Generalized multiplicative inequalities for ideal spaces

V. S. Klimov

P. G. Demidov Yaroslavl State University

Abstract: We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces $L_p^1(\Omega)$ to the function classes that result from the replacement of $L_p(\Omega)$ with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of $\Omega$. The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.

Keywords: multiplicative inequality, ideal space, domain, capacity.

UDC: 517.518.235

Received: 18.10.2001

 English version: Siberian Mathematical Journal, 2004, 45:1, 112–124

Bibliographic databases:

• 
• 
• 
•