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 \(f\in W_c(L_1(\log L)^{d-1},\ell _\infty )(\mathbb{R }^d)\) .     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

On the Action of Riesz Transforms on the Class of Bounded Functions

The paper is devoted to the \(d\) -dimensional extension of the classical identity of Stein and Weiss concerning the action of the Hilbert transform on characteristic functions. Let \((R_j)_{j=1}^d\) be the collection of Riesz transforms in \(\mathbb{R }^d\) . For \(1\le p<\infty \) , we determine the least constants \(c_{p,d}, C_{p,d}\) such that $$\begin{aligned} \int _{\mathbb{R }^d} f(x)|R_jf(x)|^p\text{ d }x&\le c_{p,d} ||f||_{L^1(\mathbb{R }^d)},\\ \int _{\mathbb{R }^d} (1-f(x))|R_jf(x)|^p\text{ d }x&\le C_{p,d} ||f||_{L^1(\mathbb{R }^d)} \end{aligned}$$ for any Borel function \(f:\mathbb{R }^d\rightarrow [0,1]\) . The proof rests on probabilistic methods and the construction of appropriate harmonic functions on \([0,1]\times \mathbb{R }\) .     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

Let \(Q\) be a fundamental domain of some full-rank lattice in \({\mathbb {R}}^d\) and let \(\mu \) and \(\nu \) be two positive Borel measures on \({\mathbb {R}}^d\) such that the convolution \(\mu *\nu \) is a multiple of \(\chi _Q\) . We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated \(L^2\) space admits an orthogonal basis of exponentials) and we show that this is the case when \(Q = [0,1]^d\) . This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede’s Conjecture for spectral measures on \({\mathbb {R}}^1\) and we show that it implies the classical Fuglede’s Conjecture on \({\mathbb {R}}^1\) .     

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 \(\mathbb {R}^d\) .     

On Radii of Spheres Determined by Subsets of Euclidean Space

In this paper the author considers the problem of how large the Hausdorff dimension of \(E\subset \mathbb {R}^d\) needs to be in order to ensure that the radii set of \((d-1)\) -dimensional spheres determined by \(E\) has positive Lebesgue measure. The author also studies the question of how often can a neighborhood of a given radius repeat. There are two results obtained in this paper. First, by applying a general mechanism developed in Grafakos et al. (2013) for studying Falconer-type problems, the author proves that a neighborhood of a given radius cannot repeat more often than the statistical bound if \(\dim _{{\mathcal H}}(E)>d-1+\frac{1}{d}\) ; In \(\mathbb {R}^2\) , the dimensional threshold is sharp. Second, by proving an intersection theorem, the author proves that for a.e \(a\in \mathbb {R}^d\) , the radii set of \((d-1)\) -spheres with center \(a\) determined by \(E\) must have positive Lebesgue measure if \(\dim _{{\mathcal H}}(E)>d-1\) , which is a sharp bound for this problem.     

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.     

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

In the framework of toroidal Pseudodifferential operators on the flat torus \({\mathbb {T}}^n := ({\mathbb {R}} / 2\pi {\mathbb {Z}})^n\) we begin by proving the closure under composition for the class of Weyl operators \(\mathrm {Op}^w_\hbar (b)\) with symbols \(b \in S^m (\mathbb {T}^n \times \mathbb {R}^n)\) . Subsequently, we consider \(\mathrm {Op}^w_\hbar (H)\) when \(H=\frac{1}{2} |\eta |^2 + V(x)\) where \(V \in C^\infty ({\mathbb {T}}^n)\) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on \(\mathbb {T}^n \times {\mathbb {R}}^n\) written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space \(H^{1} (\mathbb {T}^n; {\mathbb {C}})\) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton–Jacobi equation \(\frac{1}{2} |P+ \nabla _x v (P,x)|^2 + V(x) = \bar{H}(P)\) for \(P \in \ell {\mathbb {Z}}^n\) with \(\ell >0\) , and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of \(P+ \nabla _x v\) .     

The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere

Let \(\Delta _0\) be the Laplace–Beltrami operator on the unit sphere \(\mathbb {S}^{d-1}\) of \({\mathbb R}^d\) . We show that the Hardy–Rellich inequality of the form $$\begin{aligned} \mathop \int \limits _{\mathbb {S}^{d-1}} \left| f (x)\right| ^2 \mathrm{d}{\sigma }(x) \le c_d \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) \left| (-\Delta _0)^{\frac{1}{2}}f(x) \right| ^2 \mathrm{d}{\sigma }(x) \end{aligned}$$ holds for \(d =2\) and \(d \ge 4\) but does not hold for \(d=3\) with any finite constant, and the optimal constant for the inequality is \(c_d = 8/(d-3)^2\) for \(d =2, 4, 5,\) and, under additional restrictions on the function space, for \(d\ge 6\) . This inequality yields an uncertainty principle of the form $$\begin{aligned} \min _{e\in \mathbb {S}^{d-1}} \mathop \int \limits _{\mathbb {S}^{d-1}} (1- {\langle }x, e {\rangle }) |f(x)|^2 \mathrm{d}{\sigma }(x) \mathop \int \limits _{\mathbb {S}^{d-1}}\left| \nabla _0 f(x)\right| ^2 \mathrm{d}{\sigma }(x) \ge c'_d \end{aligned}$$ on the sphere for functions with zero mean and unit norm, which can be used to establish another uncertainty principle without zero mean assumption, both of which appear to be new.     

