We prove that if $r_i$ is the Rademacher system of functions then ${\left(ʃ\parallel {\sum }_{i=1}^n{x}_i{r}_i\left(t\right){\parallel }^{2}dt\right)}^{1/2}\le \surd 2ʃ\parallel {\sum }_{i=1}^n{x}_i{r}_i\left(t\right)\parallel dt$ for any sequence of vectors $x_i$ in any normed linear space F.

The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g{*}_{\lambda }$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts ${\mu }_S^{\varrho }$ and ${\mu }_{\lambda }^{*,\varrho }$ to S and $g{*}_{\lambda }$. The definition of ${\mu }_S^{\varrho }\left(f\right)$ is as follows: ${\mu }_S^{\varrho }\left(f\right)\left(x\right)=({ʃ}_{|y-x|<t}|1/t^{\varrho }{ʃ}_{\left|z\right|\le t}\Omega \left(z\right)/{\left(\right|z|}^{n-\varrho }{\left)f(y-z)dz\right|}^2\left(dydt\right)/\left(t^{n+1}\right){)}^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and...

Let V be an origin-symmetric convex body in ${ℝ}^n$, n≥ 2, of Gaussian measure ${\gamma }_n\left(V\right)\ge 1/2$. It is proved that for every choice $u_1,...,u_n$ of vectors in the Euclidean unit ball $B_n$, there exist signs ${\epsilon }_j\in -1,1$ with ${\epsilon }_1{u}_1+...+{\epsilon }_n{u}_n\in \left(clogn\right)V$. The method used can be modified to give simple proofs of several related results of J. Spencer and E. D. Gluskin.

We construct k-dimensional (k ≥ 3) subspaces $V^k$ of $l_1$, with a very simple structure and with projection constant satisfying $\lambda \left(V^k\right)\ge \lambda (V^k,l_1)>\lambda \left(l_2^{\left(k\right)}\right)$.

We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For $f\in L_{loc}^1({\mathbf{R}}^2)$, we determine the sharp order of local integrability obtained when the square function of $f$ is in $L^{\infty }$. The Calderón-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector-valued form of an inequality for dyadic martingales that yields the sharp dependence on p of $C_p$ in $\parallel f{\parallel }_p\le C_p\parallel Sf{\parallel }_p$.

Let G be a group generated by r elements $g_1,\dots ,g_r$. Among the reduced words in $g_1,\dots ,g_r$ of length n some, say ${\gamma }_n$, represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of ${\gamma }_{2n}$ has a limit, called the cogrowth exponent with respect to the generators $g_1,\dots ,g_r$. We show by analytic methods that the numbers ${\gamma }_n$ vary regularly, i.e. the ratio ${\gamma }_{2n+2}/{\gamma }_{2n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

The set $𝒰{𝒟}_r$ of point sets of ${ℝ}^n,n\ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r\&gt;0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $𝒰{𝒟}_{r,f}\subset 𝒰{𝒟}_r$ of the finite point sets is compatible with the restriction of this topology to $𝒰{𝒟}_{r,f}$. We show that its subsets of Delone sets of given constants in ${ℝ}^n,n\ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic...

Let $E$ be a modular elliptic curve over $\mathbb{Q}$ $L^{\left(n\right)}(s,E)$ denote the $n$-th derivative of its Hasse-Weil $L$-series. We estimate the number of twisted elliptic curves $E_d,d\le D$ such that $L^{\left(n\right)}(1,E_d)\ne 0$.

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results...