Extensions of smooth mappings into biduals and weak continuity

Choi, Y.S.,Hajek, P.,Lee, H.J. Academic Press ; Elsevier Science B.V. Amsterdam 2013 Advances in mathematics Vol.234 No.-

We study properties of uniformly differentiable mappings between real Banach spaces. Among our main results are generalizations of a number of classical results for linear operators on L<SUB>~</SUB>-spaces into the setting of uniformly differentiable mappings. Denote by B<SUB>X</SUB> the closed unit ball of a Banach space X. Let X be a L<SUB>~,λ</SUB>-space, λ≥1, and let Y be a Banach space. Let T:B<SUB>X</SUB>→Y be a continuous mapping which is uniformly differentiable in the open unit ball of X. Assuming that T is weakly compact, then T can be extended, preserving its best smoothness properties, into the mapping from the 1λ-multiple of the unit ball of any superspace of the domain space X into the same range space Y. We also show that T maps weakly Cauchy sequences from λB<SUB>X</SUB> into norm convergent sequences in Y. This is a uniformly smooth version of the Dunford-Pettis property for the L<SUB>~,λ</SUB>-spaces. We also show that a uniformly differentiable mapping T, which is not necessarily weakly compact, still maps weakly Cauchy sequences from λB<SUB>X</SUB> into norm convergent sequences in Y, provided Y<SUP>**</SUP> does not contain an isomorphic copy of c<SUB>0</SUB>. We prove that for certain pairs of Banach spaces the completion of the space of polynomials equipped with the topology of uniform convergence on the bounded sets (of the functions and their derivatives up to order k) coincides with the space of uniformly differentiable (up to order k) mappings. Our work is based on a number of tools that are of independent interest. We prove, for every pair of Banach spaces X,Y, that any continuous mapping T:B<SUB>X</SUB>→Y, which is uniformly differentiable of order up to k in the interior of B<SUB>X</SUB>, can be extended, preserving its best smoothness, into a bidual mapping T@?:B<SUB>X^*^*</SUB>→Y<SUP>**</SUP>. Another main tool is a result of Zippin's type. We show that weakly Cauchy sequences in X=C(K) can be uniformly well approximated by weakly Cauchy sequences from a certain C[0,α], α is a countable ordinal, subspace of X<SUP>**</SUP>.

Decomposition of the Witt-Burnside ring and Burnside ring of an abelian profinite group

Academic Press ; Elsevier Science B.V. Amsterdam 2009 Advances in mathematics Vol.222 No.2

Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors W<SUB>G</SUB><SUP>q</SUP>@?W<SUB>H</SUB><SUP>q</SUP>@?W<SUB>GxH</SUB><SUP>q</SUP>, where W<SUB>G</SUB><SUP>q</SUP> denotes the q-deformed Witt-Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. To do this, we first establish an isomorphism of functors B<SUB>G</SUB><SUP>q</SUP>@?B<SUB>H</SUB><SUP>q</SUP>@?B<SUB>GxH</SUB><SUP>q</SUP>, where B<SUB>G</SUB><SUP>q</SUP> denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt-Burnside rings, Math. Z. 207 (1) (2007) 151-191]. As an application, we derive a pseudo-multiplicative property of the q-Mobius function associated to the lattice of open subgroups of the direct sum of G and H.

A study on functional independence of the Iwasawa power series

Academic Press ; Elsevier Science B.V. Amsterdam 2013 Advances in mathematics Vol.238 No.-

We study a class of functional independences that the Iwasawa power series satisfy for both zero and non-zero characteristics. As results, we prove a generalization of Angles and Ranieri [B. Angles, G. Ranieri, On the linear independence of p-adic L-functions modulo p, Ann. Inst. Fourier (Grenoble) 60 (5) (2010) 1831-1855] and transcendence of the Iwasawa power series over the rational functions for non-zero characteristics. We also verify that the power series is not a solution of any non-trivial linear differential equation with the coefficients of rational functions over p-adic numbers.

