minimization for Inverse Moment Problems

We consider the problem of approximating the unknown density \(u\in L^2(\Omega ,\lambda )\) of a measure \(\mu \) on \(\Omega \subset \mathbb {R}^n\) , absolutely continuous with respect to some given reference measure \(\lambda \) , only from the knowledge of finitely many moments of \(\mu \) . Given \(d\in \mathbb {N}\) and moments of order \(d\) , we provide a polynomial \(p_d\) which minimizes the mean square error \(\int (u-p)^2d\lambda \) over all polynomials \(p\) of degree at most \(d\) . If there is no additional requirement, \(p_d\) is obtained as solution of a linear system. In addition, if \(p_d\) is expressed in the basis of polynomials that are orthonormal with respect to \(\lambda \) , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover \(p_d\rightarrow u\) in \(L^2(\Omega ,\lambda )\) as \(d\rightarrow \infty \) . In general nonnegativity of \(p_d\) is not guaranteed even though \(u\) is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing \(p_d\ge 0\) that minimizes \(\int (u-p)^2d\lambda \) now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating \(u\) .     

Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization

Let \(A\) be a compact \(d\) -rectifiable set embedded in Euclidean space \({\mathbb R}^p, d\le p\) . For a given continuous distribution \(\sigma (x)\) with respect to a \(d\) -dimensional Hausdorff measure on \(A\) , our earlier results provided a method for generating \(N\) -point configurations on \(A\) that have an asymptotic distribution \(\sigma (x)\) as \(N\rightarrow \infty \) ; moreover, such configurations are “quasi-uniform” in the sense that the ratio of the covering radius to the separation distance is bounded independently of \(N\) . The method is based upon minimizing the energy of \(N\) particles constrained to \(A\) interacting via a weighted power-law potential \(w(x,y)|x-y|^{-s}\) , where \(s>d\) is a fixed parameter and \(w(x,y)=\left( \sigma (x)\sigma (y)\right) ^{-({s}/{2d})}\) . Here we show that one can generate points on \(A\) with the aforementioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most \(r_N=C_N N^{-1/d}\) from each other, with \(C_N\) being a positive sequence tending to infinity arbitrarily slowly. To do this, we minimize the energy with respect to a varying truncated weight \(v_N(x,y)=\Phi (|x-y|/r_N)\cdot w(x,y)\) , where \(\Phi :(0,\infty )\rightarrow [0,\infty )\) is a bounded function with \(\Phi (t)=0, t\ge 1\) , and \(\lim _{t\rightarrow 0^+}\Phi (t)=1\) . Under appropriate assumptions, this reduces the complexity of generating \(N\) -point “low energy” discretizations to order \(N C_N^d\) computations.     

Global rigidity of higher rank abelian Anosov algebraic actions

We show that all \(C^\infty \) Anosov \(\mathbb {Z}^r\) -actions on tori and nilmanifolds without rank-one factor actions are, up to \(C^\infty \) conjugacy, actions by automorphisms.     

Square Function Characterization of Weak Hardy Spaces

We obtain a new square function characterization of the weak Hardy space \(H^{p,\infty }\) for all \(p\in (0,\infty )\) . This space consists of all tempered distributions whose smooth maximal function lies in weak \(L^p\) . Our proof is based on interpolation between \(H^p\) spaces. The main difficulty we overcome is the lack of a good dense subspace of \(H^{p,\infty }\) which forces us to work with general \(H^{p,\infty }\) distributions.     

