Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Well posedness for a higher order modified Camassa-Holm equation

We show that the Cauchy problem for a higher order modification of the Camassa-Holm equation is locally well posed for initial data in the Sobolev space Hs(R) for s>s^′, where 1/4?s^′<1/2 and the value of s^′ depends on the order of equation. Employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions.     

Regularity of weak solutions of Maxwell's equations with mixed boundary‐conditions

In this paper global H^s‐ and L^p‐regularity results for the stationary and transient Maxwell equations with mixed boundary conditions in a bounded spatial domain are proved. First it is shown that certain elements belonging to the fractional‐order domain of the Maxwell operator belong to H^s(Ω) for sufficiently small s > 0. It follows from this regularity result that H^s(Ω) is an invariant subspace of the unitary group corresponding to the homogeneous Maxwell equations with mixed boundary conditions. In the case that a possibly non‐linear conductivity is present a L^p‐regularity theorem for the transient equations is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

Non‐uniform dependence and persistence properties for coupled Camassa–Holm equations

This paper deals with the non‐uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well‐posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces H^s with s > 3/2. On the other hand, the persistence properties in weighted L^p spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd.     

Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation

The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H ^−7/4, using the ideas of an [`(F)] ^s{\overline F ^s}-type space [8]. Then we show that the equation is globally well posed in H ^s for s ≥ −7/4, using the ideas of the “I-method” [7].     

Forelli-Rudin type theorem in pluriharmonic Bergman spaces with small index

It is proved that the Bergman type operatorT, is a bounded projection from the pluriharmonic Bergman spaceL ^p (B)∩h(B) onto Bergman spaceL ^p (B) ∩ H(B) for 0p 1 ands (p¹-1)(n+1). As an application it is shown that the Gleason’s problem can be solved in Bergman space L^P(B)∩H(B) for 0p 1. Project supported by the National Natural Science Foundation of China (Grant No. 19871081) and the Doctoral Program Foundation of the State Education Commission of China.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

This paper is devoted to studying the initial‐value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces X_{s, b}(R²) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I‐method as well as L² conservation law, we show that this fifth‐order shallow water wave equation is globally well‐posed for the initial data in the Sobolev spaces H^s(R) with $s{>}-\frac{63}{58}$. Copyright © 2010 John Wiley & Sons, Ltd.     

Well-posedness and small data scattering for the generalized Ostrovsky equation

We consider the generalized Ostrovsky equation utx=u+(up)_xx. We show that the equation is locally well posed in Hs, s>3/2 for all integer values of p?2. For p?4, we show that the equation is globally well posed for small data in H⁵∩W3,1 and moreover, it scatters small data. The latter results are corroborated by numerical computations which confirm the heuristically expected decay of ‖uLr_‖∼t−(r−2)/(2r).     

On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1]. Supported in part by DGICYT (SAB-90-0033).     

Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity

We consider the global attractor for the weakly damped forced KdV equation in Sobolev spaces [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}for s < 0. Under the assumption that the external forcing term belongs to [(L)\dot]²(T),{\dot{L}^2({\mathbf T}),} we prove the existence of the global attractor in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})} for −1/2 ≤ s < 0, which is identical to the one in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and thus is compact in H ³(T). The argument is a combination of the I-method and decomposing the solution into two parts, one of which is uniformly bounded in [(L)\dot]²(T){\dot{L}^2({\mathbf T})} and the other decays exponentially in [(H)\dot]^s(T){\dot{H}^s({\mathbf T})}.     

Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

The paper studies the existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation arising in the model in variational form for the neo–Hookean elastomer rod where k₁, k₂>0 are real numbers, g(s) is a given nonlinear function. When g(s)=sⁿ (where n?2 is an integer), by using the Fourier transform method we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u∈C^∞((0, T]; H^∞( R ))∩C([0, T]; H³( R ))∩C¹([0, T]; H^?1( R )) as long as initial data u₀∈W^{4, 1}( R )∩H³( R ), u₁∈L¹( R )∩H^?1( R ). Moreover, when (u₀, u₁)∈H²( R ) × L²( R ), g∈C²( R ) satisfy certain conditions, the Cauchy problem has no global solution in space C([0, T]; H²( R ))∩C¹([0, T]; L²( R ))∩H¹(0, T; H²( R )). Copyright © 2009 John Wiley & Sons, Ltd.     

Isometries on the Quasi-Banach Spaces Lp （0〈p〈1）

We study the extension of isometries between the unit spheres of quasi-Banach spaces Lp for 0〈p〈1. We give some sufficient conditions such that an isometric mapping from the the unit sphere of Lp（μ） into that of another LP（ν） can be extended to be a linear isometry defined on the whole space.     

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?^*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L ^p (E), where 1 < p <  +∞. We study the same problem for the operator Δ _M  + V in L ^p (M), where 1 < p < ∞, Δ _M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.     

Large Kr‐free subgraphs in Ks‐free graphs and some other Ramsey‐type problems

In this paper we present three Ramsey‐type results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph K_s on s vertices. More than 40 years ago Erd?s and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete graph K_r. Our first result provides a new lower bound for this problem, which improves previous results of various researchers. It also allows us to solve some special cases of a closely related question posed by Erd?s. For two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any red‐blue coloring of the edges of the complete graph K_N, contains either a red copy of G or a blue copy of H. The book with n pages is the graph B_n consisting of n triangles sharing one edge. Here we study the book‐complete graph Ramsey numbers and show that R(B_n, K_n) ≤ O(n³/log^3/2n), improving the bound of Li and Rousseau. Finally, motivated by a question of Erd?s, Hajnal, Simonovits, Sós, and Szemerédi, we obtain for all 0 < δ < 2/3 an estimate on the number of edges in a K₄‐free graph of order n which has no independent set of size n^1‐δ. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation

For s < 3/2, it is shown that the Cauchy problem for the Degasperis-Procesi equation (DP) is ill-posed in Sobolev spaces H ^s . If 1/2 ≤ s < 3/2, then ill-posedness is due to norm inflation. This means that there exist DP solutions who are initially arbitrarily small and eventually arbitrarily large with respect to the H ^s norm, in an arbitrarily short time. Since DP solutions conserve a quantity equivalent to the L ²-norm, there is no norm inflation in H ⁰ for these solutions. In this case, ill-posedness is caused by failure of uniqueness. For all other s < 1/2, the situation is similar to H ⁰. Considering that DP is locally well-posed in H ^s for s > 3/2, this work establishes 3/2 as the critical index of well-posedness in Sobolev spaces.