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### Volume 61, number 1 (2020)

June 2020
 Local solvability of elliptic equations of even order with Hölder coefficients. María Amelia Muschietti and Federico Tournier We consider elliptic equations of order $2m$ with Hölder coefficients. We show local solvability of the Dirichlet problem with $m$ conditions on the boundary of the upper half space. First we consider local solvability in free space and then we treat the boundary case. Our method is based on applying the operator to an approximate solution and iterating in the Hölder spaces. A priori estimates for the approximate solution is the essential part of the paper. 1–47 On convergence of subspaces generated by dilations of polynomials. An application to best local approximation. Fabián E. Levis and Claudia V. Ridolfi We study the convergence of a net of subspaces generated by dilations of polynomials in a finite dimensional subspace. As a consequence, we extend the results given by Zó and Cuenya [Advanced Courses of Mathematical Analysis II (Granada, 2004), 193–213, World Scientific, 2007] on a general approach to the problems of best vector-valued approximation on small regions from a finite dimensional subspace of polynomials. 49–62 Perturbation of Ruelle resonances and Faure–Sjöstrand anisotropic space. Yannick Guedes Bonthonneau Given an Anosov vector field $X_0$, all sufficiently close vector fields are also of Anosov type. In this note, we check that the anisotropic spaces described by Faure and Sjöstrand and by Dyatlov and Zworski can be chosen adapted to any smooth vector field sufficiently close to $X_0$ in $C^1$ norm. 63–72 Generalized metallic structures. Adara M. Blaga and Antonella Nannicini We study the properties of a generalized metallic, a generalized product and a generalized complex structure induced on the generalized tangent bundle of a smooth manifold $M$ by a metallic Riemannian structure $(J,g)$ on $M$, providing conditions for their integrability with respect to a suitable connection. Moreover, by using methods of generalized geometry, we lift $(J,g)$ to metallic Riemannian structures on the tangent and cotangent bundles of $M$, underlying the relations between them. 73–86 A heat conduction problem with sources depending on the average of the heat flux on the boundary. Mahdi Boukrouche and Domingo A. Tarzia Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\mathbb{R}^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0, y, s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of the second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method. 87–101 On classes of finite rings. Aleksandr Tsarev A class of rings is a formation whenever it contains all homomorphic images of its members and if it is subdirect product closed. In the present paper, it is shown that the lattice of all formations of finite rings is algebraic and modular. Let $R$ be a finite commutative ring with an identity element. It is established that there is a one-to-one correspondence between the set of all invariant fuzzy prime ideals of $R$ and the set of all fuzzy prime ideals of each ring of the formation generated by $R$. 103–111 A canonical distribution on isoparametric submanifolds I. Cristián U. Sánchez We show that on every compact, connected homogeneous isoparametric submanifold $M$ of codimension $h\geq 2$ in a Euclidean space, there exists a canonical distribution which is bracket generating of step 2. An interesting consequence of this fact is also indicated. In this first part we consider only the case in which the system of restricted roots is reduced, reserving for a second part the case of non-reduced restricted roots. 113–130 Second cohomology space of $\mathfrak{sl}(2)$ acting on the space of bilinear bidifferential operators. Imed Basdouri, Sarra Hammami, and Olfa Messaoud We consider the $\mathfrak{sl}(2)$-module structure on the spaces of bilinear bidifferential operators acting on the spaces of weighted densities. We compute the second cohomology group of the Lie algebra $\mathfrak{sl}(2)$ with coefficients in the space of bilinear bidifferential operators that act on tensor densities $\mathcal{D}_{\lambda,\nu,\mu}$. 131–143 Kreĭn space unitary dilations of Hilbert space holomorphic semigroups. Stefania A. M. Marcantognini The infinitesimal generator $A$ of a strongly continuous semigroup on a Hilbert space is assumed to satisfy that $B_\beta:=A-\beta$ is a sectorial operator of angle less than $\frac{\pi}{2}$ for some $\beta \geq 0$. If $B_\beta$ is dissipative in some equivalent scalar product then the Naimark–Arocena representation theorem is applied to obtain a Kreĭn space unitary dilation of the semigroup. 145–160 Lifting vector fields from manifolds to $r$-jet prolongation of the tangent bundle. Jan Kurek and Włodzimierz M. Mikulski If $m\geq 3$ and $r\geq 0$, we deduce that any natural linear operator lifting vector fields from an $m$-manifold $M$ to the $r$-jet prolongation $J^{r}TM$ of the tangent bundle $TM$ is the composition of the flow lifting $\mathcal{J}^r$ corresponding to the $r$-jet prolongation functor $J^r$ with a natural linear operator lifting vector fields from $M$ to $TM$. If $0\leq s\leq r$ and $m\geq 3$, we find all natural linear operators transforming vector fields on $M$ into base-preserving fibred maps $J^{r}TM\to J^{s}TM$. 161–168 Localization operators and scalogram associated with the generalized continuous wavelet transform on $\mathbb{R}^d$ for the Heckman–Opdam theory. Hatem Mejjaoli and Khalifa Trimèche We consider the generalized wavelet transform $\Phi_{h}^{W}$ on $\mathbb{R}^{d}$ for the Heckman–Opdam theory. We study the localization operators associated with $\Phi_{h}^{W}$; in particular, we prove that they are in the Schatten–von Neumann class. Next we introduce some results on the scalogram for this transform. 169–196