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$%A. Fiorenza, A summability condition on the gradient ensuring BMO, Rev. Mat. Complut. 11 (1998), no. 2, 313324.%$

A Summability Condition on the Gradient Ensuring BMO
Alberto FIORENZA
Dipartimento di Matematica e Applicazioni
“R. Cacciopoli”
Via Cintia
80126 Napoli Italy

Received: April 14, 1997
Revised: December 3, 1997
ABSTRACT

It is well-known that if u  (- W 1,1(_O_)  , _O_  (-  RN  satisfies |Du| (-  LN (_O_)  , then u  belongs to BMO(_O_)  , the John-Nirenberg Space. We prove that this is no more true if |Du| belongs to an Orlicz space LA(_O_)  when the N  -function A(t)  increases less than  N
t  . In order to obtain u  (- BMO(_O_)  , we impose a suitable uniform LA  condition for |Du | .

1991 Mathematics Subject Classification: 46E30, 46E35.