We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H1(Rn) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.
Meda, S., Sjogren, P., Vallarino, M. (2008). On the H1-L1 boundedness of operators. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 136(8), 2921-2931 [10.1090/S0002-9939-08-09365-9].
On the H1-L1 boundedness of operators
MEDA, STEFANO;VALLARINO, MARIA
2008
Abstract
We prove that if q is in (1,\infty), Y is a Banach space and T is a linear operator defined on the space of finite linear combinations of (1,q)-atoms in R^n which is uniformly bounded on (1,q)-atoms, then T admits a unique continuous extension to a bounded linear operator from H1(Rn) to Y. We show that the same is true if we replace (1,q)-atoms with continuous (1,\infty)-atoms. This is known to be false for (1,\infty)-atoms.
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