Some properties of the Yamabe soliton and the related nonlinear elliptic equation

Firstly we prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot \nabla u=0$ in $\mathbb{R }^{n}$ when $n\ge 3$ , $0<m\le \frac{n-2}{n}$ , $\alpha <0$ and $\beta \le 0$ and prove various properties of the solution of the above elliptic equation for other parameter range of $\alpha $ and $\beta $ . Then these results are applied to prove some results on Yamabe solitons including the exact behaviour of the metric of the Yamabe soliton, its scalar curvature and sectional curvature, at infinity. A new proof of a result of Daskalopoulos and Sesum (The classification of locally conformally flat Yamabe solitons, http://arxiv.org/abs/1104.2242) on the positivity of the sectional curvature of Yamabe solitons is also presented.     

The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation $(-\partial _t^2+\Delta )u+u^5=0$ in ${\mathbb{R }}^{1+3}$ and prove the existence of energy class solutions which are of the form $$\begin{aligned} u(t,x)=t^\frac{\mu }{2}W(t^\mu x)+\eta (t,x) \end{aligned}$$ in the forward lightcone $\{(t,x)\in {\mathbb{R }}\times {\mathbb{R }}^3: |x|\le t, t\gg 1\}$ where $W(x)=(1+\frac{1}{3} |x|^2)^{-\frac{1}{2}}$ is the ground state soliton, $\mu $ is an arbitrary prescribed real number (positive or negative) with $|\mu |\ll 1$ , and the error $\eta $ satisfies $$\begin{aligned} \Vert \partial _t \eta (t,\cdot )\Vert _{L^2(B_t)} +\Vert \nabla \eta (t,\cdot )\Vert _{L^2(B_t)}\ll 1,\quad B_t:=\{x\in {\mathbb{R }}^3: |x|<t\} \end{aligned}$$ for all $t\gg 1$ . Furthermore, the kinetic energy of $u$ outside the cone is small. Consequently, depending on the sign of $\mu $ , we obtain two new types of solutions which either concentrate as $t\rightarrow \infty $ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

Some properties of a hypergeometric function which appear in an approximation problem

In this paper we consider properties and power expressions of the functions $f:(-1,1)\rightarrow \mathbb{R }$ and $f_L:(-1,1)\rightarrow \mathbb{R }$ , defined by $$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$ respectively, where $\gamma $ is a real parameter, as well as some properties of a two parametric real-valued function $D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$ , defined by $$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$ The inequality of Turán type $$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1<x<1, \end{aligned}$$ for $\alpha +\beta >0$ is proved, as well as an opposite inequality if $\alpha +\beta <0$ . Finally, for the partial derivatives of $D(x;\alpha ,\beta )$ with respect to $\alpha $ or $\beta $ , respectively $A(x;\alpha ,\beta )$ and $B(x;\alpha ,\beta )$ , for which $A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )$ , some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated” quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar–Rahmanov–Saff numbers.     

Some applications of the double large sieve

We sharpen a procedure of Cao and Zhai (J Théorie Nombres Bordeaux,11: 407–423, 1999) to estimate the sum $$\begin{aligned} \sum _{m\sim M} \sum _{n\sim N} a_m b_n \, e\left(\frac{F m^\alpha n^\beta }{M^\alpha N^\beta }\right) \end{aligned}$$ with $|a_m|,\ |b_n| \le 1$ . We apply this to give bounds for the discrepancy (mod 1) of the sequence $\{p^c: p\le X\}$ where $p$ is a prime variable, in the range $\frac{130}{79}\le c \le \frac{11}{5}$ . An alternative strategy is used for the range $1.48 \le c \le \frac{130}{79}$ . We use further exponential sum estimates to show that for large $R>0$ , and a small constant $\eta >0$ , the inequality $$\begin{aligned} \left| p_1^c+p_2^c+p_3^c+p_4^c+p_5^c - R\right| < R^{-\eta } \end{aligned}$$ holds for many prime tuples, provided $2<c\le 2.041$ . This improves work of Cao and Zhai (Monatsh Math, 150:173–179, 2007) and a theorem claimed by Shi and Liu (Monatsh Math, published online, 2012).     

