On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Scale invariant elliptic operators with singular coefficients

We show that a realization of the operator \({L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}}\) generates a semigroup in \({L^p(\mathbb{R}^N)}\) if and only if \({D_c=b+(N-2+c)^2/4 > 0}\) and \({s_1+\min\{0,2-\alpha\} < N/p < s_2+\max\{0,2-\alpha\}}\), where \({s_i}\) are the roots of the equation \({b+s(N-2+c-s)=0}\), or \({D_c=0}\) and \({s_0+\min\{0,2-\alpha\} < N/p < s_0+\max\{0,2-\alpha\}}\), where \({s_0}\) is the unique root of the above equation. The domain of the generator is also characterized.     

On the Chaoticity of Some Tensor Product Weighted Backward Shift Operators Acting on Some Tensor Product Fock–Bargmann Spaces

In Advances in Mathematical Physics (2011) we showed that the weighted shift \(z^{p}\frac{d^{p+1}}{dz^{p+1}} (p=0, 1, 2,\ldots )\) acting on classical Bargmann space \(\mathbb {B}_{p}\) is chaotic operator. In Journal of Mathematical physics (2014), we constructed an chaotic weighted shift \(\mathbb {M}^{*^{p}}\mathbb {M}^{p+1} (p=0, 1, 2,\ldots )\) on some lattice Fock–Bargmann \(\mathbb {E}_{p}^{\alpha }\) generated by the orthonormal basis \( {e_{m}^{(\alpha ,p)}(z) = e_{m}^{\alpha } ; m=p, p+1,\ldots }\) where \( {e_{m}^{\alpha }(z) = (\frac{2\nu }{\pi })^{1/4}e^{\frac{\nu }{2}z^{2}}e^{-\frac{\pi ^{2}}{\nu }(m +\alpha )^{2} +2i\pi (m +\alpha )z}; m \in \mathbb {N}}\) with \(\nu , \alpha \) are real numbers; \(\nu > 0\), \(\mathbb {M}\) is an weighted shift and \(\mathbb {M^{*}}\) is the adjoint of the \(\mathbb {M}\). In this paper we study the chaoticity of tensor product \(\mathbb {M}^{*^{p}}\mathbb {M}^{p+1}\otimes z^{p}\frac{d^{p}}{dz^{p+1}} (p=0, 1, 2, \ldots )\) acting on \(\mathbb {E}_{p}^{\alpha }\otimes \mathbb {B}_{p}\).     

The nonlocal Liouville-type equation in $$\mathbb {}$$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to

$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$

(1)

with \(K_k\) bounded in \(L^\infty \) and \(e^{u_k}\) bounded in \(L^1\) uniformly with respect to k, we show that up to extracting a subsequence \(u_k\) can blow-up at (at most) finitely many points \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\), or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (\(\alpha _j=\pi \) and \(\alpha _j\ge \pi \)) which are not known in dimension 2 under the weak assumption that \((K_k)\) be bounded in \(L^\infty \) and is allowed to change sign.

     

Intrinsic Structures of Certain Musielak–Orlicz Hardy Spaces

For any \(p\in (0,\,1]\), let \(H^{\Phi _p}(\mathbb {R}^n)\) be the Musielak–Orlicz Hardy space associated with the Musielak–Orlicz growth function \(\Phi _p\), defined by setting, for any \(x\in \mathbb {R}^n\) and \(t\in [0,\,\infty )\),

$$\begin{aligned}&\Phi _{p}(x,\,t)\\&\quad := {\left\{ \begin{array}{ll} \displaystyle \frac{t}{\log {(e+t)}+[t(1+|x|)^n]^{1-p}}&{} \quad \text {when}\ n(1/p-1)\notin \mathbb N \cup \{0\},\\ \displaystyle \frac{t}{\log (e+t)+[t(1+|x|)^n]^{1-p}[\log (e+|x|)]^p}&{} \quad \text {when}\ n(1/p-1)\in \mathbb N\cup \{0\}, \end{array}\right. } \end{aligned}$$

