**Abstract**

We study the regularity properties of the solutions to the fractional Laplacian equation with perturbations. The Harnack inequality of a weak solution *u* ∈*W<sup>p</sup> <sup>s</sup> <sup>R</sup><sup>l</sup>* � � to the fractional Laplacian problem is established, and the oscillation of the solution to the fractional Laplacian is estimated. We show that let 1≤*p* < ∞ and *s*∈ð Þ 0, 1 , and let *u*∈ *W<sup>p</sup> <sup>s</sup>* be a weak solution to *Lu* <sup>¼</sup> 0 in <sup>Ω</sup>, with the condition *<sup>u</sup>* <sup>¼</sup> *<sup>f</sup>* in *<sup>R</sup><sup>l</sup>* nΩ, where function *f* belongs to the Sobolev space *W<sup>p</sup> <sup>s</sup> <sup>R</sup><sup>l</sup>* � �. Then, the function *<sup>u</sup>* <sup>∈</sup>*W<sup>p</sup> <sup>s</sup>* is locally Holder continuous and oscillation of the function satisfies the estimation *osc B x*ð Þ 0,*<sup>r</sup> <sup>u</sup>*≤*C<sup>δ</sup> sp p*�1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j*u*j *<sup>p</sup>* ð Þ*B x*ð Þ 0,*<sup>ρ</sup> <sup>p</sup>* q þ þ*Cδ sp <sup>p</sup>*�<sup>1</sup> <sup>j</sup>maxf gj *<sup>u</sup>*, 0 *<sup>p</sup>*�<sup>1</sup> <sup>1</sup> j��*x*0j *<sup>l</sup>*þ*sp* D E <sup>1</sup> *p*�1 *Rl* n*B x*ð Þ 0,*ρ* holds for *<sup>δ</sup>*∈ð Þ 0, 1 and for all *<sup>r</sup>*, *<sup>ρ</sup>* such that 0 <sup>&</sup>lt;*r*<*ρ*. Also, let *<sup>u</sup>*<sup>∈</sup> *<sup>W</sup><sup>p</sup> <sup>s</sup> <sup>R</sup><sup>l</sup>* � � be a weak solution to the boundary problem for the fractional Laplacian ð Þ �<sup>Δ</sup> *<sup>s</sup> <sup>p</sup>* � *b* � ∇ � �*<sup>u</sup>* <sup>¼</sup> 0 in <sup>Ω</sup>, and on the boundary *<sup>u</sup>* <sup>¼</sup> *<sup>f</sup>* in *<sup>R</sup><sup>l</sup>* nΩ, where 1≤ *p*< ∞ and *s*∈ð Þ 0, 1 , and let *u* be nonnegative in a ball function with the center *x*<sup>0</sup> with the radius *ρ*, then the following estimation sup *B x*ð Þ 0,*<sup>r</sup> u*≤*C* inf *B x*ð Þ 0,*<sup>r</sup> <sup>u</sup>* <sup>þ</sup> *<sup>C</sup> <sup>r</sup> ρ* � � *sp p*�1 <sup>j</sup>maxf gj �*u*, 0 *<sup>p</sup>*�<sup>1</sup> <sup>1</sup> j��*x*0j *<sup>l</sup>*þ*sp* D E <sup>1</sup> *p*�1 *Rl* n*B x*ð Þ 0,*ρ* holds for *r*, *ρ* such that 0< *r*<*ρ*.

**Keywords:** Holder continuous, partial differential equation, perturbation method, analysis, calculus, mathematics, function, functional analysis, Harnack inequality, Sobolev space, singular integrals, differentiability class, semigroup, interpolation theorem, fractional Laplacian, partial differential equation, Holder inequality, Laplace operator, general solution, regularity, nonlocal model
