Map determined by rank-$$\varvec{}$$ matrice for relatively mall $$\varvec{}$$

Let n and s be integers such that \(1\le s<\frac{n}{2}\), and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\). Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\). In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\), then \(g(x)=f(x)+g(0)\), \(x\in M_n(\mathbb {K})\), for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\). Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \).     

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta <n\) and \(\tau >0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have

$$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$

Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta <2m\) and \(\tau >0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that

$$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$

for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use

$$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$

The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$

where the nonlinearity f has the critical exponential growth.

     

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.     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

A Characterization of Two-Weight Norm Inequality for Littlewood–Paley $$g_{\lambda }^{*}$$-Function

Let \(n\ge 2\) and \(g_{\lambda }^{*}\) be the well-known high-dimensional Littlewood–Paley function which was defined and studied by E. M. Stein,

$$\begin{aligned} g_{\lambda }^{*}(f)(x) =\bigg (\iint _{\mathbb {R}^{n+1}_{+}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda } |\nabla P_tf(y,t)|^2 \frac{\mathrm{d}y \mathrm{d}t}{t^{n-1}}\bigg )^{1/2}, \ \quad \lambda > 1, \end{aligned}$$

where \(P_tf(y,t)=p_t*f(y)\), \(p_t(y)=t^{-n}p(y/t)\), and \(p(x) = (1+|x|^2)^{-(n+1)/2}\), \(\nabla =(\frac{\partial }{\partial y_1},\ldots ,\frac{\partial }{\partial y_n},\frac{\partial }{\partial t})\). In this paper, we give a characterization of two-weight norm inequality for \(g_{\lambda }^{*}\)-function. We show that \(\big \Vert g_{\lambda }^{*}(f \sigma ) \big \Vert _{L^2(w)} \lesssim \big \Vert f \big \Vert _{L^2(\sigma )}\) if and only if the two-weight Muckenhoupt \(A_2\) condition holds, and a testing condition holds:

$$\begin{aligned} \sup _{Q : \text {cubes}~\mathrm{in} \ {\mathbb {R}^n}} \frac{1}{\sigma (Q)} \int _{{\mathbb {R}^n}} \iint _{\widehat{Q}} \Big (\frac{t}{t+|x-y|}\Big )^{n\lambda }|\nabla P_t(\mathbf {1}_Q \sigma )(y,t)|^2 \frac{w \mathrm{d}x \mathrm{d}t}{t^{n-1}} \mathrm{d}y < \infty , \end{aligned}$$

where \(\widehat{Q}\) is the Carleson box over Q and \((w, \sigma )\) is a pair of weights. We actually prove this characterization for \(g_{\lambda }^{*}\)-function associated with more general fractional Poisson kernel \(p^\alpha (x) = (1+|x|^2)^{-{(n+\alpha )}/{2}}\). Moreover, the corresponding results for intrinsic \(g_{\lambda }^*\)-function are also presented.

     

Existence and Multiplicity of Solutions for Impulsive Fractional Differential Equations

In this paper, we study the existence of solutions for the following impulsive fractional boundary-value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} - \frac{\mathrm{d}}{\mathrm{d}t} \Big (\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t)) \Big ) = \lambda u (t) + f (t, u (t)), &{} t \ne t_j, \;\;\text {a.e.}\;\; t \in [0, T],\\ \Delta \Big (\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j)) \Big ) = I_j (u (t_j)), &{} j = 1, 2, \ldots , n,\\ u (0) = u (T) = 0, \end{array}\right. } \end{aligned}$$

where \(\alpha \in (1/2, 1]\), \(0 = t_0< t_1< t_2< \cdots< t_n< t_{n +1} = T\), \(\lambda \) is a parameter and \(f :[0, T] \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) and \(I_j : {\mathbb {R}} \rightarrow {\mathbb {R}}\), \(j = 1, \ldots , n\) are continuous functions and

$$\begin{aligned}&\Delta \left( \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j)) \right) \\&\quad = \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^+) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^+) \\&\qquad -\, \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^-) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^-) ,\\&\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^+) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^+)) \nonumber \\&\quad = \lim _{t \rightarrow t_j^+} \left( \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t))\right) ,\\&\frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t_j^-) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t_j^-)) \\&\quad = \lim _{t \rightarrow t_j^-} \left( \frac{1}{2} {}_0D_t^{\alpha - 1} ({}_0^c D_t^\alpha u (t) ) - \frac{1}{2} {}_tD_T^{\alpha - 1} ({}_t^c D_T^\alpha u (t))\right) . \end{aligned}$$

By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution and infinitely many solutions.

