On the best constant in the Khinchin-Kahane inequality

Rafał Latała; Krzysztof Oleszkiewicz

Displaying similar documents to “On the best constant in the Khinchin-Kahane inequality”

A remark on Whittaker functions on SL $(n, ℝ)$

Taku Ishii (2005)

Annales de l’institut Fourier

Similarity:

We prove the recursive integral formula of class one $M$ -Whittaker functions on SL $(n, ℝ)$ conjectured and verified in case of $n = 3, 4$ by Stade.

The reciprocal of the beta function and $G L (n, ℝ)$ Whittaker functions

Eric Stade (1994)

Annales de l'institut Fourier

Similarity:

In this paper we derive, using the Gauss summation theorem for hypergeometric series, a simple integral expression for the reciprocal of Euler’s beta function. This expression is similar in form to several well-known integrals for the beta function itself. We then apply our new formula to the study of $G L (n, ℝ)$ Whittaker functions, which are special functions that arise in the Fourier theory for automorphic forms on the general linear group. Specifically, we deduce explicit integral...

An isomorphic Dvoretzky's theorem for convex bodies

Y. Gordon, O. Guédon, M. Meyer (1998)

Studia Mathematica

Similarity:

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in $ℝ^{n}$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of $ℝ^{n}$ satisfying $d (Y \cap K, B_{2}^{k}) \leq C (1 + \sqrt (k / l n (n / (k l n (n + 1))))$ . This formulation of Dvoretzky’s theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex. ...

Tail and moment estimates for some types of chaos

Rafał Latała (1999)

Studia Mathematica

Similarity:

Let $X_{i}$ be a sequence of independent symmetric real random variables with logarithmically concave tails. We consider a variable $X = \sum_{i \neq j} a_{i, j} X_{i} X_{j}$ , where $a_{i, j}$ are real numbers. We derive approximate formulas for the tails and moments of X and of its decoupled version, which are exact up to some universal constants.

Boundedness of Marcinkiewicz functions.

Minako Sakamoto, Kôzô Yabuta (1999)

Studia Mathematica

Similarity:

The $L^{p}$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g *_{λ}$ -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 $μ_{S}^{ϱ}$ and $μ_{λ}^{*, ϱ}$ to S and $g *_{λ}$ . The definition of $μ_{S}^{ϱ} (f)$ is as follows: $μ_{S}^{ϱ} (f) (x) = (ʃ_{| y - x | < t} | 1 / t^{ϱ} ʃ_{| z | \leq t} Ω (z) / {(| z |}^{n - ϱ} {) f (y - z) d z |}^{2} (d y d t) / (t^{n + 1}))^{1 / 2}$ , 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...

Bounded double square functions

Jill Pipher (1986)

Annales de l'institut Fourier

Similarity:

We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For $f \in L_{l o c}^{1} (R^{2})$ , 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 $∥ f ∥_{p} \leq C_{p} ∥ S f ∥_{p}$ .

Polydisc slicing in $ℂ^{n}$

Krzysztof Oleszkiewicz, Aleksander Pełczyński (2000)

Studia Mathematica

Similarity:

Let D be the unit disc in the complex plane ℂ. Then for every complex linear subspace H in $ℂ^{n}$ of codimension 1, $v o l_{2 n - 2} (D^{n - 1}) \leq v o l_{2 n - 2} (H \cap D^{n}) \leq 2 v o l_{2 n - 2} (D^{n - 1})$ . The lower bound is attained if and only if H is orthogonal to the versor $e_{j}$ of the jth coordinate axis for some j = 1,...,n; the upper bound is attained if and only if H is orthogonal to a vector $e_{j} + σ e_{k}$ for some 1 ≤ j < k ≤ n and some σ ∈ ℂ with |σ| = 1. We identify $ℂ^{n}$ with $ℝ^{2 n}$ ; by $v o l_{k} (\cdot)$ we denote the usual k-dimensional volume in $ℝ^{2 n}$ . The result is a complex counterpart of Ball’s [B1]...

On the estimation of Cesàro means of orthonormal series

J. Meder (1958)

Annales Polonici Mathematici

Similarity:

On a semigroup of measures with irregular densities

Przemysław Gadziński (2000)

Colloquium Mathematicae

Similarity:

We study the densities of the semigroup generated by the operator $- X^{2} + | Y |$ on the 3-dimensional Heisenberg group. We show that the 7th derivatives of the densities have a jump discontinuity. Outside the plane x=0 the densities are $C^{\infty}$ . We give explicit spectral decomposition of images of $- X^{2} + | Y |$ in representations.