We characterize, in terms of the Beurling-Malliavin density, the discrete spectra $\Lambda\subset\mathbb R$ for which a generator exists, that is a function $\varphi\in L^1(\mathbb R)$ such that its $\Lambda$-translates $\varphi(x-\lambda), \lambda\in\Lambda$, span $L^1(\mathbb R)$. It is shown that these spectra coincide with the uniqueness sets for certain analytic classes. We also present examples of discrete spectra $\Lambda\subset\mathbb R$ which do not admit a single generator while they admit a pair of generators.

This paper studies the on- and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs.

We consider a periodic pseudo-differential operator on the real line, which is a lower-order perturbation of an elliptic operator with a homogeneous symbol and constant coefficients. It is proved that the density of states of such an operator admits a complete asymptotic expansion at large energies. A few first terms of this expansion are found in a closed form.

Let $X$ be a Banach space, $u\in X^{**}$ and $K, Z$ two subsets of $X^{**}$. Denote by $d(u,Z)$ and $d(K,Z)$ the distances to $Z$ from the point $u$ and from the subset $K$ respectively. The Krein-Šmulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w$^*$-compact subset $K\subset X^{**}$ such that $d(K,X)=0$ satisfies $d(\overline{\text{co}}^{w^*}(K),X)=0$. We extend this result in the following way: if $Z\subset X$ is a closed subspace of $X$ and $K\subset X^{**}$ is a w$^*-$compact subset of $X^{**}$, then $$ d(\overline{\text{co}}^{w^*}(K),Z)\leq 5 d(K,Z). $$ Moreover, if $Z\cap K$ is w$^*$-dense in $K$, then $d(\overline{\text{co}}^{w^*}(K),Z)\leq 2 d(K,Z)$. However, the equality $d(K,X)=d(\overline{\text{co}}^{w^*}(K),X)$ holds in many cases, for instance, if $\ell_1\not\subseteq X^*$, if $X$ has w$^*$-angelic dual unit ball (for example, if $X$ is WCG or WLD), if $X=\ell_1(I)$, if $K$ is fragmented by the norm of $X^{**}$, etc. We also construct under $CH$ a w$^*$-compact subset $K\subset B(X^{**})$ such that $K\cap X$ is w$^*$-dense in $K$, $d(K,X)=\frac 12$ and $d(\overline{\text{co}}^{w^*}(K),X)=1$.

We use logarithmic Sobolev inequalities involving the $p$--energy functional recently derived in [Del Pino, M. and Dolbeault, J.: The optimal euclidean $\mathrm{L}^p$-Sobolev logarithmic inequality. J. Funct. Anal. 197 (2003), 151-161], [Gentil, I.: The general optimal $\mathrm{L}^p$-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations. J. Funct. Anal. 202 (2003), 591-599] to prove L$^p$-L$^q$ smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form $\dot u=\triangle_p(u^m)$ (with $(m(p-1)\ge 1$) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bounds are of the form $\Vert u(t)\Vert_q\le C\Vert u_0\Vert_r^\gamma/t^\beta$ for any $r\le q\in[1,+\infty]$ and $t>0$ and the exponents $\beta,\gamma$ are shown to be the only possible for a bound of such type.

We show that there are Hilbert spaces constructed from the Hausdorff measures $\mathcal{H}^{s}$ on the real line $\mathbb{R}$ with $0 < s < 1$ which admit multiresolution wavelets. For the case of the middle-third Cantor set $\mathbf{C}\subset \lbrack 0,1]$, the Hilbert space is a separable subspace of $L^{2}(\mathbb{R},(dx)^{s})$ where $s=\log _{3}(2)$. While we develop the general theory of multi-resolutions in fractal Hilbert spaces, the emphasis is on the case of scale $3$ which covers the traditional Cantor set $\mathbf{C}$. Introducing \begin{equation*} \psi_{1}(x)=\sqrt{2}\chi _{\mathbf{C}}(3x-1) \qquad\mbox{and}\qquad \psi _{2}(x)=\chi _{\mathbf{C}}(3x)- \chi_{\mathbf{C}}(3x-2) \end{equation*} we first describe the subspace in $L^{2}(\mathbb{R},(dx)^{s})$ which has the following family as an orthonormal basis (ONB): \begin{equation*} \psi_{i,j,k}(x)=2^{j/2}\psi_{i}(3^{j}x-k)\text{,} \end{equation*} where $i=1,2,j$, $k\in \mathbb{Z}$. Since the affine iteration systems of Cantor type arise from a certain algorithm in $\mathbb{R}^d$ which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.

