AUTHOR
Sabourova, Natalia

DATE
2004-06-08

DEPARTMENT
Mathematics /

SUMMARY
For 1 <= p,q <= ¥ the natural complexification of any bounded linear operator between real Banach spaces T:L^p(m) -> L^q(n) denoted by T_C:L_C^p(m) -> L_C^q(n) is also bounded and we are interested in the exact relation between the norm of the operator T in the real case and in the complex case. This relation can be expressed in terms of the constants g_p,q appearing in the inequality

In analysis the considered relation has, for instance, its applications in the interpolation theory and connection with the constants from the fundamental Grothendieck inequality. The work on the constants g_p,q is traced back to Marcinkiewicz, Zygmund, Verbickii-Sereda, Figiel-Iwaniec-Pelczynski, Krivine, Gasch-Maligranda, Defant and others. In this paper we try to summarize the results of these authors and show how to obtain the estimates for g_p,q as well as present a number of these estimates. We begin with the case when g_p,q=1. In addition, we show that the norms of the operators acting between spaces of functions of bounded p-variation and the norms of the positive linear operators between L^p spaces coincide with the norms of their complexifications. When calculating the biggest constant g_¥,1 Krivine described it in terms of certain tensor product, that is why this necessary complication appears in this work. Moreover, this description helps to derive some basic properties of the constants g_p,q. Finally, we specify the measures to be counting and consider the operators acting between finite-dimensional l^p spaces. We present a number of the estimates of g_p,q (l_n^p,l_m^q) in this particular case. However, the exact value of g_p,q for 1 <= q < 2 < p <= ¥ and (p,q) ¹ (¥,1) has not been found for arbitrary finite-dimensional l^p spaces as well as in more general settings.

ISSN 1402-1617 / ISRN LTU-EX--04/170--SE / NR 2004:170