Boas conjecture on the axis for the Fourier--Dunkl transform and~its generalization

D. V. Gorbachev

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 39-51.

Résumé

The question of integrability of the Fourier transform and other integral transformations $\mathcal{F}(f)$ on classes of functions in weighted spaces $L^{p}(\mathbb{R}^{d})$ is a fundamental problem of harmonic analysis. The classical Hausdorff–Young result says that if a function $f$ from $L^{p}(\mathbb{R}^{d})$ with $p\in [1,2]$, then its Fourier transform $\mathcal{F}(f)\in L^{p'}(\mathbb{R}^{d})$. For $p>2$ the Fourier transform in the general situation will be a generalized function. The Fourier transform can be defined as an usual function for $p>2$ by considering the weighted spaces $L^{p}(\mathbb{R}^{d})$. In particular, the classical Pitt inequality implies that if $p,q\in (1,\infty)$, $\delta=d(\frac{1}{q}-\frac{1}{p'})$, $\gamma\in [(\delta)_{+},\frac{d}{q})$ and function $f$ is integrable in $L^{p}(\mathbb{R}^{d})$ with power weight $|x|^{p(\gamma-\delta)}$, then its Fourier transform $\mathcal{F} (f)$ belongs to the space $L^{q}(\mathbb{R}^{d})$ with weight $|x|^{-q\gamma}$. The case $p=q$ corresponds to the well-known Hardy–Littlewood inequality. The question arises of extending the conditions for the integrability of the Fourier transform under additional conditions on the functions. In the one-dimensional case, G. Hardy and J. Littlewood proved that if $f$ is an even nonincreasing function tending to zero and $f\in L^{p}(\mathbb{R})$ for $p\in (1,\infty)$, then $\mathcal{F}(f)$ belongs to $L^{p}(\mathbb{R})$ with weight $|x|^{p-2}$. R. Boas (1972) suggested that for a monotone function $f$ the membership $|{ \cdot }|^{\gamma-\delta}f\in L^{p}(\mathbb{R})$ is equivalent to $|{ \cdot }|^{-\gamma}\mathcal{F}(f)\in L^{p}(\mathbb{R})$ if and only if $ \gamma\in (-\frac{1}{p'},\frac{1}{p})$. The one-dimensional Boas conjecture was proved by Y. Sagher (1976). D. Gorbachev, E. Liflyand and S. Tikhonov (2011) proved the multidimensional Boas conjecture for radial functions, moreover, on a wider class of general monotone non-negative radial functions $f$: $\||{ \cdot }|^{-\gamma}\mathcal{F}(f)\|_{p}\asymp \||{ \cdot }|^{\gamma-\delta}f\|_{p }$ if and only if $\gamma\in (\frac{d}{p}-\frac{d+1}{2},\frac{d}{p})$, where $\delta= d(\frac{1}{p}-\frac{1}{p'})$. For radial functions, the Fourier transform is expressed in terms of the Bessel transform of half-integer order, which reduces to the classical Hankel transform and includes the cosine and sine Fourier transforms. For the latter, the Boas conjecture was proved by E. Liflyand and S. Tikhonov (2008). For the Bessel–Hankel transform with an arbitrary order, the Boas conjecture was proved by L. De Carli, D. Gorbachev and S. Tikhonov (2013). D. Gorbachev, V. Ivanov and S. Tikhonov (2016) generalized these results to the case of $(\kappa,a)$-generalized Fourier transform. A. Debernardi (2019) studied the case of the Hankel transform and general monotone alternating functions. So far, the Boas conjecture has been considered for functions on the semiaxis. In this paper, it is studied on the entire axis. To do this, we consider the integral Dunkl transform, which for even functions reduces to the Bessel–Hankel transform. It is also shown that the Boas conjecture remains valid for the $(\kappa,a)$-generalized Fourier transform, which gives the Dunkl transform for $a=2$. As a result, we have $$ \||{ \cdot }|^{-\gamma}\mathcal{F}_{\kappa,a}(f)\|_{p,\kappa,a}\asymp \||{ \cdot }|^{\gamma-\delta}f\|_{p,\kappa,a}, $$ where $\gamma\in (\frac{d_{\kappa,a}}{p}-\frac{d_{\kappa,a}+\frac{a}{2}}{2},\frac{d_{\kappa,a} }{p})$, $\delta=d_{\kappa,a}(\frac{1}{p}-\frac{1}{p'})$, $d_{\kappa,a}=2\kappa+a-1$.

Détail

Keywords: Fourier inequality, Boas conjecture, Hardy inequality, Bellman inequality, Dunkl transform, generalized Fourier transform.

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