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.     

The theme of a vanishing period

Consider a multivalued formal function of the type 1 $$\begin{aligned} \varphi (s) : = \sum _{j=0}^k\,c_j(s).s^{\lambda + m_j}.(\mathrm{Log}\,s)^j, \end{aligned}$$ where \(\lambda \) is a positive rational number, \(c_j\) is in \({{\mathrm{\mathbb {C}}}}[[s]]\) and \(m_j \in \mathbb {N}\) for \(j \in [0,k-1]\) . The theme associated with such a \(\varphi \) is the “minimal filtered integral equation” satisfied by \(\varphi \) , in a sense which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial of \(\varphi \) to be fixed. For a given \(\lambda \) , to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated with the logarithm of the monodromy in the geometric situation described below. Our aim is to construct some analytic invariants, for instance in the following situation, let \(f : X \rightarrow D\) be a proper holomorphic function defined on a complex manifold \(X\) with values in a disc \(D\) . We assume that the only critical value is \(0 \in D\) and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \(f^{-1}(0)\) . To a smooth \((p+1)\) -form \(\omega \) on \(X\) such that \(\mathrm{d}\omega = 0 = \mathrm{d}f \wedge \omega \) and to a vanishing \(p\) -cycle \(\gamma \) chosen in the generic fiber \(f^{-1}(s_0), s_0 \in D \setminus \{0\}\) , we associated a “vanishing period” \(F_{\gamma }(s) : = \int _{\gamma _s} \omega \big /\mathrm{d}f \) which has an asymptotic expansion at \(0\) of the form \((1)\) above, when \(\gamma \) is chosen in the spectral subspace of \(H_p(f^{-1}(s_0), {{\mathrm{\mathbb {C}}}})\) for the eigenvalue \(\mathrm{e}^{2i\pi .\lambda }\) of the monodromy of \(f\) . Here \((\gamma _s)_{s \in D^*}\) is the horizontal multivalued family of \(p\) -cycles in the fibers of \(f\) obtained from the choice of \(\gamma \) . The aim of this article was to study the module generated by such a \(\varphi \) over the algebra \(\tilde{\mathcal {A}}\) , which is the \(b\) -completion of the algebra \(\mathcal {A}\) generated by the operators \(\mathrm{a} : = \times s\) and \(\mathrm{b} : = \int _{0}^{s}\) .     

Differentiability in the Sobolev space W^{1,n-1}

Let \(\Omega \subset {\mathbb {R}}^{n}\) be a domain, \(n \ge 2\) . We show that a continuous, open and discrete mapping \(f \in W_{\mathrm{loc }}^{1,n-1}(\Omega , {\mathbb {R}}^{n})\) with integrable inner distortion is differentiable almost everywhere on \(\Omega \) . As a corollary we get that the branch set of such a mapping has measure zero.     

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.     

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 extremal solution for the fractional Laplacian

We study the extremal solution for the problem \((-\Delta )^s u=\lambda f(u)\) in \(\Omega \) , \(u\equiv 0\) in \(\mathbb R ^n\setminus \Omega \) , where \(\lambda >0\) is a parameter and \(s\in (0,1)\) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions \(n<4s\) . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever \(n<10s\) . In the limit \(s\uparrow 1\) , \(n<10\) is optimal. In addition, we show that the extremal solution is \(H^s(\mathbb R ^n)\) in any dimension whenever the domain is convex. To obtain some of these results we need \(L^q\) estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with \(L^p\) data. We prove optimal \(L^q\) and \(C^\beta \) estimates, depending on the value of \(p\) . These estimates follow from classical embedding results for the Riesz potential in \(\mathbb R ^n\) . Finally, to prove the \(H^s\) regularity of the extremal solution we need an \(L^\infty \) estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.     

On the number of distinct prime factors of nj+a^hk

Let \(\omega (n)\) denote the number of distinct prime factors of \(n\) . Then for any given \(K\ge 2\) , small \(\epsilon >0\) and sufficiently large (only depending on \(K\) and \(\epsilon \) ) \(x\) , there exist at least \(x^{1-\epsilon }\) integers \(n\in [x,(1+K^{-1})x]\) such that \(\omega (nj\pm a^hk)\ge (\log \log \log x)^{\frac{1}{3}-\epsilon }\) for all \(2\le a\le K\) , \(1\le j,k\le K\) and \(0\le h\le K\log x\) .     

Selmer groups over {\mathbb {Z}}_p^d-extensions

Consider an abelian variety \(A\) defined over a global field \(K\) and let \(L/K\) be a \({\mathbb {Z}}_p^d\) -extension, unramified outside a finite set of places of \(K\) , with \({{\mathrm{Gal}}}(L/K)=\Gamma \) . Let \(\Lambda (\Gamma ):={\mathbb {Z}}_p[[\Gamma ]]\) denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the \(\Lambda (\Gamma )\) -module \(X_L\) , the dual \(p\) -primary Selmer group, varies when \(L/K\) is replaced by a strict intermediate \({\mathbb {Z}}_p^e\) -extension.