Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1183.46028
Barza, Sorina; Kolyada, Viktor; Soria, Javier
Sharp constants related to the triangle inequality in Lorentz spaces. (English)
[J] Trans. Am. Math. Soc. 361, No. 10, 5555-5574 (2009). ISSN 0002-9947; ISSN 1088-6850/e

Summary: We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le\infty$, for which the standard functional $$\|f\|_{p,s}=\left(\int^\infty_0(t^{1/p}f^*(t))^s\,\frac{dt}{t}\right)^{1/s}$$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$\|f\|_{p,s}=\inf\left\{\sum_k\|f\|_{p,s}\right\},$$ where the infimum is taken over all finite representations $f=\sum_kf_k$. We also prove that the decomposition norm and the dual norm $$\|f\|_{p,s}'=\sup\left\{\int_R fg\,d\mu:\|g\|_{p',s'}=1\right\}$$ agree for all values of $p,s>1$.

MSC 2000:: *46E30 Spaces of measurable functions
46B25 Classical Banach spaces in the general theory of normed spaces

Keywords: equivalent norms; level function; Lorentz spaces; sharp constants; quasi-norm; decomposition norm

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]