On a problem of Eidelheit from The Scottish Book concerning absolutely continuous functions

Abstract

Dedicated to the memory of M. Eidelheit on the occasion of 100th years of his birth Keywords: Absolutely continuous functions Lipschitz functions Superpositions Absolutely continuous functions of two variables Embeddings Strictly singular embeddings Superstrictly singular embeddings Beppo Levi spaces Sobolev spaces Rearrangement invariant spacesA negative solution of Problem 188 posed by Max Eidelheit in the Scottish Book concerning superpositions of separately absolutely continuous functions is presented. We discuss here this and some related problems which have also negative solutions. Finally, we give an explanation of such negative answers from the "embeddings of Banach spaces" point of view.

Diagonals of Separately Absolutely Continuous Mappings Coincide with the Sums of Absolutely Convergent Series of Continuous Functions

“…In [10] the diagonal variant of the problem of Eidelheit from the famous "Scottish Book" on a composition of absolutely continuous functions was investigated. It was proved that there exists a separately absolutely continuous function f : [0, 1] 2 → R such that its partial derivatives f ′ x and f ′ y in the degree p are integrable on [0, 1] 2 for every p > 1, and such that its diagonal g is not absolutely continuous.…”

Review of the book by Roman Duda, �Pearls from a lost city. The Lvov school of mathematics�

“…I myself have written comments to a positive solution of Problem 87 posed by Banach (pages 161-170). Moreover, jointly with A. Plichko and V. Mykhaylyuk we solved the problem of 188 posed by Eidelheit, and published the solution in [18]. The author ends this part citing three anecdotes.…”