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Search: MSC category 43A15 ( $L^p$-spaces and other function spaces on groups, semigroups, etc. )

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1. CMB Online first

Bu, Shangquan; Cai, Gang
Periodic solutions of second order degenerate differential equations with delay in Banach spaces
We give necessary and sufficient conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate differential equation with finite delays: $(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where $A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from $L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$.

Keywords:second order degenerate differential equation, Fourier multiplier theorem, well-posedness, Lebesgue-Bochner space, Besov space
Categories:34G10, 34K30, 43A15, 47D06

2. CMB 2016 (vol 60 pp. 712)

Chen, Chung-Chuan
Disjoint Hypercyclicity and Weighted Translations on Discrete Groups
Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space $\ell^p(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.

Keywords:disjoint hypercyclicity, topological transitivity, weighted translation, $\ell^p$-space
Categories:47A16, 47B38, 43A15

3. CMB 2016 (vol 59 pp. 693)

Chen, Chung-Chuan
Recurrence of Cosine Operator Functions on Groups
In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

Keywords:topologically multiple recurrence, recurrence, topological transitivity, hypercyclicity, cosine operator function
Categories:47A16, 54B20, 43A15

4. CMB 2014 (vol 57 pp. 834)

Koh, Doowon
Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields
We study $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

Keywords:finite fields, radial functions, restriction operators
Categories:42B05, 43A32, 43A15

5. CMB 2006 (vol 49 pp. 549)

Führ, Hartmut
Hausdorff--Young Inequalities for Group Extensions
This paper studies Hausdorff--Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies in particular to yield a Hausdorff--Young inequality for nonunimodular groups.

Categories:43A30, 43A15

6. CMB 2003 (vol 46 pp. 268)

Puls, Michael J.
Group Cohomology and $L^p$-Cohomology of Finitely Generated Groups
Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

Keywords:group cohomology, $L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functional
Categories:43A15, 20F65, 20F18

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