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In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

The main aim of this paper is to present some theorems in order to guarantee existence and uniqueness of best proximity points under rational contractivity conditions using very general test functions. To illustrate the variety of possible test functions, we include some examples of pairs of functions which are included in innovative papers published in the last years. As a consequence, we prove that our results unify and extend some recent results in this field.

This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion 𝒯 ¯ {\overline{\mathcal{T}}} of Teichmüller space. The boundary of 𝒯 ¯ {\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space 𝐇 ¯ {\overline{\bf H}} called the model space. The significance of 𝐇 ¯ {\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of 𝒯 ¯ {\overline{\mathcal{T}}} . The main result of the paper is that certain subsets of 𝐇 ¯ {\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on 𝐇 ¯ {\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.

This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results.

The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

In this paper, we study the L q {L^{q}} - L r {L^{r}} boundedness of bi-parameter Fourier integral operators defined by general rough Hörmander class amplitudes.

This paper studies the two-weight estimates of variation and oscillation operators for commutators of singular integrals with weighted BMO {\mathrm{BMO}} functions. A new characterization of weighted BMO {\mathrm{BMO}} spaces via the boundedness of variation and oscillation operators for the iterated commutators of Calderón–Zygmund singular integrals in the two-weight setting is given.

In this paper, the well-posedness of the higher-order Benjamin–Ono equation u t + ℋ ⁢ ( u x ⁢ x ) + u x ⁢ x ⁢ x = u ⁢ u x - ∂ x ⁡ ( u ⁢ ℋ ⁢ ∂ x ⁡ u + ℋ ⁢ ( u ⁢ ∂ x ⁡ u ) ) u_{t}+\mathcal{H}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{H}\partial_{x}% u+\mathcal{H}(u\partial_{x}u)) is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in H 3 / 4 {H^{3/4}} without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equation u t + 𝒢 δ ⁢ ( u x ⁢ x ) + u x ⁢ x ⁢ x = u ⁢ u x - ∂ x ⁡ ( u ⁢ 𝒢 δ ⁢ ∂ x ⁡ u + 𝒢 δ ⁢ ( u ⁢ ∂ x ⁡ u ) ) , 𝒢 δ = ℱ x - 1 ⁢ i ⁢ ( coth ⁡ ( δ ⁢ ξ ) ) ⁢ ℱ x , u_{t}+\mathcal{G}_{\delta}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{G}_{% \delta}\partial_{x}u+\mathcal{G}_{\delta}(u\partial_{x}u)),\quad\mathcal{G}_{% \delta}=\mathcal{F}_{x}^{-1}i(\coth(\delta\xi))\mathcal{F}_{x}, is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in H 3 / 4 {H^{3/4}} .

For the completed Riemann zeta function ξ ⁢ ( s ) {\xi(s)} , it is known that the Riemann hypothesis for ξ ⁢ ( s ) {\xi(s)} implies the Riemann hypothesis for ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} , where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence ( α ⁢ γ m ) {(\alpha\gamma_{m})} , where α is any fixed non-zero real number and γ m {\gamma_{m}} runs over the imaginary parts of the zeros of ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} . We also obtain a zero density estimate and an explicit formula for the zeros of ξ ( m ) ⁢ ( s ) {\xi^{(m)}(s)} . In particular, all our results hold uniformly for 0 ≤ m ≤ g ⁢ ( T ) {0\leq m\leq g(T)} , where the function g ⁢ ( T ) {g(T)} tends to infinity with T and g ⁢ ( T ) = o ⁢ ( log ⁡ log ⁡ T ) {g(T)=o(\log\log T)} .

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝ N {\mathbb{R}^{N}} ( N ∈ ℕ + {N\in\mathbb{N}^{+}} ), provided that the initial datum is sufficiently small. In addition, the L p {L^{p}} -norm ( 1 ≤ p ≤ ∞ {1\leq p\leq\infty} ) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

In this paper we investigate the problem of uniform continuity of nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. In particular, we give the sufficient conditions for nonautonomous superposition operators to continuously map a space of functions of bounded Λ-variation into itself. The conditions cover the generators being functions of C 1 {C^{1}} -class (in view of two variables), but also allow for less regular functions, including discontinuous generators.

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

In this paper we obtain regularity estimates in weighted Orlicz spaces for the Calderón–Zygmund singular integral operators under certain optimal conditions.

The main purpose of this paper is to establish a Calderón–Vaillancourt-type theorem on L p ${L^{p}}$ estimates for bi-parameter and bilinear pseudo-differential operators with subcritical order.

