The paper is devoted to spaces of generalized smoothness on so-called h-sets. First we find quarkonial representations of isotropic spaces of generalized smoothness on ℝⁿ and on an h-set. Then we investigate representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h-sets. We prove that both representations are equivalent, and also find the domain of some time-changed Dirichlet form on an h-set.

We use the scale of Besov spaces $B_{\tau ,\tau }^{\alpha }\left(\right)$, 1/τ = α/d + 1/p, α > 0, p fixed, to study the spatial regularity of solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains ⊂ ℝ. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Questa Nota è dedicata a mettere in evidenza alcune proprietà degli spazi $BV\left(\mathrm{\Omega }\right)=N^1\left(\mathrm{\Omega }\right)$ delle funzioni a variazione limitata e degli spazi di Nikolskii $N_1^{\lambda }\left(\mathrm{\Omega }\right)=N^{\lambda }\left(\mathrm{\Omega }\right)$ ed $N^{\lambda ,0}\left(\mathrm{\Omega }\right)$, ( $\lambda \in \left(0,1\right)$ ), che non mi risulta siano già state esposte nella forma generale qui enunciata, quali la non separabilità, l'essere il duale di uno spazio di Banach separabile, la convergenza e la compattezza debole $*$ in $L_{W^{*}}^{\mathrm{\infty }}\left(0,T;N^{\lambda }\left(\mathrm{\Omega }\right)\right)$ e le loro applicazioni al classico problema di Stefan bifase.

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small...

We prove Strichartz's conjecture regarding a characterization of Hardy-Sobolev spaces.

Given a compact manifold $N^n$, an integer $k\in {\mathbb{N}}_{*}$ and an exponent $1\le p<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k,p}(Q^m;N^n)$ when the homotopy group ${\pi }_{\lfloor kp\rfloor }\left(N^n\right)$ of order $\lfloor kp\rfloor$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m-\lfloor kp\rfloor -1$, without any restriction on the homotopy group of $N^n$.