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $\Delta ^2 u = \lambda f(u) $ in $ \Omega $ with $ \Delta u = u =0 $ on $ {\partial \Omega }$ , where $ \lambda > 0$ is a parameter, $ \Omega $ is a bounded domain in $\mathbb{R }^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$ , $ f(u)=(1+u)^p $ or $ f(u)= \frac{1}{(1-u)^p}$ where $ p>1$ . We show the extremal solution is smooth, provided $$\begin{aligned} N < 2 + 4 \sqrt{2} + 4 \sqrt{ 2 - \sqrt{2}} \approx 10.718 \text{ when} f(u)=e^u, \end{aligned}$$ and $$\begin{aligned} N < \frac{4p}{p-1} + \frac{4(p+1)}{p-1} \left( \sqrt{ \frac{2p}{p+1}} + \sqrt{ \frac{2p}{p+1} - \sqrt{ \frac{2p}{p+1}}} - \frac{1}{2} \right) \end{aligned}$$ when $ f(u)=(u+1)^p$ . New results are also obtained in the case where $ f(u)=(1-u)^{-p}$ . These are substantial improvements to various results on critical dimensions obtained recently by various authors. To do that, we derive a new stability inequality satisfied by minimal solutions of the above equation, which is more amenable to estimates as it allows a method of proof reminiscent of the second order case.     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Multibump bound states for quasilinear Schrödinger systems with critical frequency

In this paper, we are concerned with the multibump solutions for the following quasilinear Schrödinger system in ${\mathbb{R}^N}$ : $$\left\{\begin{array}{ll}-\Delta{u} + \lambda{a(x)u} - \frac{1}{2}(\Delta|u|^2)u = \frac{2\alpha}{\alpha + \beta}|u|^{\alpha-2}|\upsilon|^\beta u, \\-\Delta{\upsilon} + \lambda{b(x)\upsilon} - \frac{1}{2}(\Delta|\upsilon|^2)\upsilon = \frac{2\beta}{\alpha + \beta}|u|^\alpha|\upsilon|^{\beta-2} \upsilon, \\u(x) \rightarrow 0, \upsilon(x) \rightarrow 0 \quad as|x| \rightarrow \infty,\end{array}\right.$$ where λ > 0 is a parameter, α, β > 2 satisfying α + β < 2 · 2*, here ${2^{*} = \frac{2N}{N-2}}$ is the critical Sobolev exponent for ${N \geq 3}$ and a(x), b(x) are nonnegative potentials. Using variational methods, we prove that if the zero sets of a(x) and b(x) have several common isolated connected components ${\Omega_{1}, . . . ,\Omega_{k}}$ such that the interior of ${\Omega_{i} (i = 1, 2, . . . , k)}$ is not empty and ${\partial\Omega_{i} (i = 1, 2, . . . , k)}$ is smooth, then for λ sufficiently large, the system admits, for any nonempty subset ${J \subset \{1, 2, . . . , k\}}$ , a solution which is trapped in a neighborhood of ${\cup_{j\epsilon{J}} \Omega_{j}}$ .     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

On nonlinear elliptic equations with Hardy potential and L^{1}-data

We consider a class of nonlinear elliptic equations involving the Hardy potential and lower order terms whose simplest model is $$\begin{aligned} -\Delta u +b(|u|)|\nabla u|^{2}+\nu |u|^{s-1}u=\lambda \frac{u}{|x|^{2}}+f \end{aligned}$$ in a bounded open $\varOmega $ of $\mathbf{R }^{N}, N\ge 3,$ containing the origin, $s>\frac{N}{N-2}, \nu $ and $\lambda $ are positive real numbers. We prove that the presence of the term $\nu |u|^{s-1}u$ has an effect on the existence of solutions when $f\in L^{1}(\varOmega )$ assuming only that $b\in L^{1}(\mathbf{R })$ without any sign condition (i.e. $b(s)s\ge 0$ ).     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

