On the square function associated with generalized Bochner-Riesz means

We consider generalized Bochner-Riesz multipliers of the form $(1-ρ(ξ))_+^λ$ where $ρ:\mathbb{R}^2\to\mathbb{R}$ belongs to a class of rough distance functions homogeneous with respect to a nonisotropic dilation group. We prove a critical $L^4$ estimate for the associated square function, which we use to derive multiplier theorems for multipliers of the form $m\circρ$ where $m:\mathbb{R}\to\mathbb{C}$.

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