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The 1-Reduction for Removable Singularities, and the Negative Hölder Spaces

Anthony G. O'Farrell
Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences
Vol. 88A, No. 2 (1988), pp. 133-151
Published by: Royal Irish Academy
Stable URL: http://www.jstor.org/stable/20489298
Page Count: 19
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The 1-Reduction for Removable Singularities, and the Negative Hölder Spaces
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Abstract

Let L be a pseudodifferential operator of (possibly non-integral) order m on ${\Bbb R}^{d}$, and let F be a topological vector space of distributions on ${\Bbb R}^{d}$. We say that a compact set $K\subset {\Bbb R}^{d}$ is L--F-null if, given an open set $U\subset {\Bbb R}^{d}$, and a distribution f ∈ F such that Lf is $C^{\infty}$ on U ∼ K, it follows that Lf is $C^{\infty}$ on U. We also say that K is a set of removable singularities for solutions of Lf = g(g smooth) that belong to F. We describe the 1-reduction, a method for reducing problems about singularities for general elliptic operators to problems about supports. Applying the 1-reduction and a theorem of Dahlberg, we identify the $L-T_{\beta}-\text{null}$ sets for all elliptic L and all real β such that β≠order L. The scale of $T_{\beta}$ is essentially an extension of the scale of Hölder--Zygmund spaces. These null-sets are the compact sets K with $M^{d+\beta -\text{order}L}(K)=0$, where $M^{\alpha}$ denotes α-dimensional Hausdorff content. We indicate the further application of the 1-reduction to Sobolev and Besov spaces, and dual approximation problems.

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