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Pointwise multipliers of Besov spaces of smoothness zero and spaces of continuous functions. (English) Zbl 1036.46024

Let \(B^s_{pq} (\mathbb{R}^n)\) with \(1 \leq p,q \leq \infty\) and \(s \in \mathbb{R}\) be the well-known Besov spaces in Euclidean \(n\)-space \(\mathbb{R}^n\). A function \(m \in L_\infty (\mathbb{R}^n)\) is said to be a pointwise multiplier for \(B^s_{pq} (\mathbb{R}^n)\) if \( f \mapsto mf\) is a bounded map in \(B^s_{pq} (\mathbb{R}^n)\). The characterisation of the collection of all pointwise multipliers, denoted by \(M(B^s_{pq})\), has attracted a lot of attention for decades. The present paper deals with the characterisations of \(M(B^0_{\infty, \infty})\) (Theorem 4) and \(M(B^0_{\infty, 1} )\) (Theorem 5) which are especially complicated. Applications to regularity assertions for elliptic partial differential equations are given.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J15 Second-order elliptic equations
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References:

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