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THE HARDY OPERATOR AND THE GAP BETWEEN L∞ AND BMO

Published online by Cambridge University Press:  01 February 1998

JAN LANG  and
LUBOš PICK
JAN LANG
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic. E-mail: pick@math.cas.cz
LUBOš PICK
Affiliation:
Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic. E-mail: pick@math.cas.cz
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Abstract

We study boundedness and compactness properties of the Hardy integral operator Tf(x)=∫xAf from a weighted Banach function space X(v) into L∞ and BMO. We give a new simple characterization of compactness of T from X(v) into BMO. We construct examples of spaces X(v) such that T(X(v)) is (a) bounded in L∞ but not compact in BMO; (b) compact in BMO but not bounded in L∞; (c) bounded in BMO but neither bounded in L∞ nor compact in BMO; (d) bounded in L∞, compact in BMO and yet not compact in L∞. In order to obtain the last of the counterexamples we construct a new weighted Banach function space.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 196 - 208
Copyright
The London Mathematical Society 1998

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