Given a bounded Lipschitz domain $\Omega\subset {\mathbb R}^n$, $n\geq 3$, we prove~that the Poisson's problem for the Laplacian with right-hand side in $L^p_{-t}(\Omega)$, Robin-type boundary datum in the Besov space $B^{1-1/p-t,p}_{p}(\partial \Omega)$ and non-negative, non-everywhere vanishing Robin coefficient $b\in L^{n-1}(\partial \Omega)$, is uniquely solvable in the class $L^p_{2-t}(\Omega)$ for $(t,\frac{1}{p})\in {\mathcal V}_{\epsilon}$, where ${\mathcal V}_{\epsilon}$ ($\epsilon\geq 0$) is an open ($\Omega$,$b$)-dependent plane region and ${\mathcal V}_{0}$ is to be interpreted ad the common (optimal) solvability region for all Lipschitz domains. We prove a similar regularity result for the Poisson's problem for the 3-dimensional Lamé System with traction-type Robin boundary condition. All solutions are expressed as boundary layer potentials.

The Sierpinski gasket and other self-similar fractal subsets of $\mathbb R^d$, $d\ge 2$, can be mapped by quasiconformal self-maps of $\mathbb R^d$ onto sets of Hausdorff dimension arbitrarily close to one. In $\mathbb R^2$ we construct explicit mappings. In $\mathbb R^d$, $d\ge 3$, the results follow from general theorems on the equivalence of invariant sets for iterated function systems under quasisymmetric maps and global quasiconformal maps. More specifically, we present geometric conditions ensuring that (i) isomorphic systems have quasisymmetrically equivalent invariant sets, and (ii) one-parameter isotopies of systems have invariant sets which are equivalent under global quasiconformal maps.

We prove Fefferman's SAK Principle for a class of hypoelliptic operators on $\mathbb R^2$ whose nonnegative symbol vanishes anisotropically on the characteristic manifold.

In this paper we obtain bounds on $h^0(E)$ where $E$ is a semistable bundle of rank 3 over a smooth irreducible projective curve $X$ of genus $g \geq 2$ defined over an algebraically closed field of characteristic 0. These bounds are expressed in terms of the degrees of stability $s_1(E)$, $s_2(E)$. We show also that in some cases the bounds are best possible. These results extend recent work of J. Cilleruelo and I. Sols for bundles of rank 2.

Let $\Omega$ be a bounded domain in $\mathbb R^n$ and denote by $id_\Omega$ the restriction operator from the Besov space $B_{pq}^{1+n/p}(\mathbb R^n)$ into the generalized Lipschitz space $Lip^{(1,-\alpha)}(\Omega)$. We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like $e_k(id_\Omega) \sim k^{-1/p}$ if $\alpha > \max (1+2/p-1/q,1/p)$. Our estimates improve previous results by Edmunds and Haroske.

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.

We study a multilinear fixed-point equation in a closed ball of a Banach space where the application is $1$-Lipschitzian: existence, uniqueness, approximations, regularity.

H.P.F. Swinnerton-Dyer determined the structure of the ring of modular forms modulo $p$ in the elliptic modular case. In this paper, the structure of the ring of Hilbert modular forms modulo $p$ is studied. In the case where the discriminant of corresponding quadratic field is 8 (or 5), the explicit structure is determined.