     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

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 \(\mathbb {E}_{p}^{\alpha }\otimes \mathbb {B}_{p}\).     

On the classification of 4-dimensional $$(m,\rho )$$-quasi-Einstein manifolds with harmonic Weyl curvature

In this paper we study four-dimensional \((m,\rho )\)-quasi-Einstein manifolds with harmonic Weyl curvature when \(m\notin \{0,\pm 1,-2,\pm \infty \}\) and \(\rho \notin \{\frac{1}{4},\frac{1}{6}\}\). We prove that a non-trivial \((m,\rho )\)-quasi-Einstein metric g (not necessarily complete) is locally isometric to one of the following: (i) \({\mathcal {B}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {B}}^2_\frac{R}{2(m+2)}\) is the northern hemisphere in the two-dimensional (2D) sphere \({\mathbb {S}}^2_\frac{R}{2(m+2)}\), \({\mathbb {N}}_\delta \) is a 2D Riemannian manifold with constant curvature \(\delta \), and R is the constant scalar curvature of g. (ii) \({\mathcal {D}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\), where \({\mathcal {D}}^2_\frac{R}{2(m+2)}\) is half (cut by a hyperbolic line) of hyperbolic plane \({\mathbb {H}}^2_\frac{R}{2(m+2)}\). (iii) \({\mathbb {H}}^2_\frac{R}{2(m+2)}\times {\mathbb {N}}^2_\frac{R(m+1)}{2(m+2)}\). (iv) A certain singular metric with \(\rho =0\). (v) A locally conformal flat metric. By applying this local classification, we obtain a classification of the complete \((m,\rho )\)-quasi-Einstein manifolds given the condition of a harmonic Weyl curvature. Our result can be viewed as a local classification of gradient Einstein-type manifolds. A corollary of our result is the classification of \((\lambda ,4+m)\)-Einstein manifolds, which can be viewed as (m, 0)-quasi-Einstein manifolds.     

Sign-changing blowing-up solutions for a non-homogeneous elliptic equation at the critical exponent

We consider the equation \(-\Delta u = |u| ^{\frac{4}{n-2}}u + \varepsilon f(x) \) under zero Dirichlet boundary conditions in a bounded domain \(\Omega \) in \(\mathbb {R}^{n}\), \(n \ge 3\), with \(f\ge 0\), \(f\ne 0\). We find sign-changing solutions with large energy. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem

$$\begin{aligned} -\Delta w = |w|^{\frac{4}{n-2}} w, \quad w \in {\mathcal D}^{1,2} (\mathbb {R}^n) \end{aligned}$$

recently constructed in del Pino et al. (J Differ Equ 251(9):2568–2597, 2011).

     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

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.     

Eichler’s commutation relation and some other invariant subspaces of Hecke operators

Let k be an odd positive integer, L a lattice on a regular positive definite k-dimensional quadratic space over \(\mathbb {Q}\), \(N_L\) the level of L, and \(\mathscr {M}(L)\)  be the linear space of \(\theta \)-series attached to the distinct classes in the genus of L. We prove that, for an odd prime \(p|N_L\), if \(L_p=L_{p,1}\,\bot \, L_{p,2}\), where \(L_{p,1}\) is unimodular, \(L_{p,2}\) is (p)-modular, and \(\mathbb {Q}_pL_{p,2}\) is anisotropic, then \(\mathscr {M}(L;p):=\) \(\mathscr {M}(L)\) \(+T_{p^2}.\) \(\mathscr {M}(L)\)  is stable under the Hecke operator \(T_{p^2}\). If \(L_2\) is isometric to \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle 2\varepsilon \rangle \) or \(\left( \begin{array}{ll}0&{}1\\ 1&{}0\end{array}\right) ^{\kappa }\,\bot \, \langle \varepsilon \rangle \) with \(\varepsilon \in \mathbb {Z}_2^{\times }\) and \(\kappa :=\frac{k-1}{2}\), then \(\mathscr {M}(L;2):=T_{2^2}.\mathscr {M}(L)+T_{2^2}^2.\,\mathscr {M}(L)\) is stable under the Hecke operator \(T_{2^2}\). Furthermore, we determine some invariant subspaces of the cusp forms for the Hecke operators.     

