Bounded translation operator for the $(k,1)$-generalized Fourier transform

In spaces with a Dunkl weight $v_k (x)$ of power type on $\mathbb{R}^ d$, defined by a root system and a nonnegative multiplicity function $k$ invariant with respect to a finite reflection group, a meaningful harmonic analysis is constructed. that generalizes the Fourier analysis in the Euclidean space. The classical Fourier analysis on the Euclidean space corresponds to the case $k\equiv0$. In 2012, Salem Ben Sa\v{i}d, Kobayashi, and Orsted defined the two-parameteric $(k, a)$-generalized Fourier transform, acting in spaces with weight $|x|^{a-2}v_k(x)$, $a>0$. The most interesting cases are $a =2$ and $a =1$. For $a =2$ the generalized Fourier transform coincides with the Dunkl transform and it is well studied. In case $a=1$ harmonic analysis, which is important, in particular, in problems of quantum mechanics, has not yet been sufficiently studied. One of the essential elements of harmonic analysis is the bounded translation operator, which allows one to determine the convolution and structural characteristics of functions. For $a=1$, there is a translation operator $\tau^yf(x)$. Its $L^p$-boundedness was recently established by Salem Ben Sa\v{i}d and Deleaval, but only on radial functions and for $1\le p\le 2$. In this paper, a new generalized translation operator $T^tf(x)$ is proposed. It is obtained by integrating of the operator $\tau^yf(x)$ over the unit Euclidean sphere with respect to the variable $y'$, $|y'|=1$, $y=ty'$. We prove that it is positive on functions from the Schwartz space $\mathcal{S}(\mathbb{R}^d)$, for it $T^t1=1$ and it admits a representation with a probability measure. From this we deduce its $L^p$-boundedness for all $1\le p<\infty$ and boundedness on the space $C_b(\mathbb{R}^d)$ of continuous bounded functions.