Each measurable map of an open set U ⊂ R n to R n is equal almost everywhere to the gradient of a continuous almost everywhere differentiable function defined on R n that vanishes, together with its gradient, outside U .
Abstract. In this work we investigate families of translation invariant differentiation bases B of rectangles in R n , for which L log n−1 L(R n ) is the largest Orlicz space that B differentiates. In particular, we improve on techniques developed by Stokolos in [11] and [13].
In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.
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