A logarithmically improved regularity criterion for the supercritical quasi-geostrophic equations in Besov space

In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space \(\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)\). The result shows that if θ is a weak solutions satisfies

$$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha }{{\alpha - r}}} }}{{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2}{r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$

then θ is regular at t = T. In view of the embedding \({L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}\) with \(2 \leqslant p < \frac{2}{r}\) and 0 ≤ r < 1, we see that our result extends the results due to [20] and [31].

     

Behaviour at the non-positive integers of Dirichlet series associated to polynomials of several variables

In this paper, we relate the special values at a non-positive integer \({\underline{\mathbf{s}}=(s_{1},\ldots, s_{r})= -\underline{\mathbf{N}}= (-N_{1},\ldots, -N_{r})}\) obtained by meromorphic continuation of the multiple Dirichlet series \({{Z(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\sum_{\underline{m}\in {\mathbb{N}}^{*n}}{\frac{1}{\prod_{i=1}^{r}{P_{i}^{ s_{i}}(\underline{m})}}}}}\) to special values of the function \({Y(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\int_{[1, +\infty[^{n}} {\prod_{i=1}^{r}{P_{i}^{- s_{i}}(\underline{\mathbf{x}})}\; d{\underline{\mathbf{x}}}}}\) where \({\underline{\mathbf{P}}=(P_{1},..., P_{r}),\; (r\geq 1)}\) are elliptic polynomials in “\({n}\) ” variables. We prove a simple relation between \({Z(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\) and \({Y(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\), such that for all \({\underline{\mathbf{a}} \in {\mathbb{R}}^{n}_{+}}\), we denote \({\underline{\mathbf{P}}_{\underline{\mathbf{a}}}:=(P_{1 \underline{\mathbf{a}}},\ldots, P_{r \underline{\mathbf{a}}})}\), where \({P_{i\;\underline{\mathbf{a}}}(\underline{\mathbf{x}}):= P_i(\underline{\mathbf{x}}+ \underline{\mathbf{a}})\; (1\leq i\leq r)}\) is the shifted polynomial.     

Parity check matrices and product representations of squares

Let \(N_{\mathbb{F}} \)(n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field \(\mathbb{F}\) in which each column has at most r nonzero entries and every k columns are linearly independent over \(\mathbb{F}\). We obtain near-optimal upper bounds for \(N_{\mathbb{F}} \)(n,k,r) in the case k > r. Namely, we show that \(N_\mathbb{F} (n,k,r) \ll n^{\frac{r}{2} + \frac{{cr}}{k}} \) where \(c \approx \frac{4}{3}\) for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.     

Markoff–Lagrange spectrum and extremal numbers

Let \( \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) \) denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant c > 0 with the property that, for arbitrarily large values of X, the inequalities\( \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} \)admit no non-zero solution \( \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} \). Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X ≥ 1. Such extremal numbers are transcendental and their set is stable under the action of \( {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) \) on \( \mathbb{R}\backslash \mathbb{Q} \) by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf _q→∞ q||qξ|| is \( \frac{1}{3} \), the largest possible value for a non-quadratic number, and that there is a natural bijection between the \( {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) \)-equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.     

Infinitely many non-radial solutions for fractional Nirenberg problem

In this paper, the prescribed \(\sigma \)-curvature problem

$$\begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned}$$

is considered. When \({\tilde{K}}(x)\) is some axis symmetric function on \({\mathbb {S}}^N\), by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for \(0<\sigma \le 1\) and \(N> 2\sigma +2\).

     

On limit theorems for Banach-space-valued linear processes

Let \( {\left( {{\epsilon_i}} \right)_{i \in \mathbb{Z}}} \) be i.i.d. random elements in a separable Banach space \( \mathbb{E} \), and let \( \mathop {\left( {{a_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) be continuous linear operators from \( \mathbb{E} \) to a Banach space \( \mathbb{F} \) such that \( \sum\nolimits_{i \in \mathbb{Z}} {\left\| {{a_i}} \right\|} \) is finite. We prove that the linear process \( \mathop {\left( {{X_n}} \right)}\nolimits_{n \in \mathbb{Z}} \) defined by \( {X_n}: = \sum\nolimits_{i \in \mathbb{Z}} {{a_i}} \left( {{\epsilon_{n - i}}} \right) \) inherits from \( \mathop {\left( {{\epsilon_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) the central limit theorem and functional central limit theorems in various Banach spaces of \( \mathbb{F} \)-valued functions, including Hölder spaces.