Riesz means associated with certain product type convex domain

In this paper we study the maximal operators and the convolution operators Tδ associated with multipliers of the form

     

Marcinkiewicz multiplier theorem for the Weyl functional calculus

Let {X _j , Y _j , T : 1 ≤ j ≤ n} be a basis satisfying the commutation relation for the Heisenberg Lie algebra . Then we obtain a multi-parameter Marcinkiewicz multiplier theorem for the operator defined by m(X ₁,..., X _n , Y ₁,..., Y _n , T).     

Singular Radon transforms along odd curves on the Heisenberg group

In this article we study the singular integral operators along the curve on the Heisenberg group. It is a variable coefficient extension of the singular integrals along the odd curves on the Euclidean space ℝ². The proof is based on the generalized Calderon-Zygmund theory on the space of homogeneous type.     

Hilbert Transforms Along Curves in the Heisenberg Group

We consider Hilbert transforms along curves on the Heisenberggroup. In particular, necessary and sufficient conditions forthe L²-boundedness are given for (t)=(t,(t),0) when is evenor odd and convex on R⁺. 1991 Mathematics Subject Classification:42B20, 42B25.     

Electrochemical synthesis and one step modification of PMProDot nanotubes and their enhanced electrochemical properties

Poly (3,4-(2-methylene)propylenedioxythiophene) (PMProDot) nanotubes were synthesized within the pores of polycarbonate and were further modified with styrene and vinylcarbazole by a one step electrochemical method through the methylene functional group. The enhanced electrochemical and electrochromic properties of composite nanotubes were investigated using FTIR, UV/Vis absorbance spectroscopy, and AFM.     

Triple Hilbert Transforms Along Polynomial Surfaces

Given W ì \mathbbZ₊³\Omega \subset {\mathbb{Z}}_{+}^{3}, we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form ($t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m$t_1, t_2, t_3,\sum_{m \in \Omega} a_{m} t^m) is bounded in L^p(\mathbbR⁴)L^p({\mathbb{R}}^4).     

Scattering Theory Below Energy for a Class of Hartree Type Equations

We prove the global existence and scattering for the Hartree-type equation in H ^s (?³) the low regularity space s < 1. We follow the ideas in Colliander et al. (2004 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2004 ). Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ?³ . Comm. Pure Appl. Math. 57 : 987 – 1014 .[Crossref], [Web of Science ®] , [Google Scholar]) to the Hartree-type nonlinearity, and also develop the theory of the classical multilinear operator modifying the L ^p estimate in Coifman and Meyer (1978 Coifman , R. , Meyer , Y. ( 1978 ). Au delá des opérateurs pseudo-differentiel . Astérisque, Société Mathématique de France 57 . [Google Scholar]).     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Direct Observation of Dynamic Mechanical Regulation of DNA Condensation by Environmental Stimuli

Gene delivery is a promising way to treat hereditary diseases and cancer; however, there is little understanding of DNA:carrier complex mechanical properties, which may be critical for the protection and release of nucleic acids. We applied optical tweezers to directly measure single‐molecule mechanical properties of DNA condensed using 19‐mer poly‐L ‐lysine (PLL) or branched histidine–lysine (HK) peptides. Force–extension profiles indicate that both carriers condense DNA actively, showing force plateaus during stretching and relaxation cycles. As the environment such as carrier concentration, pH, and the presence of zinc ions changes, DNA:HK complexes showed dynamically regulated mechanical properties at multiple force levels. The fundamental knowledge from this study can be applied to design a mechanically tailored complex which may enhance transfection efficiency by controlling the stability of the complex temporally and spatially.     

Cylinder Multipliers Associated with a Convex Polygon

In this paper we prove that sharp weak-type estimates for the maximal operators associated with a cylindric distance function associated with a convex polygon on when 2/3 < p < 1 and , or when p = 2/3 and . This work was supported (in part) by research funds from Chosun University, 2007.