In this paper, we study the L^p boundedness of a class of maximal operators T_{Ωj}^(γ) and a related class of rough singular integrals on product spaces. We obtain appropriate L^p estimates for such maximal operators and singular integrals. These estimates are used in an extrapolation argument and allow us to obtain some new and improved results for certain maximal integral operators and singular integrals on product spaces under certain sharp conditions on the kernel functions. Also, one of our main results in this paper is a corrigendum of a result obtained by Ding-Lin.

We investigate a removable singularity theorem and other some basic properties of a J-holomorphic mapping for strongly pseudo-convex manifolds, which are necessary for constructing the moduli space of J-holomorphic mappings.

We prove several properties of finite p-groups which are generated by two elements of prime order p and which have a fixed-point-free automorphism of order seven.

The purpose of this paper is twofold. Let R_j (j = 1,2, ... , n) be Riesz transforms on $\mathbb{R}$ⁿ. First we prove the convergence of truncated operators of R_iR_j in generalized Hardy spaces. Our first result is an extension of the convergence in L^p($\mathbb{R}$^ⁿ) (1 < p < ∞). Secondly, as an application of the first result, we show a uniqueness theorem for the Navier-Stokes equation. J. Kato (2003) established the uniqueness of solutions of the Navier-Stokes equations in the whole space when the velocity field is bounded and the pressure field is a BMO-valued locally integrable-in-time function for bounded initial data. We extend the part "BMO-valued" in his result to "generalized Campanato space valued". The generalized Campanato spaces include L¹, BMO and homogeneous Lipschitz spaces of order α (0 < α < 1).

P. G. Casazza, O. Christensen, S. Li, and A. Lindner proved in [3] that some families of complex exponentials were either Riesz bases or not frames in L²[-π,π]. First, we shall advance their results in this note. Sedletskii constructed in [9] an exponential system which is complete, minimal and not uniformly minimal with separable spectrum in L²[-π,π]. Next, we shall construct a similar example with nonseparable spectrum in L²[-π,π].

Let Mⁿ (n ≥ 3) be a complete δ (> $\frac{(n-1)^2}{n^2}$)-stable minimal hypersurface in an (n + 1)-dimensional Euclidean space $\mathbb{R}$ⁿ⁺¹. We prove that there are no nontrivial L² harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, M has only one end. This implies that if M has finite total curvature, then M is a hyperplane.

In this paper we introduce a new function space which unifies and generalizes the Besov-type and the Triebel-Lizorkin-type function spaces introduced by S. Jaffard and D. Yang- W. Yuan. This new function space covers the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, and further the Morrey spaces. We define the new function space through wavelet expansions. We establish characterizations of the new function space such as the ϕ-transform characterization in the sense of Frazier-Jawerth, the atomic and molecular decomposition characterization. Moreover, in the inhomogeneous case, we give a characterization by local polynomial approximation. As application, we investigate the boundedness of the Calderòn-Zygmund operator and the trace theorem on the new function space.