On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

We study a multilinear version of the Hörmander multiplier theorem, namely

$$ \Vert T_{\sigma}(f_{1},\dots,f_{n})\Vert_{L^{p}}\lesssim \sup_{k\in\mathbb{Z}}{\Vert \sigma(2^{k}\cdot,\dots,2^{k}\cdot)\widehat{\phi^{(n)}}\Vert_{L^{2}_{(s_{1},\dots,s_{n})}}}\Vert f_{1}\Vert_{H^{p_{1}}}\cdots\Vert f_{n}\Vert_{H^{p_{n}}}. $$

We show that the estimate does not hold in the limiting case \(\min \limits {(s_{1},\dots ,s_{n})}=d/2\) or \({\sum}_{k\in J}{({s_{k}}/{d}-{1}/{p_{k}})}=-{1}/{2}\) for some \(J \subset \{1,\dots ,n\}\). This provides the necessary and sufficient condition on \((s_{1},\dots ,s_{n})\) for the boundedness of T_σ.

     

Functional Matrix Multipliers for Parseval Gabor Multi-frame Generators

We consider the problem of characterizing the bounded linear operator multipliers on \(L^{2}(\mathbb{R})\) that map Gabor frame generators to Gabor frame generators. We prove that a functional matrix \(M(t)=[f_{ij}(t)]_{m \times m}\) (where \(f_{ij}\in L^{\infty}(\mathbb{R})\)) is a multiplier for Parseval Gabor multi-frame generators with parameters \(a, b >0\) if and only if \(M(t)\) is unitary and \(M^{*}(t)M(t+\frac{1}{b})= \lambda(t)I\) for some unimodular \(a\)-periodic function \(\lambda(t)\). As a special case (\(m =1\)) this recovers the characterization of functional multipliers for Parseval Gabor frames with single function generators.

     

Mixed Wavelet Leaders Multifractal Formalism in a Product of Critical Besov Spaces

In this paper, we will prove (resp. study) the Baire generic validity of the upper-Hölder (resp. iso-Hölder) mixed wavelet leaders multifractal formalism on a product of two critical Besov spaces \(B_{t_{1}}^{\frac{m}{t_{1}},q_{1}}(\mathbb {R}^m) \times B_{t_{2}}^{\frac{m}{t_{2}},q_{2}}(\mathbb {R}^m)\), for \(t_1,t_2>0\), \(q_1 \le 1\) and \(q_2 \le 1\). Contrary to product spaces \(B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \times B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \) with \(s_{1} > \frac{m}{t_{1}}\) and \(s_{2} >\frac{m}{t_{2}}\) (Ben Slimane in Mediterr J Math, 13(4):1513–1533, 2016) and \((B_{t_{1}}^{s_{1},\infty }(\mathbb {R}^m) \cap C^{\gamma _{1}}(\mathbb {R}^m)) \times (B_{t_{2}}^{s_{2},\infty }(\mathbb {R}^m) \cap C^{\gamma _{2}}(\mathbb {R}^m)\) with \(0<\gamma _{1}<s_{1}<\frac{m}{t_{1}}\) and \(0<\gamma _{2}<s_{2}<\frac{m}{t_{2}}\) (Ben Abid et al. in Mediterr J Math, 13(6):5093–5118, 2016), all pairs of functions in the obtained generic set are not uniform Hölder. Nevertheless, the characterization of the upper bound of the Hölder exponent by decay conditions of local wavelet leaders suffices for our study.     

Frames and Finite-Rank Integral Representations of Positive Operator-Valued Measures

Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form

$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$

for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.

     

Ground state solutions for a class of fractional Hamiltonian systems

In this paper we study the following fractional Hamiltonian systems

$$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))- L(t).x(t)+\nabla W(t,x(t))=0, \\ x\in H^{\alpha }(\mathbb {R}, \mathbb {R}^{N}), \end{array} \right. \end{aligned}$$

where \(\alpha \in \left( {1\over {2}}, 1\right] ,\ t\in \mathbb {R}, x\in \mathbb {R}^N,\ _{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are the left and right Liouville–Weyl fractional derivatives of order \(\alpha \) on the whole axis \(\mathbb {R}\) respectively, \(L:\mathbb {R}\longrightarrow \mathbb {R}^{2N}\) and \(W: \mathbb {R}\times \mathbb {R}^{N}\longrightarrow \mathbb {R}\) are suitable functions. One ground state solution is obtained by applying the monotonicity trick of Jeanjean and the concentration-compactness principle in the case where the matrix L(t) is positive definite and \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition.

