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On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

Let Λ n × m and k be a positive integer. Let f : n m be a locally bounded map such that for each ( ξ , η ) Λ , the derivatives D ξ j f ( x ) : = d j d t j f ( x + t ξ ) | t = 0 , j = 1 , 2 , k , exist and are continuous. In order to conclude that any such map f is necessarily of class C k it is necessary and sufficient that Λ be not contained in the zero-set of a nonzero homogenous polynomial Φ ( ξ , η ) which is linear in η = ( η 1 , η 2 , , η m ) and homogeneous of degree k in ξ = ( ξ 1 , ξ 2 , , ξ n ) . This generalizes a result of J. Boman for the case k = 1 . The statement and the proof of a theorem of Boman for the case k = is also extended...

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