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1. CMB 2006 (vol 49 pp. 82)

Gogatishvili, Amiran; Pick, Luboš
Embeddings and Duality Theorem for Weak Classical Lorentz Spaces
We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$ \left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t), $$ where $$ f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds. $$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$ \|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t), $$ where $$ f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds. $$

Keywords:Discretizing sequence, antidiscretization, classical Lorentz spaces, weak Lorentz spaces, embeddings, duality, Hardy's inequality
Categories:26D10, 46E20

2. CMB 2002 (vol 45 pp. 349)

Coppens, Marc
Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces
Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some algebraically closed field $k$ of characteristic $0$. For an integer $t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the dimension of the space of global sections of $O(t)$, and let $k$ be an integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$ be general points on $\mathbf{P}^n$ and let $\pi \colon X \to \mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points. Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 - \cdots -E_k)$ is a very ample invertible sheaf on $X$.

Keywords:blowing-up, projective space, very ample linear system, embeddings, Veronese map
Categories:14E25, 14N05, 14N15

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