Existence and concentration of solution for a class of fractional Hamiltonian systems with subquadratic potential

In this article, we consider the following fractional Hamiltonian systems:

$$\begin{aligned} {_{t}}D_{\infty }^{\alpha }({_{-\infty }}D_{t}^{\alpha }u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb {R}, \end{aligned}$$

where \(\alpha \in (1/2, 1)\), \(\lambda >0\) is a parameter, \(L\in C(\mathbb {R}, \mathbb {R}^{n\times n})\) and \(W \in C^{1}(\mathbb {R} \times \mathbb {R}^n, \mathbb {R})\). Unlike most other papers on this problem, we require that L(t) is a positive semi-definite symmetric matrix for all \(t\in \mathbb {R}\), that is, \(L(t) \equiv 0\) is allowed to occur in some finite interval \(\mathbb {I}\) of \(\mathbb {R}\). Under some mild assumptions on W, we establish the existence of nontrivial weak solution, which vanish on \(\mathbb {R} \setminus \mathbb {I}\) as \(\lambda \rightarrow \infty ,\) and converge to \(\tilde{u}\) in \(H^{\alpha }(\mathbb {R})\); here \(\tilde{u} \in E_{0}^{\alpha }\) is nontrivial weak solution of the Dirichlet BVP for fractional Hamiltonian systems on the finite interval \(\mathbb {I}\). Furthermore, we give the multiplicity results for the above fractional Hamiltonian systems.

     

Boundedness for Higher Order Commutators of Fractional Integrals on Variable Exponent Herz–Morrey Spaces

In this paper, the boundedness for higher order commutators of fractional integrals is obtained on variable exponent Herz–Morrey spaces \( M\dot{K}_{p, q(\cdot )}^{\alpha (\cdot ), \lambda }(\mathbb {R}^{n})\) applying some properties of variable exponent and \(\mathrm {BMO}\) function, where \(\alpha (x)\in L^{\infty }(\mathbb {R}^{n})\) are log-Hölder continuous both at the origin and at infinity, and q(x) satisfies the logarithmic continuity condition.     

Efficient construction of 2-chains representing a basis of $H_{2}(\overline {\Omega }, \partial {\Omega }; \mathbb {Z})$

We present an efficient algorithm for the construction of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of \(\overline {\Omega }\) the idea is to find, first g different 1-boundaries of \(\overline {\Omega }\) with supports contained in ?Ω whose homology classes in \(\mathbb {R}^{3} \setminus {\Omega }\) form a basis of \(H_{1}(\mathbb {R}^{3} \setminus {\Omega };\mathbb {Z})\), and then to construct a set of 2-chains in \(\overline {\Omega }\) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in \(\overline {\Omega }\) modulo ?Ω form a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\). We devise a simple procedure for the construction of the required set of 1-boundaries of \(\overline {\Omega }\) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms.     

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.

     

Construction of sign-changing solutions for a harmonic equation with subcritical exponent

In this paper, we study the harmonic equation involving subcritical exponent \((P_{\varepsilon })\): \( \Delta u = 0 \), in \(\mathbb {B}^n\) and \(\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }\) on \( \mathbb {S}^{n-1}\) where \(\mathbb {B}^n \) is the unit ball in \(\mathbb {R}^n\), \(n\ge 5\) with Euclidean metric \(g_0\), \(\partial \mathbb {B}^n = \mathbb {S}^{n-1}\) is its boundary, K is a function on \(\mathbb {S}^{n-1}\) and \(\varepsilon \) is a small positive parameter. We construct solutions of the subcritical equation \((P_{\varepsilon })\) which blow up at two different critical points of K. Furthermore, we construct solutions of \((P_{\varepsilon })\) which have two bubbles and blow up at the same critical point of K.     

Duality of Non-Homogeneous Central Herz-Morrey-Musielak-Orlicz Spaces

We determine the associate space of non-homogeneous central Herz-Morrey-Musielak-Orlicz space \(\mathcal {H}^{\Phi ,q,\omega }(\textbf {R}^{N})\). We also determine the associate spaces of the space \(\underline {\mathcal {H}}^{\Phi ,q,\omega }(\textbf {R}^{N})\) and its complementary space \(\overline {\mathcal {H}}^{\Phi ,q,\omega }(\textbf {R}^{N})\).     

Fractional Sobolev metrics on spaces of immersed curves

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) and on its Sobolev completions \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\). We prove local well-posedness of the geodesic equations both on the Banach manifold \({\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) and on the Fréchet-manifold \(\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)\) provided the order of the metric is greater or equal to one. In addition we show that the \(H^s\)-metric induces a strong Riemannian metric on the Banach manifold \({\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})\) of the same order s, provided \(s>\frac{3}{2}\). These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.     

Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables

The main object of study in this paper is the double holomorphic Eisenstein series \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) having two complex variables \(\mathbf{s}=(s_1,s_2)\) and two parameters \(\mathbf{z}= (z_1,z_2)\) which satisfies either \(\mathbf{z}\in (\mathfrak {H}^+)^2\) or \(\mathbf{z}\in (\mathfrak {H}^-)^2\), where \(\mathfrak {H}^{\pm }\) denotes the complex upper and lower half-planes, respectively. For \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\), its transformation properties and asymptotic aspects are studied when the distance \(|z_2-z_1|\) becomes both small and large under certain natural settings on the movement of \(\mathbf{z}\in (\mathfrak {H}^{\pm })^2\). Prior to the proofs our main results, a new parameter \(\eta \), which plays a pivotal role in describing our results, is introduced in connection with the difference \(z_2-z_1\). We then establish complete asymptotic expansions for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) when \(\mathbf{z}\) moves within the poly-sector either \((\mathfrak {H}^+)^2\) or \((\mathfrak {H}^-)^2\), so as to \(\eta \rightarrow 0\) through \(|\arg \eta |<\pi /2\) in the ascending order of \(\eta \) (Theorem 1). This further leads us to show that counterpart expansions exist for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in the descending order of \(\eta \) as \(\eta \rightarrow \infty \) through \(|\arg \eta |<\pi /2\) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in closed forms for integer lattice point \(\mathbf{s}\in \mathbb {Z}^2\) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) at \(\mathbf{s}\in \mathbb {Z}^2\) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases \(2\pi (1, z_j)\), \(j=1,2\). The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric functions.     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).

     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.

     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.     

Littlewood-Paley Characterizations of Hajłasz-Sobolev and Triebel-Lizorkin Spaces via Averages on Balls

Let \(p\in (1,\infty )\) and \(q\in [1,\infty )\). In this article, the authors characterize the Triebel-Lizorkin space \({F}^{\alpha }_{p,q}(\mathbb {R}^{n})\) with smoothness order α ∈ (0, 2) via the Lusin-area function and the \(g_{\lambda }^{*}\)-function in terms of difference between f(x) and its ball average \(B_{t}f(x):=\frac 1{|B(x,t)|}{\int }_{B(x,t)}f(y)\,dy\) over the ball B(x, t) centered at \(x\in \mathbb {R}^{n}\) with radius t ∈ (0, 1). As an application, the authors obtain a series of characterizations of \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) via pointwise inequalities, involving ball averages, in spirit close to Haj?asz gradients, here some interesting phenomena naturally appear that, in the end-point case when α = 2, some of these pointwise inequalities characterize the Triebel-Lizorkin spaces \(F^{2}_{p,2}(\mathbb {R}^{n})\), while not \(F^{2}_{p,\infty }(\mathbb {R}^{n})\), and that some of other obtained pointwise characterizations are only known to hold true for \(F^{\alpha }_{p,\infty }(\mathbb {R}^{n})\) with \(p\in (1,\infty )\), α ∈ (0, 2) or α ∈ (n/p, 2). In particular, some new pointwise characterizations of Haj?asz-Sobolev spaces via ball averages are obtained. Since these new characterizations only use ball averages, they can be used as starting points for developing a theory of Triebel-Lizorkin spaces with smoothness orders not less than 1 on spaces of homogeneous type.     

On $$L^p$$-Boundedness of Pseudo-Differential Operators of Sjöstrand’s Class

We extended the known result that symbols from modulation spaces \(M^{\infty ,1}(\mathbb {R}^{2n})\), also known as the Sjöstrand’s class, produce bounded operators in \(L^2(\mathbb {R}^n)\), to general \(L^p\) boundedness at the cost of loss of derivatives. Indeed, we showed that pseudo-differential operators acting from \(L^p\)-Sobolev spaces \(L^p_s(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\) spaces with symbols from the modulation space \(M^{\infty ,1}(\mathbb {R}^{2n})\) are bounded, whenever \(s\ge n|1/p-1/2|.\) This estimate is sharp for all \(1< p<\infty \).     

Complexity of oscillatory integrals on the real line

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space \(H^{s} (\mathbb {R})\) and from the space \(C^{s}(\mathbb {R})\) with an arbitrary integer s ≥ 1. We find tight upper and lower bounds for the worst case error of optimal algorithms that use n function values. More specifically, we study integrals of the form

$$ I_{k}^{\varrho} (f) = {\int}_{\mathbb{R}} f(x) \,\mathrm{e}^{-i\,kx} \varrho(x) \, \mathrm{d} x\ \ \ \text{for}\ \ f\in H^{s}(\mathbb{R})\ \ \text{or}\ \ f\in C^{s}(\mathbb{R}) $$

(1)

with \(k\in {\mathbb {R}}\) and a smooth density function ρ such as \( \rho (x) = \frac {1}{\sqrt {2 \pi }} \exp (-x^{2}/2)\). The optimal error bounds are \({\Theta }((n+\max (1,|k|))^{-s})\) with the factors in the Θ notation dependent only on s and ?.     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.     

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 \(2^{n-1}\).     

Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary

We consider the problem

$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$

where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\ge 3,\) \(V_{\infty }>0,\) \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),\) \(\inf _{\mathbb {R}^{N}}V>-V_{\infty }\) and \(2<p<\frac{2N}{N-2}\). We assume V is periodic in the first m variables, and decays exponentially to zero in the remaining ones. We also assume that \(\Omega \) is periodic in the first m variables and has bounded complement in the other ones. Then, assuming that \(\Omega \) and V are invariant under some suitable group of symmetries on the last \(N-m\) coordinates of \(\mathbb {R}^{N}\), we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\)

     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

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 \(\alpha \in {\mathbb {Q}}\setminus \{{0}\}\).