Let $\Omega \subset {ℝ}^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1{L}^n{log}^{\alpha }L\left(\Omega \right)$ into an Orlicz space corresponding to a Young function which behaves like $exp\left(t^{n/(n-1-\alpha )}\right)$ for large $t$. We also give the result for the embedding into multiple exponential spaces. Our main result is Theorem where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very...

We characterize those Baire one functions f for which the diagonal product x → (f(x), g(x)) has a connected graph whenever g is approximately continuous or is a derivative.

A new construction method for aggregation operators based on a composition of aggregation operators is proposed. Several general properties of this construction method are recalled. Further, several special cases are discussed. It is also shown, that this construction generalizes a recently introduced twofold integral, which is exactly a composition of the Choquet and Sugeno integral by means of a min operator.

We construct a set B and homeomorphism f where f and $f^{-1}$ have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.

We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence...

Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.

If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\parallel f\parallel ={sup}_I\left|{\int }_{I}f\right|$ where the supremum is taken over all intervals $I\subset ℝ$. Define the translation ${\tau }_x$ by ${\tau }_{x}f\left(y\right)=f(y-x)$. Then $\parallel {\tau }_{x}f-f\parallel$ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\parallel {\tau }_{x}f-f\parallel$ can tend to 0 arbitrarily slowly. In general, $\parallel {\tau }_{x}f-f\parallel \ge \text{osc}f\left|x\right|$ as $x\to 0$, where $\text{osc}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\parallel {\tau }_{x}F-F\parallel \le \parallel f\parallel \left|x\right|$. An example shows that the function $y\mapsto {\tau }_{x}F\left(y\right)-F\left(y\right)$ need not be in $L^1$. However, if $f\in L^1$ then $\parallel {\tau }_x{F-F\parallel }_1\le {\parallel f\parallel }_1\left|x\right|$....

It is shown that a monotone function acting between euclidean spaces $R^n$ and $R^m$ is continuous almost everywhere with respect to the Lebesgue measure on $R^n$.