On the resonant Carleson-Radon transform in all dimensions. The degree one resonant case

In this paper, we provide the resolution of the degree one resonant case in all dimensions. Our main result reads as follows: for any dimension $D\ge 1$ set $\mathbf{X}(\mathbf{t})=(\mathbf{t},\mathrm{\mid }\mathbf{t}{\mathrm{\mid }}^2),\mathbf{t}\in {\mathbb{R}}^D$, and let $K(\mathbf{t})$ be any suitable translation invariant Calderón–Zygmund kernel. If $\mathbb{V}\le {\mathbb{R}}^{D+1}$ is any linear subspace such that $\mathrm{\exists }{\mathbf{v}}_0\in {\mathbb{R}}^D\times \{0\}$ nontrivial with ${\mathbf{v}}_0\perp \mathbb{V}$ then the following (maximal) Carleson-Radon transform $C{R}_{\mathbb{V}}^{\ast }$ is $L^p({\mathbb{R}}^{D+1})-$bounded in the maximal range $1<p<\mathrm{\infty }$, where $$C{R}_{\mathbb{V}}^{\ast }f(\mathbf{x}):=\underset{\begin{array}{c}0<r<R<\mathrm{\infty }\\ \mathbf{a}\in \mathbb{V}\end{array}}{\sup }\mid {\int }_{r<\mathrm{\mid }\mathbf{t}\mathrm{\mid }\le R}f(\mathbf{x}-\mathbf{X}(\mathbf{t}))e(\mathbf{a}\cdot \mathbf{X}(\mathbf{t}))K(\mathbf{t})d\mathbf{t}\mid \mathrm{.}$$ The above choice for $\mathbb{V}$ creates a maximal linear subspace of ${\mathbb{R}}^{D+1}$ closed under parabolic scaling for which - $C{R}_{\mathbb{V}}^{\ast }$ is degree one resonant, and - $C{R}_{\mathbb{V}}^{\ast }$ is not degree two (or higher) resonant. The proof of the above result unravels several new manifestations and ideas meant to capture the remarkable features of the resonant Carleson-Radon behavior.