Extremal geometry of a Brownian porous medium

The path \(W[0,t]\) of a Brownian motion on a \(d\) -dimensional torus \(\mathbb T ^d\) run for time \(t\) is a random compact subset of \(\mathbb T ^d\) . We study the geometric properties of the complement \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) for \(d\ge 3\) . In particular, we show that the largest regions in \(\mathbb T ^d{{\setminus }} W[0,t]\) have a linear scale \(\varphi _d(t)=[(d\log t)/(d-2)\kappa _d t]^{1/(d-2)}\) , where \(\kappa _d\) is the capacity of the unit ball. More specifically, we identify the sets \(E\) for which \(\mathbb T ^d{{\setminus }} W[0,t]\) contains a translate of \(\varphi _d(t)E\) , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) and the \(\varepsilon \) -cover time of \(\mathbb T ^d\) as \(\varepsilon \downarrow 0\) . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in \(\mathbb T ^d{{\setminus }} W_{\rho (t)}[0,t]\) , where \(W_{\rho (t)}[0,t]\) is the Wiener sausage of radius \(\rho (t)\) , with \(\rho (t)\) chosen much smaller than \(\varphi _d(t)\) but not too small. The idea behind this choice is that \(\mathbb T ^d {{\setminus }} W[0,t]\) consists of “lakes”, whose linear size is of order \(\varphi _d(t)\) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of \(\mathbb T ^d {{\setminus }} W_{\rho (t)}[0,t]\) as \(t\rightarrow \infty \) . Our results give a complete picture of the extremal geometry of \(\mathbb T ^d{{\setminus }} W[0,t]\) and of the optimal strategy for \(W[0,t]\) to realise extreme events.     

Estimates for vector valued Dirichlet polynomials

We estimate the \(\ell _1\) -norm \(\sum _{n=1}^N \Vert a_n\Vert \) of finite Dirichlet polynomials \(\sum _{n=1}^N a_n n^{-s},\,s \in {\mathbb {C}}\) with coefficients \(a_n\) in a Banach space. Our estimates quantify several recent results on Bohr’s strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.     

Optimal initial value conditions for the existence of strong solutions of the Boussinesq equations

Consider the instationary Boussinesq equations in a smooth bounded domain \(\Omega \subseteq \mathbb {R}^3\) with initial values \(u_0 \in L^2_{\sigma }(\Omega )\) , \( \theta _0 \in L^2(\Omega )\) and gravitational force \(g\) . We call \((u,\theta )\) strong solution if \((u,\theta )\) is a weak solution and additionally Serrin’s condition \(u \in L^s(0,T; L^q(\Omega ))\) holds where \( 1 satisfy \(\frac{2}{s} + \frac{3}{q} =1\) . In this paper we show that \(\int _0^{\infty } \Vert e^{-tA} u_0 \Vert _q^s \, dt < \infty \) is necessary and sufficient for the existence of such a strong solution \((u,\theta )\) in a sufficiently small interval \([0,T[\, , 0 < T\le \infty \) . Furthermore we show that strong solutions are uniquely determined and that they are smooth if the data are smooth. The crucial point is the fact that we have required no additional integrability condition for \(\theta \) in the definition of a strong solution \((u,\theta )\) .     

Shape derivatives for minima of integral functionals

For \(\Omega \) varying among open bounded sets in \(\mathbb R ^n\) , we consider shape functionals \(J (\Omega )\) defined as the infimum over a Sobolev space of an integral energy of the kind \(\int _\Omega [ f (\nabla u) + g (u) ]\) , under Dirichlet or Neumann conditions on \(\partial \Omega \) . Under fairly weak assumptions on the integrands \(f\) and \(g\) , we prove that, when a given domain \(\Omega \) is deformed into a one-parameter family of domains \(\Omega _\varepsilon \) through an initial velocity field \(V\in W ^ {1, \infty } (\mathbb R ^n, \mathbb R ^n)\) , the corresponding shape derivative of \(J\) at \(\Omega \) in the direction of \(V\) exists. Under some further regularity assumptions, we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of \(V\) on \(\partial \Omega \) . Our approach to obtain the shape derivative is new, and it is based on the joint use of Convex Analysis and Gamma-convergence techniques. It allows to deduce, as a companion result, optimality conditions in the form of conservation laws.     

Recognition of \text {PSL}(2,p) by order and some information on its character degrees where p is a prime

Let \(G\) be a finite group and \(\text {cd}(G)\) be the set of irreducible character degrees of \(G\) . In this paper we prove that if \(p\) is a prime number, then the simple group \(\text {PSL}(2,p)\) is uniquely determined by its order and some information about its character degrees. In fact we prove that if \(G\) is a finite group such that (i) \(|G|=|\text {PSL}(2,p)|\) , (ii) \(p\in \text {cd}(G)\) , (iii) \(\text {cd}(G)\) has an even integer, and (iv) there does not exist any element \(a\in \text {cd}(G)\) such that \(2p\mid a\) , then \(G\cong \text {PSL}(2,p)\) . As a consequence of our result we get that \(\text {PSL}(2,p)\) is uniquely determined by its order and the largest and the second largest character degrees.