On the doubling condition for non-negative positive definite functions on on the half-line with power weight

Abstract

Continuous non-negative positive definite functions satisfy the following
property:
\[
\int_{-R}^{R}f(x)\,dx\le C(R)\int_{-1}^{1}f(x)\,dx,\quad R\ge 1,
\tag{$*$}
\]
where the smallest positive constant $C(R)$ does not depend on $f$. For $R=2$,
this property is well known as the doubling condition at zero. These
inequalities have applications in number theory.

In the one-dimensional case, the inequality ($*$) was studied by B.F.~Logan
(1988), as well as recently by A.~Efimov, M.~Ga\'al, and Sz.~R\'ev\'esz (2017).
It has been proven that $2R-1\le C(R)\le 2R+1$ for $R=2,3,\ldots$, whence it
follows that $C(R)\sim 2R$. The question of exact constants is still open.

A multidimensional version of the inequality ($*$) for the Euclidean space
$\mathbb{R}^{n}$ was investigated by D.V.~Gorbachev and S.Yu.~Tikhonov (2018).
In particular, it was proved that for continuous positive definite functions
$f\colon \mathbb{R}^{n}\to \mathbb{R}_{+}$
\[
\int_{|x|\le R}f(x)\,dx\le c_{n}R^{n}\int_{|x|\le 1}f(x)\,dx,
\]
where $c_{n}\le 2^{n}n\ln n\,(1+o(1))(1+R^{-1})^{n}$ при $n\to \infty$. For
radial functions, we obtain the one-dimensional weight inequality
\[
\int_{0}^{R}f(x)x^{n-1}\,dx\le c_{n}R^{n}\int_{0}^{1}f(x)x^{n-1}\,dx,\quad n\in
\mathbb{N}.
\]

We study the following natural weight generalization of such inequalities:
\[
\int_{0}^{R}f(x)x^{2\alpha+1}\,dx\le
C_{\alpha}(R)\int_{0}^{1}f(x)x^{2\alpha+1}\,dx,\quad \alpha\ge -1/2,
\]
where $f\colon \mathbb{R}_{+}\to \mathbb{R}_{+}$ is an even positive definite
function with respect to the weight $x^{2\alpha+1}$. This concept has been
introduced by B.M.~Levitan (1951) and means that for arbitrary
$x_{1},\ldots,x_{N}\in \mathbb{R}_{+}$ matrix
$(T_{\alpha}^{x_i}f(x_j))_{i,j=1}^{N}$ is semidefinite. Here $T_{\alpha}^{t}$
is the Bessel--Gegenbauer generalized translation. Levitan proved an analogue
of the classical Bochner theorem for such functions according to which $f$ has
the nonnegative Hankel transform (in the measure sense).

We prove that for every $\alpha\ge -1/2$
\[
c_{1}(\alpha)R^{2\alpha+2}\le C_{\alpha}(R)\le c_{2}(\alpha)R^{2\alpha+2},\quad
R\ge 1.
\]
The lower bound is trivially achieved on the function $f(x)=1$. To prove the
upper bound we apply lower estimates of the sums
$\sum_{k=1}^{m}a_{k}T^{x_{k}}\chi(x)$, where $\chi$ is the characteristic
function of the segment $[0,1]$, and also we use properties of the Bessel
convolution.