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Academic Press ; Elsevier Science B.V. Amsterdam 2016 Advances in mathematics Vol.293 No.-

<P>Quantization of universal Teichmuller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group T. This yields certain central extensions of T by Z, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension (T) over cap (Kash) of T resulting from the Kashaev quantization, and show that it corresponds to 6 times the Euler class in H-2 (T; Z). Meanwhile, the braided Ptolemy-Thompson groups T*, T-# of Funar-Kapoudjian are extensions of T by the infinite braid group B-infinity and by abelianizing the kernel B-infinity one constructs central extensions T-ab*, T-ab(#) of T by Z, which are of topological nature. We show (T) over cap (Kash) congruent to T-ab(#) Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension (T) over cap (CF) of T resulting from the Chekhov-Fock (-Goncharov) quantization and thus showed that it corresponds to 12 times the Euler class and that (T) over cap (CF) congruent to T-ab*. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations. (C) 2016 Elsevier Inc. All rights reserved.</P>

The Webster scalar curvature flow on CR sphere. Part II

Academic Press ; Elsevier Science B.V. Amsterdam 2015 Advances in mathematics Vol.268 No.-

This is the second of two papers, in which we study the problem of prescribing Webster scalar curvature on the CR sphere as a given function f. Using the Webster scalar curvature flow, we prove an existence result under suitable assumptions on the Morse indices of f.

The existence of greedy bases in rank 2 quantum cluster algebras

Lee, K.,Li, L.,Rupel, D.,Zelevinsky, A. Academic Press ; Elsevier Science B.V. Amsterdam 2016 Advances in mathematics Vol.300 No.-

<P>We establish the existence of a quantum lift of the greedy basis. (C) 2016 Elsevier Inc. All rights reserved.</P>

Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols

Curto, R.E.,Hwang, I.S.,Kang, D.O.,Lee, W.Y. Academic Press ; Elsevier Science B.V. Amsterdam 2014 Advances in mathematics Vol.255 No.-

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos's Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic? We first prove: Let Φ@?L<SUB>M'n</SUB><SUP>~</SUP> be a matrix-valued rational function having a ''matrix pole'', i.e., there exists α@?D for which kerH<SUB>Φ</SUB>@?(z-α)H<SUB>C^n</SUB><SUP>2</SUP>, where H<SUB>Φ</SUB> denotes the Hankel operator with symbol Φ. If(i)T<SUB>Φ</SUB> is hyponormal; (ii)ker[T<SUB>Φ</SUB><SUP>@?</SUP>,T<SUB>Φ</SUB>] is invariant for T<SUB>Φ</SUB>, then T<SUB>Φ</SUB> is normal. Hence in particular, if T<SUB>Φ</SUB> is subnormal then T<SUB>Φ</SUB> is normal. Next, we show that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.

The Webster scalar curvature flow on CR sphere. Part I

Academic Press ; Elsevier Science B.V. Amsterdam 2015 Advances in mathematics Vol.268 No.-

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, L<SUP>p</SUP> convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The results in this paper will be used to prove a result of prescribing Webster scalar curvature on the CR sphere, which is the main result of the second paper.

Restriction estimates for space curves with respect to general measures

Ham, S.,Lee, S. Academic Press ; Elsevier Science B.V. Amsterdam 2014 Advances in mathematics Vol.254 No.-

In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn@?t rely on the offspring curve method, which has been extensively used in the previous works. Our work was inspired by the recent argument due to Bourgain and Guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces.

Localized energy equalities for the Navier-Stokes and the Euler equations

Academic Press ; Elsevier Science B.V. Amsterdam 2014 Advances in mathematics Vol.254 No.-

Let (v,p) be a smooth solution pair of the velocity and the pressure for the Navier-Stokes (Euler) equations on R<SUP>N</SUP>x(0,T), N≥3. We set the Bernoulli function Q=12|v|<SUP>2</SUP>+p. Under suitable decay conditions at infinity for (v,p) we prove that for almost all α(t) and β(t) defined on (0,T) there holds∫{α(t)<Q(x,t)<β(t)}(12@?@?t|v|<SUP>2</SUP>+ν|ω|<SUP>2</SUP>)dx=ν∫{Q(x,t)=β(t)}|@?Q|dS-ν∫{Q(x,t)=α(t)}|@?Q|dS, where ω=curlv is the vorticity. This shows that, in each region squeezed between two levels of the Bernoulli function, besides the energy dissipation due to the enstrophy, the energy flows into the region through the level hypersurface having the higher level, and the energy flows out of the region through the level hypersurface with the lower level. Passing α(t)↓inf<SUB>x@?R^N</SUB>Q(x,t) and β(t)↑sup<SUB>x@?R^N</SUB>Q(x,t), we recover the well-known energy equality, 12ddt∫<SUB>R^N</SUB>|v|<SUP>2</SUP>=-ν∫<SUB>R^N</SUB>|ω|<SUP>2</SUP>dx. A weaker version of the above equality under the weaker decay assumption of the solution at spatial infinity is also derived. The stationary version of the equality implies the previous Liouville type results on the Navier-Stokes equations.