     

Topological Properties of the Set of Functions Generated by Neural Networks of Fixed Size

We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to \(L^p\)-norms, \(0< p < \infty \), for all practically used activation functions, and also not closed with respect to the \(L^\infty \)-norm for all practically used activation functions except for the ReLU and the parametric ReLU. Finally, the function that maps a family of weights to the function computed by the associated network is not inverse stable for every practically used activation function. In other words, if \(f_1, f_2\) are two functions realized by neural networks and if \(f_1, f_2\) are close in the sense that \(\Vert f_1 - f_2\Vert _{L^\infty } \le \varepsilon \) for \(\varepsilon > 0\), it is, regardless of the size of \(\varepsilon \), usually not possible to find weights \(w_1, w_2\) close together such that each \(f_i\) is realized by a neural network with weights \(w_i\). Overall, our findings identify potential causes for issues in the training procedure of deep learning such as no guaranteed convergence, explosion of parameters, and slow convergence.

     

Existence,Nonexistence and Multiplicity Results of a Chern-Simons-Schrödinger System

We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$

where \(\lambda >0\) is a parameter, \(p\in (2,4)\) and

$$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$

We prove that the system has no solutions for \(\lambda \) large and has two radial solutions for \(\lambda \) small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of \(\lambda \). Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).

     

Regularity of Weak Solutions to the 3D Magneto-Micropolar Equations in Besov Spaces

This paper deals with the regularity of weak solutions to the 3D magneto-micropolar fluid equations in Besov spaces. It is shown that for \(0\le\alpha\le1\) if \(u\in L^{\frac{2}{1+\alpha}}(0,T; \dot{B}_{\infty,\infty}^{\alpha})\), then the weak solution \((u,\omega ,b)\) is regular on \((0,T]\).

     

Weighted $$L^2$$ L 2 -norms of Gegenbauer polynomials

We study integrals of the form

$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$

where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).

     

On the Fourier coefficients of certain Hilbert modular forms

We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).

     

On Certain Integral Operators of Fractional Type

In this paper we study integral operators of the form $$T\,f\left( x \right) = \int {k_1 \left( {x - a_1 y} \right)k_2 \left( {x - a_2 y} \right)...k_m \left( {x - a_m y} \right)f\left( y \right)dy} ,$$ $$k_i \left( y \right) = \sum\limits_{j \in Z} {2^{\frac{{jn}}{{q_i }}} } \varphi _{i,j} \left( {2^j y} \right),\,1 \leqq q_i < \infty ,\frac{1}{{q_1 }} + \frac{1}{{q_2 }} + ... + \frac{1}{{q_m }} = 1 - r,$$ $0 \leqq r < 1$ , and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T:L^p \left( {R^n } \right) \to T:L^q \left( {R^n } \right)$ for $1 < p < \frac{1}{r}$ and $\frac{1}{q} = \frac{1}{p} - r$ .     

On the regular summability of multiple function series and Lebesgue functions

Основной целью работ ы является обобщение одного результата Кратца и Т раутнера [4], известного для одном ерных функциональны х рядов, на кратные ряды. Этот рез ультат касается суммируемо сти функционального ряда почти всюду при слабых пред положениях. В частности, он примен им к суммируемости по Чезаро и по Риссу. Мы рассматриваемd-кр атный ряд $$\mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x), \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d }^2< \infty $$ и предполагается, что функции \(f_{k_1 ,...,k_d } (x)\) интегрируе мы по пространству с полож ительной мерой и имеют почти вс юду ограниченные фун кции Лебега для метода суммирова ния Т. Метод Т определяетсяd-мерной матрицей \(T = \{ a_{m_1 ,...,m_d ;k_1 ,...,k_d } \} \) сл едующим образом: $$t_{m_1 ,...,m_d } (x) = \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty a_{m_1 ,...,m_d ;k_1 ,...,k_d } c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x).$$ Эти средние существу ют, поскольку мы предп олагаем, что \(a_{m_1 ,...,m_d ;k_1 ,...,k_d } = 0\) ,если max(k ₁,...,k _d) достаточно вели к (в зависимости, конеч но, отm ₁,...,m _d). При некоторых дополнительных усло виях на матрицуТ (см. (7)– (9) в разделе 3) устанавлива ется почти всюду регулярная схо димость средних \(t_{m_1 ,...,m_d } (x) \user2{} \user2{(}m_1 \user2{,}...\user2{,}m_d \user2{)} \to \infty \) . Как вспомогательный результат, в работе об общается теорема Алексича [1] о сх одимости почти всюду некоторы х подпоследовательн остей частных сумм функцио нального ряда.     

Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case

We consider the generalized Gagliardo-Nirenberg inequality in $\Bbb{R}^{n}$ including homogeneous Besov space $\dot{B}^{s}_{r,\rho}(\Bbb{R}^{n})$ with the critical order s=n/r, which describes the continuous embedding such as $L^{p}(\Bbb{R}^{n})\cap\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})\subset L^{q}(\Bbb{R}^{n})$ for all q with p ≦ q<∞, where 1≦ p ≦ r<∞ and 1<ρ ≦∞. Indeed, the following inequality holds: $$\|u\|_{L^{q}(\Bbb{R}^{n})}\leqq C\,q^{1-1/\rho}\|u\|_{L^{p}(\Bbb{R}^{n})}^{p/q}\|u\|_{\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})}^{1-p/q},$$ where C is a constant depending only on r. In this inequality, we have the exact order 1?1/ρ of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space $\dot{F}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ , which implies the usual Sobolev imbedding with the critical Sobolev space $\dot{H}^{n/r}_{r}(\Bbb{R}^{n})$ . Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in $\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ .     

A class of inequalities for non-negative sequences

В работе для неотрица тельных последовате льностей (...,a _?1 ⁱ ), aa ₀ ⁱ ),a ₁ ⁱ ), ...), удовлетв оряющих условию \(0< \mathop {\sup }\limits_k a_k^{(i)}< \infty\) (i=1,...,т), доказ а но неравенство (1) $$\begin{gathered} \mathop \sum \limits_{k = - \infty }^\infty \mathop {\sup }\limits_{k \leqq k_1 + \ldots + k_m \leqq k + l} (a_{k_1 }^{(1)} \ldots a_{k_m }^{(m)} ) \geqq \hfill \\ \geqq \mathop \prod \limits_{i = 1}^m (\mathop {\sup }\limits_{ - \infty< k< \infty } a_k^{(i)} )\left[ {\mathop \sum \limits_{i = 1}^m \frac{{\mathop \sum \limits_{k = - \infty }^\infty (a_k^{(i)} )^{p_i } }}{{(\mathop {\sup }\limits_{ - \infty< k< \infty } a_k^{(i)} )^{p_i } }} + l - m + 1} \right], \hfill \\ \end{gathered}$$ гдеl произвольное не отрицательное целое число, 1≦p ₁, ...,p _m ≦∞ и \(\mathop \sum \limits_{i = 1}^m p_i^{ - 1} = 1\) . Это неравенство явля ется обобщением и уто чнением неравенств А. Прекопа, Ш. Данча и Л. Лейндлера. Доказано также, что ес ли все последователь ности содержат только коне чное число ненулевых членов, то н еобходимым условием для равенства в (1) является существование такого числа α>0, чтоa _k( ⁱ )=а илиa _k( ⁱ )=0 для всехi=1,...,m;?∞<k<∞.     

