2 results
Nonlinear fractional Laplacian problems with nonlocal ‘gradient terms’
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 5 / October 2020
- Published online by Cambridge University Press:
- 04 February 2020, pp. 2682-2718
- Print publication:
- October 2020
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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.
$$\mathbb{D}_s^2 (u) = \frac{a_{N,s}}{2} \int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,{\rm d}y.$$
Estimates for Compositions of Maximal Operators with Singular Integrals
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 4 / 01 December 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 801-813
- Print publication:
- 01 December 2013
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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.
