Pour trois fonctions non-négatives intégrables sur ${\mathbf{R}}^n$, $f,g$ et $h$, telles que $\left(h\right(x+y){)}^{-1/n}\le (f\left(x\right){)}^{-1/n}+\left(g\right(y){)}^{-1/n}$, Borelll a établi l’inégalité $\int h\left(z\right)dz\ge \min \int f\left(x\right)dx,\int g\left(y\right)dy)$. Nous déterminons les conditions précises où l’inégalité sera stricte. La clef de cette analyse est une nouvelle caractérisation des fonctions convexes mesurables.

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${\left\{P_i\right\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i\left(x\right)+Q_1\left(x\right)P_{i-1}\left(x\right)+Q_2\left(x\right)P_{i-2}\left(x\right)=0$ with the standard initial conditions $P_0\left(x\right)=1,P_{-1}\left(x\right)=0,$ where $Q_1\left(x\right)$ and $Q_2\left(x\right)$ are arbitrary real polynomials.

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\mathrm{BV}}$ parametrization (i.e., a $C^1$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X={ℝ}^n$. We present examples which show applicability of our characterizations. For example, we show that the $C^{1,\mathrm{BV}}$ and $C^2$ parametrization problems are equivalent for $X=ℝ$ but are not equivalent for $X={ℝ}^2$.

In this paper we consider the Darboux type properties for the paratingent. We review some of the standard facts on the multivalued functions and the paratingent. We prove that the paratingent has always the Darboux property but the property D* holds only when the paratingent is a multivalued function.

This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{\text{'}\text{'}\text{'}}+q\left(t\right)x^{-\gamma }=0$, by means of regularly varying functions, where $\gamma$ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\to \infty$ and to acquire...

Let F be a multifunction with values in Lₚ(Ω, X). In this note, we study which regularity properties of F are preserved when we consider the decomposable hull of F.

It is shown to be consistent that every function of first Baire class can be decomposed into ${\aleph }_1$ continuous functions yet the least cardinal of a dominating family in ${}^{\omega }\omega$ is ${\aleph }_2$. The model used in the one obtained by adding ${\omega }_2$ Miller reals to a model of the Continuum Hypothesis.

Using the idea of the optimal decomposition developed in recent papers (Edmunds-Krbec, 2000) and in Cruz-Uribe-Krbec we study the boundedness of the operator Tg(x) = ∫x1 g(u)du / u, x ∈ (0,1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and Lorentz-Orlicz spaces. These operators are naturally linked with Moser's lemma, O'Neil's convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs...

The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.