which is the sharp target space of the bilinear decomposition of the product of the Hardy space \(H^p(\mathbb {R}^n)\) and its dual. Moreover, \(H^{\Phi _1}(\mathbb {R}^n)\) is the prototype appearing in the real-variable theory of general Musielak–Orlicz Hardy spaces. In this article, the authors find a new structure of the space \(H^{\Phi _p}(\mathbb {R}^n)\) by showing that, for any \(p\in (0,\,1]\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{\phi _0}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\) and, for any \(p\in (0,\,1)\), \(H^{\Phi _p}(\mathbb {R}^n)=H^{1}(\mathbb {R}^n) +H_{W_p}^p({{{\mathbb {R}}}^n})\), where \(H^1(\mathbb {R}^n)\) denotes the classical real Hardy space, \(H^{\phi _0}({{{\mathbb {R}}}^n})\) the Orlicz–Hardy space associated with the Orlicz function \(\phi _0(t):=t/\log (e+t)\) for any \(t\in [0,\infty )\), and \(H_{W_p}^p(\mathbb {R}^n)\) the weighted Hardy space associated with certain weight function \(W_p(x)\) that is comparable to \(\Phi _p(x,1)\) for any \(x\in \mathbb {R}^n\). As an application, the authors further establish an interpolation theorem of quasilinear operators based on this new structure.

     

On a transport problem and monoids of non-negative integers

A problem about how to transport profitably a group of cars leads us to studying the set T formed by the integers n such that the system of inequalities, with non-negative integer coefficients,

$$\begin{aligned} a_1x_1 +\cdots + a_px_p + \alpha \le n \le b_1x_1 +\cdots + b_px_p - \beta \end{aligned}$$

has at least one solution in \({\mathbb N}^p\). We prove that \(T\cup \{0\}\) is a submonoid of \(({\mathbb N},+)\) and, moreover, we give algorithmic processes to compute T.

     

Quadratic Hamilton–Poisson Systems on $\mathfrak{se}(1,1)^{*}_{-}$: The Inhomogeneous Case

The blow-up of smooth solution to the isentropic compressible Navier-Stokes-Poisson (NSP) system on \(\mathbb{R}^{d}\) is studied in this paper. We obtain that if the initial density is compactly supported, the spherically symmetric smooth solution to the NSP system on \(\mathbb{R}^{d}\ (d\geq 2)\) blows up in finite time. In the case \(d=1\), if \(2\mu +\lambda >0\), then the NSP system only exits a zero smooth solution on ? for the compactly supported initial density.     

Spectrality of infinite convolutions with three-element digit sets

Let \(0< \rho <1\) and let \(\{a_n, b_n\}_{n=1}^\infty \) be a sequence of integers with bounded from upper and lower. Associated with them there exists a unique Borel probability measure \(\mu _{\rho , \{0, a_n, b_n\}}\) generated by the following infinite convolution product

$$\begin{aligned} \mu _{\rho , \{0, a_n, b_n\}}=\delta _{\rho \{0, a_1, b_1\}} *\delta _{\rho ^2 \{0, a_2, b_2\}} *\delta _{\rho ^3 \{0, a_3, b_3\}} *\cdots \end{aligned}$$

in the weak convergence, where \(\delta _E=\frac{1}{\# E}\sum _{e \in E} \delta _e\) and \(\hbox {gcd}(a_n, b_n)=1\) for all \(n \in {{\mathbb {N}}}\). In this paper, we show that \(L^2(\mu _{\rho , \{0, a_n, b_n\}})\) admits an exponential orthonormal basis if and only if \(\rho ^{-1} \in 3{{\mathbb {N}}}\) and  \(\{a_n, b_n\} \equiv \{1, 2\} \ (\mathrm {mod} \ 3)\) for all \(n \in {{\mathbb {N}}}\).

     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as

$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.

     

Weak type (1, 1) of some operators for the Laplacian with drift

Let \(v = (v_1, \ldots , v_n)\) be a vector in \(\mathbb {R}^n {\setminus } \{ 0 \}\). Consider the Laplacian on \(\mathbb {R}^n\) with drift \(\Delta _{v} = \sum _{i = 1}^n \Big ( \frac{\partial ^2}{\partial x_i^2} + 2 v_i \frac{\partial }{\partial x_i} \Big )\) and the measure \(d\mu (x) = e^{2 \langle v, x \rangle } dx\), with respect to which \(\Delta _{v}\) is self-adjoint. Let d and \(\nabla \) denote the Euclidean distance and the gradient operator on \(\mathbb {R}^n\). Consider the space \((\mathbb {R}^n, d, d\mu )\), which has the property of exponential volume growth. We obtain weak type (1, 1) for the Riesz transform \(\nabla (- \Delta _{v} )^{-\frac{1}{2}}\) and for the heat maximal operator, with respect to \(d\mu \). Further, we prove that the uncentered Hardy–Littlewood maximal operator is bounded on \(L^p\) for \(1 < p \le +\infty \) but not of weak type (1, 1) if \(n \ge 2\).     

