1,278,678 research outputs found

### Maximal theorems and square functions for analytic operators on Lp-spaces

Let T : Lp --> Lp be a contraction, with p strictly between 1 and infinity,
and assume that T is analytic, that is, there exists a constant K such that
n\norm{T^n-T^{n-1}} < K for any positive integer n. Under the assumption that T
is positive (or contractively regular), we establish the boundedness of various
Littlewood-Paley square functions associated with T. As a consequence we show
maximal inequalities of the form \norm{\sup_{n\geq 0}\, (n+1)^m\bigl
|T^n(T-I)^m(x) \bigr |}_p\,\lesssim\, \norm{x}_p, for any nonnegative integer
m. We prove similar results in the context of noncommutative Lp-spaces. We also
give analogs of these maximal inequalities for bounded analytic semigroups, as
well as applications to R-boundedness properties

### Trees, linear orders and G\^ateaux smooth norms

We introduce a linearly ordered set Z and use it to prove a necessity
condition for the existence of a G\^ateaux smooth norm on C(T), where T is a
tree. This criterion is directly analogous to the corresponding equivalent
condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits
a G\^ateaux smooth lattice norm then it also admits a lattice norm with
strictly convex dual norm.Comment: A different version of this paper is to appear in J. London Math. So

### Gruenhage compacta and strictly convex dual norms

We prove that if K is a Gruenhage compact space then C(K)* admits an
equivalent, strictly convex dual norm. As a corollary, we show that if X is a
Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage
compact in the w*-topology and |.| is equivalent to a coarser, w*-lower
semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual
norm. We give a partial converse to the first result by showing that if T is a
tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if
T is a Gruenhage space. Finally, we present some stability properties satisfied
by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect
images

### Polynomial bounds on the Sobolev norms of the solutions of the nonlinear wave equation with time dependent potential

We consider the Cauchy problem for the nonlinear wave equation $u_{tt} -
\Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) \geq 0$ having
compact support with respect to $x$. The linear equation without the nonlinear
term $u^3$ and potential periodic in $t$ may have solutions with exponentially
increasing as $t \to \infty$ norm $H^1({\mathbb R}^3_x)$. In [2] it was
established that adding the nonlinear term $u^3$ the $H^1({\mathbb R}^3_x)$
norm of the solution is polynomially bounded for every choice of $q$. In this
paper we show that $H^k({\mathbb R}^3_x)$ norm of this global solution is also
polynomially bounded. To prove this we apply a different argument based on the
analysis of a sequence $\{Y_k(n\tau_k)\}_{n = 0}^{\infty}$ with suitably
defined energy norm $Y_k(t)$ and $0 < \tau_k <1.

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