In the present work we prove that, in the space of Pettis integrable functions, any subset that is decomposable and closed with respect to the topology induced by the so-called Alexiewicz norm $\left|∥·∥\right|$ (where $\left|∥f∥\right|={sup}_{[a,b]\subset [0,1]}\parallel {\int }_a^{b}f\left(s\right)ds\parallel$) is convex. As a consequence, any such family of Pettis integrable functions is also weakly closed.

We establish a Trudinger inequality for functions that satisfy a suitable Poincarè inequality in a Euclidean space equipped with a Borel measure that need not be doubling.

The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $-\Delta u-{\Lambda }_N\frac{u}{{\left|x\right|}^2}={\left|\nabla u+\frac{N-2}{2}\frac{u}{{\left|x\right|}^2}x\right|}^2{\left|x\right|}^{\left(N-2\right)/2}+\lambda f\left(x\right)$ in $\Omega$, $u=0$ on $\partial \Omega$, ${\Lambda }_N={(\left(N-2\right)/2)}^2$. This problem is a particular case of problem (2). Notice that $\left(N-2\right)/2$ is optimal as coefficient and exponent on the right hand side.

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.

If $(\Omega ,\Sigma ,\mu )$ is a finite measure space and $X$ a Banach space, in this note we show that $L_{w^{*}}^1(\mu ,X^{*})$, the Banach space of all classes of weak* equivalent $X^{*}$-valued weak* measurable functions $f$ defined on $\Omega$ such that $\parallel f\left(\omega \right)\parallel \le g\left(\omega \right)$ a.e. for some $g\in L_1\left(\mu \right)$ equipped with its usual norm, contains a copy of $c_0$ if and only if $X^{*}$ contains a copy of $c_0$.

2000 Mathematics Subject Classification: 42B30, 46E35, 35B65.We prove two results concerning the div-curl lemma without assuming any sort of exact cancellation, namely the divergence and curl need not be zero, and $$div\left(u^{-}{v}^{\to }\right)\in H^1\left(R^d\right)$$ which include as a particular case, the result of [3].