On Summation of Nonharmonic Fourier Series

Let a sequence \(\Lambda \subset {\mathbb {C}}\) be such that the corresponding system of exponential functions \({\mathcal {E}}(\Lambda ):=\left\{ {\text {e}}^{i\lambda t}\right\} _{\lambda \in \Lambda }\) is complete and minimal in \(L^2(-\pi ,\pi )\), and thus each function \(f\in L^2(-\pi ,\pi )\) corresponds to a nonharmonic Fourier series in \({\mathcal {E}}(\Lambda )\). We prove that if the generating function \(G\) of \(\Lambda \) satisfies the Muckenhoupt \((A_2)\) condition on \({\mathbb {R}}\), then this series admits a linear summation method. Recent results show that the \((A_2)\) condition cannot be omitted.     

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.

     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

Ratio sets of random sets

We study the typical behaviour of the size of the ratio set A / A for a random subset \(A\subset \{1,\dots , n\}\). For example, we prove that \(|A/A|\sim \frac{2\text {Li}_2(3/4)}{\pi ^2}n^2 \) for almost all subsets \(A\subset \{1,\dots ,n\}\). We also prove that the proportion of visible lattice points in the lattice \(A_1\times \cdots \times A_d\), where \(A_i\) is taken at random in [1, n] with \(\mathbb P(m\in A_i)=\alpha _i\) for any \(m\in [1,n]\), is asymptotic to a constant \(\mu (\alpha _1,\dots ,\alpha _d)\) that involves the polylogarithm of order d.     

Duals of Weighted Exponential Systems

The paper considers the basis and frame properties of the system of weighted exponentials \({\mathcal{E}}(g,\mathbb{Z}\backslash F) = \{e^{2\pi i n x} g(x)\}_{n\in\mathbb{Z}\backslash F}\) in \(L^{2}({\mathbb{T}})\), where \(g \in L^{2}({\mathbb{T}}) \backslash\{0\}\) and F??. It is shown that many of the frame properties of \({\mathcal {E}}(g,\mathbb{Z}\backslash F)\) are affected by the cardinalities of F and the behavior of the zeros of g.     

On a Fractional $$ p \& q$$ Laplacian Problem with Critical Sobolev–Hardy Exponents

We consider the following fractional \( p \& q\) Laplacian problem with critical Sobolev–Hardy exponents

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p} u + (-\Delta )^{s}_{q} u = \frac{|u|^{p^{*}_{s}(\alpha )-2}u}{|x|^{\alpha }}+ \lambda f(x, u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^{N}{\setminus } \Omega , \end{array} \right. \end{aligned}$$

where \(0<s<1\), \(1\le q<p<\frac{N}{s}\), \((-\Delta )^{s}_{r}\), with \(r\in \{p,q\}\), is the fractional r-Laplacian operator, \(\lambda \) is a positive parameter, \(\Omega \subset \mathbb {R}^{N}\) is an open bounded domain with smooth boundary, \(0\le \alpha <sp\), and \(p^{*}_{s}(\alpha )=\frac{p(N-\alpha )}{N-sp}\) is the so-called Hardy–Sobolev critical exponent. Using concentration-compactness principle and the mountain pass lemma due to Kajikiya [23], we show the existence of infinitely many solutions which tend to be zero provided that \(\lambda \) belongs to a suitable range.

     

Unique representation bi-basis for rational numbers field

For \(A\subseteq {\mathbb {Q}}\), \(\alpha \in {\mathbb {Q}}\), let \(r_{A}(\alpha )=\#\{(a_{1}, a_{2})\in A^{2}: \alpha =a_{1}+a_{2}, a_{1}\le a_{2}\},\) \(\delta _{A}(\alpha )=\#\{(a_{1}, a_{2})\in A^{2}: \alpha =a_{1}-a_{2} \}.\) In this paper, we construct a set \(A\subset {\mathbb {Q}}\) such that \(r_{A}(\alpha )=1\) for all \(\alpha \in {\mathbb {Q}}\) and \(\delta _{A}(\alpha )=1\) for all \(\alpha \in {\mathbb {Q}}\setminus \{{0}\}\).