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Carlson type inequalities and embeddings of interpolation spaces

Author(s): Leo Larsson
Journal: Proc. Amer. Math. Soc. 132 (2004), 2351-2356.
MSC (2000): Primary 46B70; Secondary 46E30, 26D15
Posted: January 20, 2004
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Abstract: We discuss the close relation between Carlson type inequalities

\begin{displaymath}\Vert a\Vert _X\leq K\Vert a\Vert _{A_0}^{1-\theta}\Vert a\Vert _{A_1}^\theta\end{displaymath}

and interpolation, and prove embedding results for real interpolation spaces, in particular into weighted $L_p$-spaces.

References:

1.
Barza, S., Burenkov, V., Pecaric, J., and Persson, L.-E.: Sharp Multidimensional Multiplicative Inequalities for Weighted $L_p$ Spaces with Homogeneous Weights, Math. Inequal. Appl. 1 (1998), 53-67. MR 98k:26031

2.
Bergh, J. and Löfström, J.: Interpolation Spaces: An Introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1976. MR 58:2349

3.
Carlson, F.: Une inégalité, Ark. Mat. Astr. Fysik 25B (1934), No. 1.

4.
Gustavsson, J. and Peetre, J.: Interpolation of Orlicz Spaces, Studia Math. 60 (1977), 33-59. MR 55:11021

5.
Kruglyak, N. Ya., Maligranda, L., and Persson, L.-E.: A Carlson Type Inequality with Blocks and Interpolation, Studia Math. 104 (1993), 161-180. MR 94k:46153

6.
Larsson, L.: A New Carlson Type Inequality, Math. Inequal. Appl. 6 (2003), 55-79. MR 2003k:26026

7.
Levin, V. I.: Sharp Constants in Carlson Type Inequalities, Doklady Akad. Nauk, SSSR 59 (1948), 635-638 (Russian). MR 9:415b

8.
Maligranda, L. and Persson, L.-E.: Real Interpolation between Weighted $L^p$ and Lorentz Spaces, Bull. Polish Acad. Sci. Math. 35 (1987), No. 11-12, 765-778. MR 89m:46055

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Additional Information:

Leo Larsson
Affiliation: Department of Mathematics, Uppsala University, Box 480, SE-751, 06 Uppsala, Sweden
Email: leo@math.uu.se

DOI: 10.1090/S0002-9939-04-07357-5
PII: S 0002-9939(04)07357-5
Keywords: Inequalities, real interpolation spaces, Carlson's inequality, embeddings, weighted Lebesgue spaces
Received by editor(s): December 11, 2002
Received by editor(s) in revised form: April 30, 2003
Posted: January 20, 2004
Communicated by: N. Tomczak-Jaegermann
Copyright of article: Copyright 2004, American Mathematical Society

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