Some Berezin Number Inequalities for Operator Matrices

The Berezin symbol à of an operator A acting on the reproducing kernel Hilbert space H = H(Ω) over some (nonempty) set is defined by \(\tilde A(\lambda ) = \left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle \), λ ∈ Ω, where \(\hat k_\lambda = k_\lambda /\left\| {k_\lambda } \right\|\) is the normalized reproducing kernel of H. The Berezin number of the operator A is defined by \(ber(A) = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\tilde A(\lambda )} \right| = \mathop {\sup }\limits_{\lambda \in \Omega } \left| {\left\langle {A\hat k_\lambda ,\hat k_\lambda } \right\rangle } \right|\). Moreover, ber(A) ? w(A) (numerical radius). We present some Berezin number inequalities. Among other inequalities, it is shown that if \(T = \left[ {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right] \in \mathbb{B}(\mathcal{H}(\Omega _1 ) \oplus \mathcal{H}(\Omega _2 ))\), then

$$ber(T) \leqslant \frac{1}{2}(ber(A) + ber(D)) + \frac{1}{2}\sqrt {(ber(A) - ber(D))^2 + \left( {\left\| B \right\| + \left\| C \right\|} \right)^2 } .$$

     

Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data

An existence result is established for a class of quasilinear parabolic problem which is a diffusion type equations having continuous coefficients blowing up for a finite value of the unknown, a second hand \(\mu \in \mathcal {M}_{b}(Q)\) and an initial data \(u_{0}\in L^{1}(\Omega )\). We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some compactness results are also shown under additional hypotheses.

     

$$\pi $$π-Formulas and Gray code

In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula

$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$

related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).

     

Superlinear Elliptic Equations with Variable Exponent via Perturbation Method

We are concerned with the following \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\)

$$ -\triangle _{p(x)} u+|u|^{p(x)-2}u= f(x,u)\quad \mbox{in } \mathbb{R} ^{N}. $$

The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution \(u_{\lambda }\) of perturbation equations. First, we prove that \(u_{\lambda }\rightharpoonup u\) as \(\lambda \rightarrow 0\). Second, by using vanishing lemma, we get that \(u\) is a nontrivial solution of the original problem.

     

$$L^2_f$$ L f 2 -harmonic 1-forms on smooth metric measure spaces with positive $$\lambda _1(\Delta _f)$$ λ 1 ( Δ f )

In this paper, we study vanishing and splitting results on a complete smooth metric measure space \((M^n,g,\mathrm {e}^{-f}\mathrm {d}v)\) with various negative m-Bakry-Émery Ricci curvature lower bounds in terms of the first eigenvalue \(\lambda _1(\Delta _f)\) of the weighted Laplacian \(\Delta _f\), i.e., \(\mathrm {Ric}_{m,n}\ge -a\lambda _1(\Delta _f)-b\) for \(0<a\le \dfrac{m}{m-1}, b\ge 0\). In particular, we consider three main cases for different a and b with or without conditions on \(\lambda _1(\Delta _f)\). These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung, and Vieira.

     

Blow Up Criteria for the Incompressible Nematic Liquid Crystal Flows

In this paper, we investigate blow up criteria for the local smooth solutions to the 3D incompressible nematic liquid crystal flows via the components of the gradient velocity field \(\nabla u\) and the gradient orientation field \(\nabla d\). More precisely, we show that \(0< T_{ \ast}<+\infty\) is the maximal time interval if and only if

$$\begin{aligned} & \int_{0}^{T_{\ast}} \bigl\Vert \Vert \partial_{i}u\Vert _{L_{x_{i}} ^{\gamma}} \bigr\Vert _{L_{x_{j}x_{k}}^{\alpha}}^{\beta}+ \|\nabla d\| _{L^{\infty}}^{\frac{8}{3}}\mathrm{d}t=\infty, \\ &\quad\text{ with } \frac{2}{\alpha}+\frac{2}{\beta}\leq\frac{3\alpha +2}{4\alpha}, \text{ and } 1\leq\gamma\leq\alpha,2< \alpha\leq+\infty, \end{aligned}$$

or

$$\begin{aligned} \int_{0}^{T_{\ast}}\|\partial_{3}u_{3} \|^{\beta}_{L^{\alpha}}+\| \nabla d\|^{\frac{8}{3}}_{L^{\infty}} \mathrm{d}t=\infty,\quad\text{with } \frac{3}{\alpha}+\frac{2}{\beta}\leq \frac{3(\alpha+2)}{4 \alpha}, \text{ and } 2< \alpha\leq\infty, \end{aligned}$$

where \(i,j,k\in\{1,2,3\}\), \(i\neq j\), \(i\neq k\), and \(j\neq k\).