Littlewood--Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood--Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood--Paley analysis and obtain the corresponding ``renormalization'' for the convergence of this multiparameter Littlewood--Paley analysis.

Soit $F$ un corps local non archim\'edien, et $G$ le groupe des $F$-points d'un groupe r\'eductif connexe quasi-d\'eploy\'e d\'efini sur $F$. Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact biinvariantes par un sous-groupe d'Iwahori $I$ donn\'e. On montre tout d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$ sont enti\`erement d\'etermin\'ees par sa restriction au sous-espace de dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la r\'eunion des sous-groupes parahoriques de $G$ contenant $I$. On utilise ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions sur $G$, que cet espace est engendr\'e d'une part par certaines int\'egrales orbitales semi-simples, d'autre part par les int\'egrales orbitales unipotentes, en montrant tout d'abord des r\'esultats analogues sur les groupes finis.

This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

The infinitely divisible distributions on $\mathbb{R}^+$ of random variables $C_t$, $S_t$ and $T_t$ with Laplace transforms $$ \left( \frac{1}{\cosh \sqrt{2\lambda}} \right)^t, \quad \left( \frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}} \right)^t, \quad \text{and} \quad \left( \frac{\tanh \sqrt{2\lambda}}{\sqrt{2\lambda}} \right)^t $$ respectively are characterized for various $t>0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L\'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t=1$ or $2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of $C_1$ and $S_2$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo~4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from $S_t$ and $C_t$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by ``analytic'' and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let $(T_1,\ldots,T_{K})$ be $L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if $\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where $Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned by $(T_1f,\ldots,T_kf)$. Call a function $h$ a $(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$, $\|h\|_1<\varepsilon$, and if $\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of measure greater than $1-\varepsilon$. The following is the main technical result, which is obtained by elementary estimates. There is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a $\delta$-spanning function, and if the joint distribution of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge M)\vee(-M)\bigr)/M$ is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$. Furthermore, if $T_k$s are the averages of operators induced by the iterates of a measure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence $(T_i)$ of these averages. Assume that for each $K\geq1$ there is a subsequence $(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning function $f_K$ for this subsequence. Then for each $\varepsilon>0$ there is a function $h$, $0\leq h\leq1$, $\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e.. Another application of the main result gives a refinement of a part of Bourgain's ``Entropy Theorem'', resulting in a different, self contained proof of that theorem.