### $\mathcal{O}$-regularly varying functions in approximation theory.

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In the paper, binary 1-Lipschitz aggregation operators and specially quasi-copulas are studied. The characterization of 1-Lipschitz aggregation operators as solutions to a functional equation similar to the Frank functional equation is recalled, and moreover, the importance of quasi-copulas and dual quasi-copulas for describing the structure of 1-Lipschitz aggregation operators with neutral element or annihilator is shown. Also a characterization of quasi-copulas as solutions to a certain functional...

2000 Mathematics Subject Classification: 26A33, 33C60, 44A20In this survey we present a brief history and the basic ideas of the generalized fractional calculus (GFC). The notion “generalized operator of fractional integration” appeared in the papers of the jubilarian Prof. S.L. Kalla in the years 1969-1979 when he suggested the general form of these operators and studied examples of them whose kernels were special functions as the Gauss and generalized hypergeometric functions, including arbitrary...

We study the Poincaré inequality in Sobolev spaces with variable exponent. Under a rather mild and sharp condition on the exponent p we show that the inequality holds. This condition is satisfied e.g. if the exponent p is continuous in the closure of a convex domain. We also give an essentially sharp condition for the exponent p as to when there exists an imbedding from the Sobolev space to the space of bounded functions.

An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre’s interpolation functor $\u27e8{\u27e9}_{\phi}$ (see [26]) and its Gagliardo closure on couples of...

We state and prove a chain rule formula for the composition $T\left(u\right)$ of a vector-valued function $u\in {W}^{1,r}\left(\mathrm{\Omega};{\mathbb{R}}^{M}\right)$ by a globally Lipschitz-continuous, piecewise ${C}^{1}$ function $T$. We also prove that the map $u\to T\left(u\right)$ is continuous from ${W}^{1,r}\left(\mathrm{\Omega};{\mathbb{R}}^{M}\right)$ into ${W}^{1,r}\left(\mathrm{\Omega}\right)$ for the strong topologies of these spaces.