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Let
$\Omega \subset \mathbb{R}^{N} $, N ≽ 2, be a smooth bounded domain. For s ∈ (1/2, 1), we consider a problem of the form
where λ > 0 is a real parameter, f belongs to a suitable Lebesgue space, 
$$\left\{\begin{array}{@{}ll} (-\Delta)^s u = \mu(x)\, \mathbb{D}_s^{2}(u) + \lambda f(x), & {\rm in}\,\Omega, \\ u= 0, & {\rm in}\,\mathbb{R}^{N} \setminus \Omega,\end{array}\right.$$
$\mu \in L^{\infty}$ and 
$\mathbb {D}_s^2$ is a nonlocal ‘gradient square’ term given by
Depending on the real parameter λ > 0, we derive existence and non-existence results. The proof of our existence result relies on sharp Calderón–Zygmund type regularity results for the fractional Poisson equation with low integrability data. We also obtain existence results for related problems involving different nonlocal diffusion terms.
$$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$
We prove weak-type 
$\left( 1,\,1 \right)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator 
$\Delta *\Psi $ where 
$\Delta *$ is Bourgain’s maximal multiplier operator and 
$\Psi $ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the 
${{L}^{q}}$ operator norm when 
$1\,<\,q\,<\,2$. We also consider associated variation-norm estimates.
