Suppose that \[ \vec{ \gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t)) = (a_1 t^{ d_1},\dots,a_n t^{ d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{ 1}{ p} = \sum_{ j=1}^n \frac{ 1}{ p_j} \leq 1$, there exists an absolute constant $0 < C = C_{ p_1,\dots,p_n;\vec{ \gamma}} < \infty$ so that \[ \| \sup_{ r > 0} \ \frac{ 1}{ r} \int_{ 0}^r \prod_{ i=1}^n |f_i(x-\gamma_i(t))| \ dt \|_{ L^p(\mathbb{ R})} \leq C \cdot \prod_{ i=1}^n \| f_j \|_{ L^{ p_j}(\mathbb{ R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.
Categories: math.CA