In this paper, we give estimates of the L p ⁢ ( ℝ n ) ${L^{p}(\mathbb{R}^{n})}$ ( 1 < p < ∞ ${1 ) and weak type ( 1 , 1 ) ${(1,1)}$ norm of singular integral operator T Ω ${T_{\Omega}}$ with homogeneous kernel defined by T Ω ⁢ f ⁢ ( x ) = p . v . ∫ ℝ n Ω ⁢ ( x - y ) | x - y | n ⁢ f ⁢ ( y ) ⁢ 𝑑 y , $T_{\Omega}f(x)=\mathrm{p.v.}\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}% f(y)\,dy,$ where Ω satisfies L 1 ${L^{1}}$ -Dini condition with mean value zero on 𝕊 n - 1 ${\mathbb{S}^{n-1}}$ . More precisely, we show that the following two estimates hold: ∥ T Ω ⁢ f ∥ p ≲ ( ( log ⁡ n ) ⁢ ∥ Ω ∥ 1 + ∫ 0 1 n ω ~ 1 ⁢ ( δ ) δ ⁢ 𝑑 δ ) ⁢ ∥ f ∥ p , $\|T_{\Omega}f\|_{p}\lesssim\Biggl{(}(\log n)\|\Omega\|_{1}+\int_{0}^{\frac{1}{% n}}\frac{\tilde{\omega}_{1}(\delta)}{\delta}\,d\delta\Biggr{)}\|f\|_{p},$ and for any λ > 0 ${\lambda>0}$ , λ ⁢ m ⁢ ( { x ∈ ℝ n : | T Ω ⁢ f ⁢ ( x ) | > λ } ) ≲ ( ( log ⁡ n ) ⁢ ∥ Ω ∥ 1 + ∫ 0 1 n ω ~ 1 ⁢ ( δ ) δ ⁢ 𝑑 δ ) ⁢ ∥ f ∥ 1 , $\lambda m(\{x\in\mathbb{R}^{n}:|T_{\Omega}f(x)|>\lambda\})\lesssim{\Bigg{(}(% \log n)\|\Omega\|_{1}+\int_{0}^{\frac{1}{n}}\frac{\tilde{\omega}_{1}(\delta)}{% \delta}\,d\delta\Biggr{)}}\|f\|_{1},$ where ω ~ 1 ${\tilde{\omega}_{1}}$ denotes the L 1 ${L^{1}}$ integral continuous modulus of Ω defined by translation in ℝ n ${\mathbb{R}^{n}}$ . Moreover, some similar estimates are also established for the Marcinkiewicz integral μ Ω ${\mu_{\Omega}}$ , which was introduced by Stein in [13].

The aim of this paper is to derive explicitly a connection between the Zagier elliptic trilogarithm and Mahler measures of a certain family of three-variable polynomials defining K3 surfaces. In addition, we prove some linear relations satisfied by the elliptic trilogarithm evaluated at torsion points on elliptic curves. This result can be viewed as a higher dimensional analogue of exotic relations of the elliptic dilogarithm., 25 pages

Minkowski's second theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of an o-symmetric convex body in terms of its successive minima. In this paper we study analogs of this theorem from two different points of view: either relaxing the symmetry condition, assuming that the centroid of the body lies at the origin, or replacing the volume functional by the surface area.

Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0 < p , q < ∞ ${0 and α = ( α 1 , α 2 ) ∈ ℝ 2 ${\alpha =(\alpha _1,\alpha _2)\in \mathbb {R}^2}$ . In this paper, we introduce multi-parameter Triebel–Lizorkin spaces F ˙ p α , q ( ℝ m ) ${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ associated with different homogeneities arising from the composition of two singular integral operators whose weak (1,1) boundedness was first studied by Phong and Stein [Amer. J. Math. 104 (1982), 141–172]. We then establish its atomic decomposition which is substantially different from that for the classical one-parameter Triebel–Lizorkin spaces. As an application of our atomic decomposition, we obtain the necessary and sufficient conditions for the boundedness of an operator T on the multi-parameter Triebel–Lizorkin type spaces. In the special case of α 1 = α 2 = 0 ${\alpha _1=\alpha _2=0}$ , q = 2 and 0 < p ≤ 1 ${0 , our spaces F ˙ p α , q ( ℝ m ) ${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ coincide with the Hardy spaces H com p ${H^p_{\mathrm {com}}}$ associated with the composition of two different singular integrals (see []). Therefore, our results also give an atomic decomposition of H com p ${H^p_{\mathrm {com}}}$ . Our work appears to be the first result of atomic decomposition in the Triebel–Lizorkin spaces in the multi-parameter setting.

