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A summability condition on the gradient ensuring BMO. (English) Zbl 0926.46028

Summary: It is well-known that if \(u\in W^{1,1}(\Omega)\), \(\Omega\subset \mathbb{R}^N\) satisfies \(| Du|\in L^N(\Omega)\), then \(u\) belongs to \(\text{BMO}(\Omega)\), the John-Nirenberg space. We prove that this is no more true if \(| Du|\) belongs to an Orlicz space \(L_A(\Omega)\) when the N-function \(A(t)\) increases less than \(t^N\). In order to obtain \(u\in \text{BMO}(\Omega)\), we impose a suitable uniform \(L_A\) condition for \(| Du|\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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