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On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon (2011)

Commentationes Mathematicae Universitatis Carolinae

Let Λ ⊂ ℝ n × ℝ m and k be a positive integer. Let f : ℝ n → ℝ 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...

On local properties of functions and singular integrals in terms of the mean oscillation

Rahim Rzaev, Lala Aliyeva (2008)

Open Mathematics

This paper is devoted to research on local properties of functions and multidimensional singular integrals in terms of their mean oscillation. The conditions guaranteeing existence of a derivative in the L p-sense at a given point are found. Spaces which remain invariant under singular integral operators are considered.

On some representations of almost everywhere continuous functions on ℝ m

Ewa Strońska (2006)

Colloquium Mathematicae

It is proved that the following conditions are equivalent: (a) f is an almost everywhere continuous function on ℝ m ; (b) f = g + h, where g,h are strongly quasicontinuous on ℝ m ; (c) f = c + gh, where c ∈ ℝ and g,h are strongly quasicontinuous on ℝ m .

On the analytic approximation of differentiable functions from above

Alessandro Tancredi, Alberto Tognoli (2002)

Bollettino dell'Unione Matematica Italiana

We determine conditions in order that a differentiable function be approximable from above by analytic functions, being left invariate on a fixed analytic subset which is a locally complete intersection.

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