Sparse pointwise bounds for maximal truncations of rough singular integrals and Sobolev-type inequalities

Abstract

Let $1<\rho <n$ and let Omega be in $L^{\rho }(S^{(n-1)})$ with vanishing mean. We prove that the maximal truncation $T_{\Omega }^{\ast }$ of the rough singular integral $T_{\Omega }$ is pointwise dominated by finitely many sparse potentials of the form: ${\sum }_{Q\in S}l(Q)\ast ((1/|Q|)\ast {\int }_Q|\nabla f{|}^p{)}^{1/p}$, where $1/\rho =1/{\rho }^{\prime }+1/n$ and $\rho \le p<n$. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for $T_{\Omega }$ to $T_{\Omega }^{\ast }$. Since the argument does not require the boundedness of $T_{\Omega }^{\ast }$ on the target space, it yields two-weight Sobolev inequalities: $\parallel T_{\Omega }^{\ast }f\parallel L^q(u)\le C\ast \parallel \nabla f\parallel L^p(v)$, under joint two-weight conditions, while the target weight u itself is only required to belong to $A_{\infty }$. Additionally, we prove a Hedberg-type estimate involving a Morrey norm of the gradient and apply it to weighted grand Lebesgue spaces. Further consequences are obtained in weighted Lebesgue, Orlicz, and variable Lebesgue spaces.