A Baire function not countably decomposable into continuous functions
A bornological approach to rotundity and smoothness applied to approximation.
A bound for certain bibasic sums and applications.
A Brief Story about the Operators of the Generalized Fractional Calculus
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...
A C1 function which is nowhere strongly paraconvex and nowhere semiconcave
A Cantor regular set which does not have Markov's property
A capacity approach to the Poincaré inequality and Sobolev imbeddings in variable exponent Sobolev spaces.
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.
A Carlson type inequality with blocks and interpolation
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 (see [26]) and its Gagliardo closure on couples of...
A category analogue of the density topology
A Cauchy type condition and integrability
A Cauchy-Pompeiu formula in super Dunkl-Clifford analysis
Using a distributional approach to integration in superspace, we investigate a Cauchy-Pompeiu integral formula in super Dunkl-Clifford analysis and several related results, such as Stokes formula, Morera's theorem and Painlevé theorem for super Dunkl-monogenic functions. These results are nice generalizations of well-known facts in complex analysis.
A certain modification of Hardy's inequality
A chain rule formula for the composition of a vector-valued function by a piecewise smooth function
We state and prove a chain rule formula for the composition of a vector-valued function by a globally Lipschitz-continuous, piecewise function . We also prove that the map is continuous from into for the strong topologies of these spaces.
A characterization of a two-weight inequality for discrete two-dimensional Hardy operators.
A characterization of functions via lower directional derivatives
The notion of -stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of -stable functions coincides with the class of C functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp. 383–387.
A characterization of chaotic functions with entropy zero via their maximal scrambled sets
In this note we characterize chaotic functions (in the sense of Li and Yorke) with topological entropy zero in terms of the structure of their maximal scrambled sets. In the interim a description of all maximal scrambled sets of these functions is also found.
A characterization of midconvex set-valued functions
A characterization of monomial functions.
A characterization of sets in with DC distance function
We give a complete characterization of closed sets whose distance function is DC (i.e., is the difference of two convex functions on ). Using this characterization, a number of properties of such sets is proved.
A characterization of Sobolev spaces via local derivatives
Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function possesses an derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space . Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.