Boundedness of non-local operators with spatially dependent coefficients and L_p-estimates for non-local equations

2021/11/07 by Hongjie Dong, Dong, Hongjie, Pilgyu Jung +3 · 1 citation
Mathematics · #35B65 #35R11 #47B38 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B65 #msc:35R11 #msc:47B38

paper · pdf · doi:10.48550/arxiv.2111.04029

arxiv created 2021/11/07 · arxiv updated 2021/11/09

Abstract

We prove the boundedness of the non-local operator L^a u(x)=∫_ℝ^d (u(x+y)-u(x)-χ_α(y)(∇ u(x),y)) a(x,y)\fracdy|y|^d+α from H_p,w^α(ℝ^d) to L_p,w(ℝ^d) for the whole range of p ∈ (1,∞), where w is a Muckenhoupt weight. The coefficient a(x,y) is bounded, merely measurable in y, and Hölder continuous in x with an arbitrarily small exponent. We extend the previous results by removing the largeness assumption on p as well as considering weighted spaces with Muckenhoupt weights. Using the boundedness result, we prove the unique solvability in L_p spaces of the corresponding parabolic and elliptic non-local equations.

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• A note on the L^p-solvability of a strongly-coupled nonlocal system of equations

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