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t) + q(t)u′(t) ∈ Au(t) + f (t) for a.a. t ∈ ℝ+ = [0,∞), subject to the condition , where A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p, q ∈ L∞(ℝ+), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0. Eq. (E) has been investigated under various assumptions by Barbu [1]-[2], Brezis [4], Véron [9]-[10] and others. Recently (see Moroşanu [8]), we proved existence and uniqueness of a strong solution to Eq. (E) subject to u(0) = x ∈ D(A) in the weighted space X = Lb 2(ℝ+; H), where b(t) = a(t)/p(t), a(t) = exp(∫0 t q(s)/p(s) ds), under our weak conditions on p and q (see above). Here we extend this result to the general case , while keeping the other assumptions unchanged, thus solving a long standing problem. In addition, the proof of our recent result, which is the starting point of the present paper, is considerably simplified. Furthermore, some qualitative properties of solutions are pointed out, an application to a minimization problem is given, and some open problems are formulated.

In this paper, we show the uniqueness of smooth embedded self-shrinkers asymptotic to infinite order to a generalized cylinder. Also, we construct non-rotationally symmetric self-shrinking ends asymptotic to a generalized cylinder with rate as fast as any given polynomial.

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in C^{1,1} open sets D in R^d, of a large class of subordinate Brownian motions with Gaussian components. When D is bounded, our sharp two-sided Dirichlet heat kernel estimates hold for all t>0. Integrating the heat kernel estimates with respect to the time variable t, we obtain sharp two-sided estimates for the Green functions, in bounded C^{1,1} open sets, of such subordinate Brownian motions with Gaussian components., 27 pages. arXiv admin note: text overlap with arXiv:1303.6449

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension $2n+2$ which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on $\partial W$, which we call \emph{convex open book}, induced by a compact convex Lefschetz fibration on $W$ carries the contact structure induced by the convex symplectic structure (i.e., Liouville structure) on $W$. Here we show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on $W$, any simply connected embedded Lagrangian submanifold of a page in a convex open book on $\partial W$ can be assumed to be Legendrian in $\partial W$ with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one correspondence between convex stabilizations of a convex open book and convex stabilizations of the corresponding compact convex Lefschetz fibration. We also show that the convex stabilization of a compact convex Lefschetz fibration on $W$ yields a compact convex Lefschetz fibration on a Liouville domain $W'$ which is exact symplectomorphic to a \emph{positive expansion} of $W$. In particular, with the induced structures $\partial W$ and $\partial W'$ are contactomorphic., Comment: 13 pages, 1 figure, minor corrections made

The purpose of this paper is to study convergence of a newly defined modified S-iteration process to a common fixed point of two asymptotically quasi-nonexpansive type mappings in the setting of CAT(0) space. We give a suffcient condition for convergence to a common fixed point and establish some strong convergence theorems for the said iteration process and mappings under suitable conditions. Our results extend and improve many known results from the existing literature.

This paper is concerned with the following nonlinear Schrödinger equation: where f(x, t)t-1 < μ* := inf σ(-Δ + V) < f(x, t)t-1 < V(∞) = V(x). The existence of a positive solution and a least energy solution for problem (P) are discussed. Here, we assume neither that f(x, t) ≡ f(t), nor that f(x, t)t-1 is non-decreasing on t ≥ 0. Furthermore, we do not require that f(x, t) have a limit as |x| → +∞.

In this paper we consider magnetic Schr\"odinger operators in R^n, n \ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not a corollary of the free case as the differentiated free resolvent does not have small operator norm on any weighted L^2 spaces. We instead show that the spectral radius of such operators decreases to zero, hence their perturbation of the identity is still invertible. The key estimates are based on an angular decomposition of the free resolvent, or rather a bound that holds uniformly for all possible angular decompositions. The proof avoids the Fourier transform and instead uses H\"ormander's variable coefficient Plancherel theorem for oscillatory integrals., Comment: 36 pages

In this paper, we address the issue of motion planning for the control system L R that results from the rolling without slipping nor spinning of a two dimensional Riemannian manifold M 1 onto another one M 2 . We present two procedures to tackle the motion planning problem when M 1 is a plane and M 2 a convex surface. The first approach rests on the Liouvillian character of L R . More precisely, if just one of the manifolds has a symmetry of revolution, then L R is shown to be a Liouvillian system. If, in addition, that manifold is convex and the other one is a plane, then a maximal linearizing output is explicitely computed. The second approach consists of the use of a continuation method. Even though L R admits nontrivial abnormal extremals, we are still able to successfully apply the continuation method if M 2 admits a stable periodic geodesic.