The Hardy inequality ${\int }_{\Omega }{\left|u\left(x\right)\right|}^{p}d{\left(x\right)}^{-p}\ddot{x}\le c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^p\ddot{x}$ with $d\left(x\right)=dist(x,\partial \Omega )$ holds for $u\in C_0^{\infty }\left(\Omega \right)$ if $\Omega \subset {ℝ}^n$ is an open set with a sufficiently smooth boundary and if $1<p<\infty$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.

We prove an approximation theorem in generalized Sobolev spaces with variable exponent $W^{1,p\left(·\right)}\left(\Omega \right)$ and we give an application of this approximation result to a necessary condition in the calculus of variations.

We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.

Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities ${\int }_{ℝ₊}M\left(\omega \left(x\right)\right|u\left(x\right)\left|\right)exp(-\phi \left(x\right))dx\le C{\int }_{ℝ₊}M\left(\right|u^{\text{'}}\left(x\right)\left|\right)exp(-\phi \left(x\right))dx$, where u belongs to some set of locally absolutely continuous functions containing $C{₀}^{\infty }\left(ℝ₊\right)$. We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants.

Define $h^{\infty }\left(E\right)$ as the subspace of $C^{\infty }(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-\infty }(E*)$ consisting of all harmonic E*-valued functions g such that ${(1-|x\left|\right)}^{m}f$ is bounded for some m>0. Then the dual $h^{\infty }(E*)$ is represented by $h^{-\infty }(E*)$ through $⟨f,g{⟩}_0=li{m}_{r\to 1}{ʃ}_B⟨f\left(rx\right),g\left(x\right)⟩dx$, $f\in h^{-\infty }(E*),g\in h^{\infty }\left(E\right)$. This extends the results of S. Bell in the scalar case.

In this paper the absolute convergence of the group Fourier transform for the Heisenberg group is investigated. It is proved that the Fourier transform of functions belonging to certain Besov spaces is absolutely convergent. The function spaces are defined in terms of the heat semigroup of the full Laplacian of the Heisenberg group.

We are concerned with a relation between parabolicity and coerciveness in Besov spaces for a higher order linear evolution equation in a Banach space. As proved in a preceding work, a higher order linear evolution equation enjoys coerciveness in Besov spaces under a certain parabolicity condition adopted and studied by several authors. We show that for a higher order linear evolution equation coerciveness in Besov spaces forces the parabolicity of the equation. We thus conclude that parabolicity...