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

RESUMO

Let $1 < \rho < n$ and let Omega be in $L^\rho(S^{(n-1)})$ with vanishing mean. We prove that the maximal truncation $T^*_\Omega$ of the rough singular integral $T_\Omega$ is pointwise dominated by finitely many sparse potentials of the form: $\sum_{Q \in S} l(Q) * ( (1/|Q|) * \int_Q |\nabla f|^p )^{1/p}$, where $1/\rho~ = 1/\rho' + 1/n$ and $\rho~ \leq p < n$. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for $T_\Omega$ to $T^*_\Omega$. Since the argument does not require the boundedness of $T^*_\Omega$ on the target space, it yields two-weight Sobolev inequalities: $\parallel T^*_\Omega f \parallel L^q(u) \leq C * \parallel \nabla f \parallel L^p(v)$, 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.