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The operators $L$ we consider are infinitessimal generators of stable L\\'evy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that $u$ is $C^{2s+\\alpha}$ in $x$ and $C^{1+\\frac{\\alpha}{2s}}$ in $t$, whenever $f$ is $C^{\\alpha}$ in $x$ and $C^{\\frac{\\alpha}{2s}}$ in $t$. 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