New pointwise bounds by Riesz potential type operators

TL;DR

This work addresses sharp pointwise control of rough integral operators $T_{\Omega,\alpha}$ for $0<\alpha<n$ by sparse potential operators that encode the size of the kernel $\Omega$. The authors establish three regimes based on $\Omega$'s size: a critical $L^{n,\infty}$ case giving a bound by $I_{\alpha}( |\nabla f| )$, a subcritical $L^{r,r^*}$ case leading to sparse bounds by $I_{\alpha,L^s}^{\mathscr S}$, and an endpoint $L(\log L)^{1/n'}$ case with a similar sparse bound for $0<\alpha<1$. These pointwise bounds yield Sobolev-type mappings $T_{\Omega,\alpha}:\dot W^{1,p} \to L^{p^*}$ for $1<p<n$, with endpoint weak-type estimates, and they extend to robust weighted estimates via sparse operator domination. The results unify the treatment of rough singular, hypersingular, and rough fractional operators, and impose a framework for weighted and endpoint Sobolev estimates, providing new tools for PDEs with rough kernels. Overall, the paper advances the understanding of Sobolev regularity for rough operators and offers a versatile approach through sparse domination and dyadic analysis.

Abstract

We investigate new pointwise bounds for a class of rough integral operators, $T_{Ω,α}$, for a parameter $0<α<n$ that includes classical rough singular integrals of Calderón and Zygmund, rough hypersingular integrals, and rough fractional integral operators. We prove that the rough integral operators are bounded by a sparse potential operator that depends on the size of the symbol $Ω$. As a result of our pointwise inequalities, we obtain several new Sobolev mappings of the form $T_{Ω,α}:\dot W^{1,p}\rightarrow L^q$

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1