1,278,678 research outputs found

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

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    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

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    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

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    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

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    We consider the Cauchy problem for the nonlinear wave equation uttβˆ’Ξ”xu+q(t,x)u+u3=0u_{tt} - \Delta_x u +q(t, x) u + u^3 = 0 with smooth potential q(t,x)β‰₯0q(t, x) \geq 0 having compact support with respect to xx. The linear equation without the nonlinear term u3u^3 and potential periodic in tt may have solutions with exponentially increasing as tβ†’βˆž t \to \infty norm H1(Rx3)H^1({\mathbb R}^3_x). In [2] it was established that adding the nonlinear term u3u^3 the H1(Rx3)H^1({\mathbb R}^3_x) norm of the solution is polynomially bounded for every choice of qq. In this paper we show that Hk(Rx3)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 {Yk(nΟ„k)}n=0∞\{Y_k(n\tau_k)\}_{n = 0}^{\infty} with suitably defined energy norm Yk(t)Y_k(t) and $0 < \tau_k <1.
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