| Abstract: |
| We study the interior $L^{p}$ estimates to the equation $(-\triangle)^{\sigma/2}u=f$ in $B_{1}$ by a geometric method, where $(-\triangle)^{\sigma/2}$ stands for the fractional Laplace operator. Our results generalize the classical $L^{p}$ estimates established by Calder\`{o}n and Zgymund. The analytical tools used in this paper are the Hardy-Liittlewood maximum functions, energy estimates and the Besicovich`s covering lemma. The main difficulty of our works is that the fractional Laplace operator is a non-local elliptic operator. |
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