Applications of strongly convergent sequences to Fourier series by means of modulus functions

The aim of this work is to estimate sums involving P(n), the largest prime factor of an integer \({n \geqq 2}\) under digital constraints \({{f(P(n)) \equiv a}{\rm mod} b}\), for every \({a \in \mathbb{Z}}\) and an integer \({b \geqq 2}\) where f is a strongly q-additive function with integer values (i.e. \({f(aq^j + b) = f(a) + f(b)}\), with \({(a, b, j) \in \mathbb{N}^3}\), \({{0 \leqq b} < q^j}\)). We also estimate the cardinality of the set \({\{{n \leqq x, f(P(n) + c)} \equiv {a {\rm mod} b}, P(n) \equiv l {\rm mod} k\}}\), where \({c \in \mathbb{Z}}\), \({k \geqq 2}\).     

{L^{p}} stability for the Boltzmann equation near vacuum

The paper is concerned with the uniform time stability in the Lebesgue space \({L^{p}(\mathbb{R}^{3} \times \mathbb{R}^{3})}\) of solutions to the Boltzmann equation near vacuum. Precisely, for the soft potential case \({-2 < \gamma < 0}\), there exists \(p_{\gamma} > 1\) such that the nonnegative solution with algebraic decay rate in x, v at infinity is stable with respect to small initial data uniformly in time in \({L^{p}}\) with \({1 \leq p < p_{\gamma}}\).     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B′in separable complex Hilbert spaces H and H′, respectively. It is found that if the mapping f ∈ C~(1+α)at z_0∈ ?B with f(z_0) = w_0∈ ?B′, then the Fr′echet derivative operator Df(z_0) maps the tangent space Tz_0(?B~n) to Tw_0(?B′), the holomorphic tangent space T_(z_0)~(1,0)(?B~n) to T_(w_0)~(1,0)(?B′),respectively.     

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 \({{\mathbf W}^m_p ({\mathbb R}^n)}\).     

On polynomial congruences

We deal with functions which fulfil the condition \({\Delta_h^{n+1} \varphi(x)\in\mathbb{Z}}\) for all x, h taken from some linear space V. We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions \({f\colon V\rightarrow \mathbb{R},\ g\colon V \rightarrow \mathbb{Z}}\) such that \({\varphi = f + g}\) and \({\Delta_h^{n+1}f(x)=0}\) for all \({x, h\in V}\).     

Complete real Kähler submanifolds in codimension two

Minimal isometric immersions \(f : M^{2n} \rightarrow {\mathbb{R}}^{2n+2}\) in codimension two from a complete Kähler manifold into Euclidean space had been classified in Dajczer and Gromoll (Invent Math 119:235–242, 1995) for n ≥  3. In this note we describe the non-minimal situation showing that, if f is real analytic but not everywhere minimal, then f is a cylinder over a real Kähler surface \(g : N^4 \rightarrow {\mathbb{R}}^6\) , that is, \(M^{2n} = N^4 \times {\mathbb{C}}^{n-2}\) and f = g × id split, where \({id} : {\mathbb{C}}^{n-2} \cong {\mathbb{R}}^{2n-4}\) is the identity map. Moreover, g can be further described.     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem

$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$

where f is a polynomial and \({\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}\) as well as \({{\Omega}\subset\mathbb{R}^p}\) are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations \({(J_i)\subset\mathbb{R}[\mathbf{y}]}\) of the optimal value function \({J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}\). The polynomial \({J_i\in\mathbb{R}[\mathbf{y}]}\) is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem \({\mathbf{y}\mapsto J(\mathbf{y})}\), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem \({J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}\) and prove that \({\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}\) converges to J* as i → ∞. Finally, for fixed ? ≤ i, each \({J^*_\ell}\) (and hence \({\hat{J}^*_i}\)) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).

     

Riesz Transform Characterizations of Hardy Spaces Associated to Degenerate Elliptic Operators

Let \({L_{w}}{:=-w^{-1}{\rm div}(A\nabla)}\) be the degenerate elliptic operator on the Euclidean space \({{\mathbb{R}^{n}}}\), where w is a Muckenhoupt \({A_{2}({\mathbb{R}^{n}})}\) weight. In this article, the authors establish the Riesz transform characterization of the Hardy space \({H^{p}_{L_{w}}({\mathbb{R}}^{n})}\) associated with L_w, for \({w \in A_{q}({\mathbb{R}}^{n}) \cap RH_{\frac{n}{n-2}}({\mathbb{R}^{n}})}\) with \({n \geq 3}\), \({q \in [1,2]}\) and \({p \in (q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]}\) if, for some \({r \in (1,\,2]}\), \({{\{tL_w e^{-tL_w}\}}_{t\geq 0}}\) satisfies the weighted \({L^{r}-L^{2}}\) full off-diagonal estimates.     

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.     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

Solutions of perturbed Schrödinger equations with critical nonlinearity

We consider the perturbed Schrödinger equation

$\left\{\begin{array}{ll}{- \varepsilon ^2 \Delta u + V(x)u = P(x)|u|^{p - 2} u + k(x)|u|^{2* - 2} u} &; {\text{for}}\, x \in {\mathbb{R}}^N\\ \qquad \qquad \quad {u(x) \rightarrow 0} &; \text{as}\, {|x| \rightarrow \infty} \end{array} \right.$

where \(N\geq 3, \ 2^*=2N/(N-2)\) is the Sobolev critical exponent, \(p\in (2, 2^*)\) , P(x) and K(x) are bounded positive functions. Under proper conditions on V we show that it has at least one positive solution provided that \(\varepsilon\leq{\mathcal{E}}\) ; for any \(m\in{\mathbb{N}}\) , it has m pairs of solutions if \(\varepsilon\leq{\mathcal{E}}_{m}\) ; and suppose there exists an orthogonal involution \(\tau:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}\) such that V(x), P(x) and K(x) are τ -invariant, then it has at least one pair of solutions which change sign exactly once provided that \(\varepsilon\leq{\mathcal{E}}\) , where \({\mathcal{E}}\) and \({\mathcal{E}}_{m}\) are sufficiently small positive numbers. Moreover, these solutions \(u_\varepsilon\to 0\) in \(H^1({\mathbb{R}}^N)\) as \(\varepsilon\to 0\) .

     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)