Rectangular differentiation of integrals of Besov functions

We study the differentiation of integrals of functions in the Besov spaces $B^{\alpha,1}_{p}(\mathbb{R}^n),$ $\alpha >0,$ $1\le p<\infty,$ with respect to the basis of arbitrarily oriented rectangular parallelepipeds in $\mathbb{R}^n.$ We show that positive results hold if $\alpha\ge \textstyle \frac{n-1}{p}$ and we give counterexamples for the case $0<\alpha<\textstyle \frac{n}{p}-1.$ Similar results hold for $B^{\alpha,q}_{p}(\mathbb{R}^n),$ $q >1.$ For more general bases we can also prove negative results for $\textstyle \frac{n}{p}-1\le\alpha<\textstyle \frac{n-1}{p}.$

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Published 1 January 2002