Critical exponent for semilinear wave equation with critical potential

We consider the Cauchy problem for the semilinear wave equation ${u_{tt} - \Delta u + V(x)u_t = |u|^p}$ .When ${V(x) = V_0(1 + |x|^2)^{-1/2}, V_0 \geq n}$ , we prove that the critical exponent for the problem is ${p_c(n)=\left\{\begin{array}{ll} 1+\frac{2}{n-1},& n \geq 2,\ +\infty,& n=1. \end{array}\right.}$      

Liouville theorems for stable solutions of biharmonic problem

We prove some Liouville type results for stable solutions to the biharmonic problem $\Delta ^2 u= u^q, \,u>0$ in $\mathbb{R }^n$ where $1 < q < \infty $ . For example, for $n \ge 5$ , we show that there are no stable classical solution in $\mathbb{R }^n$ when $\frac{n+4}{n-4} < q \le \left(\frac{n-8}{n}\right)_+^{-1}$ .     

Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α?>?0 such that ${|\Delta(E)| \gtrsim q}$ whenever ${|E| \gtrsim q^{\alpha}}$ , where ${E \subset {\mathbb {F}}_q^d}$ , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ${\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}$ . Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold ${\frac{d+1}{2}}$ , and in Hart et?al. (Trans Am Math Soc 363:3255–3275, 2011) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to ${\tfrac{4}{3}}$ , consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set ${\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}$ for a pin ${y\in E}$ has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000) showed that if the Hausdorff dimension of a set E is greater than ${\tfrac{d+1}{2}}$ , then the Lebesgue measure of Δ _y (E) is positive for almost every pin y. In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set ${\Pi_y(E)=\{x\cdot y: x\in E\}}$ . Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to ${\frac{d^2}{2d-1}}$ . The pinned dot product result for Cartesian products implies the following sum-product result. Let ${A\subset \mathbb F_q}$ and ${z\in \mathbb F^*_q}$ . If ${|A|\geq q^{\frac{d}{2d-1}}}$ then there exists a subset ${E'\subset A\times \dots \times A=A^{d-1}}$ with ${|E'|\gtrsim |A|^{d-1}}$ such that for any ${(a_1,\dots, a_{d-1}) \in E'}$ , $$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$ where ${a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}$ . A generalization of the Falconer distance problem is to determine the minimal α?>?0 such that E contains a congruent copy of a positive proportion of k-simplices whenever ${|E| \gtrsim q^{\alpha}}$ . Here the authors improve on known results (for k?>?3) using Fourier analytic methods, showing that α may be taken to be ${\frac{d+k}{2}}$ .     

Spectrum of Functions and Distributions for the Jacobi-Dunkl Transform on {\mathbb{R}}

We establish real Paley-Wiener theorems for the Jacobi-Dunkl transform on ${\mathbb{R}}$ . More precisely, we characterize the functions in the generalized Schwartz space ${\mathcal{S}^{r}_{\alpha , \beta}(\mathbb{R})}$ and in ${L^{p}_{{A}_{\alpha , \beta}} \mathbb{R})}$ whose Jacobi-Dunkl transform has bounded, unbounded, convex and nonconvex support. Finally, we study the spectral problem on the generalized tempered distributions ${\mathcal{S}^{'r}_{\alpha , \beta}(\mathbb{R})}$ .     

The circle method and bounds for L-functions - I

Let $f$ be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let $\chi $ be a primitive character of conductor $M$ . For the twisted $L$ -function $L(s, f\otimes \chi )$ we establish the hybrid subconvex bound $$\begin{aligned} L\left( \frac{1}{2}+it, f\otimes \chi \right) \ll (M(3+|t|))^{\frac{1}{2}-\frac{1}{18}+\varepsilon }, \end{aligned}$$ for $t\in \mathbb{R }$ . The implied constant depends only on the form $f$ and $\varepsilon $ .