In the paper, we show that the space of functions of bounded variation and the space of regulated functions are, in some sense, the dual space of each other, involving the Henstock-Kurzweil-Stieltjes integral.

Let H(Q) be the space of all the functions which are holomorphic on an open neighbourhood of a convex locally closed subset Q of CN, endowed with its natural projective topology. We characterize when the topology of the weighted inductive limit of Fréchet spaces which is obtained as the Laplace transform of the dual H(Q)' of H(Q) can be described by weighted sup-seminorms. The behaviour of the corresponding inductive limit of spaces of continuous functions is also investigated.

For $1\<p\<\infty$, a characterization is given of the dual space of weak $L^p$ taken over a non atomic measure space.

We prove some exact formulas for the E and K functionals for pairs of the type (X(A),l sub ∞ (B)) where X has the lattice property. These formulas are extensions of their well-known counterparts in the scalar valued case. In particular we generalize formulas by Pisier and by the present author.

We will consider the following problem$$-\Delta u-\lambda \frac{u}{{\left|x\right|}^2}={\left|\nabla u\right|}^p+cf,u\>0\text{in}\Omega ,$$where $\Omega \subset {ℝ}^N$ is a domain such that $0\in \Omega$, $N\ge 3$, $c\>0$ and $\lambda \>0$. The main objective of this note is to study the precise threshold $p_{+}=p_{+}\left(\lambda \right)$ for which there is novery weak supersolutionif $p\ge p_{+}\left(\lambda \right)$. The optimality of $p_{+}\left(\lambda \right)$ is also proved by showing the solvability of the Dirichlet problem when $1\le p\<p_{+}\left(\lambda \right)$, for $c\>0$ small enough and $f\ge 0$ under some hypotheses that we will prescribe.

We establish the Euler-Lagrange inclusion of a nonsmooth integral functional defined on Orlicz-Sobolev spaces. This result is achieved through variational techniques in nonsmooth analysis and an integral representation formula for the Clarke generalized gradient of locally Lipschitz integral functionals defined on Orlicz spaces.

Let Φ be an N-function, then the Jung constants of the Orlicz function spaces LΦ[0,1] generated by Φ equipped with the Luxemburg and Orlicz norms have the exact value:(i) If FΦ(t) = tφ(t)/Φ(t) is decreasing and 1 < CΦ < 2, then JC(L(Φ)[0,1]) = JC(LΦ[0,1]) = 21/CΦ-1;(ii) If FΦ(t) is increasing and CΦ > 2, then JC(L(Φ)[0,1]) = JC(LΦ[0,1])=2-1/CΦ,where CΦ= limt→+∞ tφ(t)/Φ(t).

Let $D$ be a domain in $C^2$. Given $w\in C$, set $D_w=\left\{z\in C\mid \left(z,w\right)\in D\right\}$. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E\left(D,f\right)$ of all $w$ such that $f(\cdot ,w)$ is not square-integrable in $D_w$ has measure zero. We call this set the exceptional set for $f$. In this Note we prove that whenever $0<r<1$ there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $C^2$ such that $E\left(B,f\right)$ is the circle $C\left(0,r\right)=\left\{z\in C\mid \left|z\right|=r\right\}$.

We discuss recent developments in the study of the homotopy classes for the Sobolev spaces $W^{1,p}\left(M;N\right)$. In particular, we report on the work of H. Brezis - Y. Li [5] and F.B. Hang - F.H. Lin [9].

In this article, we prove the first mean value theorem for integrals [16]. The formalization of various theorems about the properties of the Lebesgue integral is also presented.MML identifier: MESFUNC7, version: 7.8.09 4.97.1001

We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on ${ℝ}^n$ for the relativistic Schrödinger operator $\sqrt{-\Delta }+Q$ to be bounded as an operator from the Sobolev space $W_2^{1/2}\left({ℝ}^n\right)$ to its dual $W_2^{-1/2}\left({ℝ}^n\right)$.