The Tingley's Probelm, which is named after the pioneering contribution of Tingley, is also known as the extension problem. It is nowadays a central topic for those researchers working on preservers. Up to the present, no negative counterexample is known and the general problem remains open even for two dimensional Banach spaces. The efforts gave rise to a wide list of positive answers to Tingley's problem for concrete classical Banach spaces and for some classes of Banach spaces. In this paper, we solved the Tingley's problem between the complex Banach spaces ℓp(Γ) (1 ≤ p < ∞) and complex Banach space E, i.e., we show that the complex Banach spaces ℓp(Γ) (1 ≤ p < ∞) satisfy the Mazur-Ulam property.
Let A be a C*-algebra with the unit I, Φ:A → A be a surjective map. We show that Φ preserves strong k-skew commutativity, that is, Φ satisfies *[Φ(A), Φ(B)]k=*[A, B]k, ∀ A, B ∈ A if and only if Φ(A)=Φ(I)A, ∀ A ∈ A, where Φ(I) ∈ Z (A) with Φ(I)*=Φ(I) and Φ(I)k+1=I, Z (A) is the center of A. In particular, if Z (A)=CI, then Φ:A → A preservers strong k-skew commutativity if and only if Φ(A)=A, ∀ A ∈ A or Φ(A)=-A, ∀ A ∈ A. The latter case does not occur if k is even.
We study the exact asymptotic behavior of large solutions to the following equation Δpu=b(x)f(u), u(x) > 0, x ∈ Ω, where b ∈ C(Ω) is non-negative and nontrivial in Ω, f ∈ C[0, ∞) ∩ C1(0, ∞) is positive and non-decreasing on (0, ∞). When Ω=RN (N ≥ 3), by using truncation technique and the upper and lower solution methods, we establish the exact asymptotic behavior of entire large solutions to the above equation. When Ω is a C4-bounded domain, we revel the influence of the mean curvature H(x(x)) of ∂Ω to boundary behavior of large solutions. Since (Δp) (p≠2) is a non-linear operator and H(x(x)) is a variable function on ∂Ω, the calculation of the result is quite different from the one p=2.
We characterize commuting dual Toeplitz operators with bounded measurable symbols on the orthogonal complement of the Fock space, giving the necessary and sufficient conditions. Moreover, we give a Brown-Halmos Theorem for dual Toeplitz operators.
This paper is concerned with the existence and qualitative properties of cylindrically symmetric traveling fronts for nonlocal delayed diffusion equations. Recently, the existence and stability of V-shaped traveling fronts and pyramidal traveling fronts for nonlocal delayed diffusion equation have been established. Using the limit of a sequence of pyramidal traveling fronts, we establish the existence and qualitative properties of cylindrically symmetric traveling fronts. Moreover, the asymptotic behaviors of level set and the nonexistence of the cylindric symmetric traveling are also proved.
We study viscosity method for the general implicit double midpoint rule for finding the fixed points of asymptotically non-expansive mappings in real Banach spaces. Under suitable conditions imposed on the parameters, some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in this article extend and improve the main results of other authors.
Let μ be a normal function on[0, 1). In this paper, the authors give the bidirectional estimates of the volume integral of a single variable point in the unit ball under the normal weight measure in some cases, and give the bidirectional estimates for all indicators in special cases. As an application, the authors give some necessary and sufficient conditions for the boundedness or compactness of Cesàro type operator on normal weight Dirichlet spaces.
The Bedford-McMullen carpet plays an important role in fractal geometry. Although any self-affine carpet is not self-similar, we can obtain the average geodesic distance on the carpet using the technique named finite pattern.
This paper is devoted to studying the Lp-mapping properties of the rough Marcinkiewicz integrals with rough kernels along real-analytic submanifolds. Under assuming that the radial kernel h∈Δγ(R+) for some γ ∈ (1, ∞] and the sphere kernel Ω ∈ Lq(Sn-1) for some q ∈ (1, 2], the Lp-boundedness for such operators are established. Furthermore, by the extrapolation arguments, the corresponding Lp-bounds are obtained under some optimal size conditions on the unit sphere Ω ∈ L(log L) 1/2 (Sn-1) or Ω ∈ Bq(0,-1/2)(Sn-1) for some q > 1. Meanwhile, the Lp estimates for the related maximal rough Marcinkiewicz integrals are also considered.
