The aim of this paper is to establish a few qualitative uncertainty principles for the windowed Opdam--Cherednik transform on weighted modulation spaces associated with this transform. In particular, we obtain the Cowling--Prices, Hardys and Morgans
uncertainty principles for this transform on weighted modulation spaces. The proofs of the results are based o
In this paper, we introduce the Hausdorff operator associated with the Opdam--Cherednik transform and study the boundedness of this operator in various Lebesgue spaces. In particular, we prove the boundedness of the Hausdorff operator in Lebesgue spa
ces, in grand Lebesgue spaces, and in quasi-Banach spaces that are associated with the Opdam--Cherednik transform. Also, we give necessary and sufficient conditions for the boundedness of the Hausdorff operator in these spaces.
We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type [ |A f|_{mathcal{X}}^2 leq C |f|_{mathcal{X}} big|A^2 fbig|_{mathcal{X}}, quad f in dombig(A^2big), ] and recall that under excee
dingly stronger hypotheses on the operator $A$ and/or the Banach space $mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.