Methods of the Perturbation Theory for Fundamental Solutions to the Generalization of the Fractional Laplaciane

Abstract

We study the regularity properties of the solutions to the fractional Laplacian equation with perturbations. The Harnack inequality of a weak solution u∈WspRl 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∈01, and let u∈Wsp be a weak solution to Lu=0inΩ, with the condition u=finRl\\Ω, where function f belongs to the Sobolev space WspRl. Then, the function u∈Wsp is locally Holder continuous and oscillation of the function satisfies the estimation oscBx0ru≤Cδspp−1upBx0ρp+ +Cδspp−1maxu0p−11⋅−x0l+spRl\\Bx0ρ1p−1 holds for δ∈01 and for all r,ρ such that 0<r<ρ. Also, let u∈WspRl be a weak solution to the boundary problem for the fractional Laplacian −Δps−b⋅∇u=0inΩ, and on the boundary u=finRl\\Ω, where 1≤p<∞ and s∈01, and let u be nonnegative in a ball function with the center x0 with the radius ρ, then the following estimation supBx0ru≤CinfBx0ru+Crρspp−1max−u0p−11⋅−x0l+spRl\\Bx0ρ1p−1 holds for r,ρ such that 0<r<ρ.

1. Introduction

The fractional Laplacian is an integrodifferential operator, which can be defined by a formula

$$\mathit{Lu}\left(x\right)=c_{l,s}⟨\frac{u\left(x\right)-u\left(\cdot \right)}{{\left|x-\cdot \right|}^{l+2s}}⟩,$$E1

where $c_{l,s}=\frac{4^s\mathrm{\Gamma }\left(s+\frac{l}{2}\right)}{{\pi }^{\frac{l}{2}}\left|\mathrm{\Gamma }\left(-s\right)\right|}$ is a constant dependent on the dimension of the space $l>2.$ [1, 2, 3].

Let $\mathrm{\Omega }$ be a bounded domain, we consider the integrodifferential problem

$$\begin{array}{l}\mathit{Lu}=0\text{in}\mathrm{\Omega },\\ u=f\text{in}{R}^l\backslash \mathrm{\Omega },\end{array}$$E2

where function $f$ belongs to a certain functional class, for instance, to the Sobolev space $W_s^p\left(R^l\right)$.

Similarly, we can consider the problem

$$\begin{array}{l}{\left(-\mathrm{\Delta }\right)}_s^{p}u=0\text{in}\mathrm{\Omega },\\ u=f\text{in}{R}^l\backslash \mathrm{\Omega },\end{array}$$E3

where the ${\left(-\mathrm{\Delta }\right)}_s^p$ sits for the fractional p-Laplace operator. The weak solutions to these problems coincide with the class of the minimizers to the functionals

$$\mathit{Fu}={\tilde{c}}_{l,s}{⟨{⟨\frac{{\left|u\left(x\right)-u\left(y\right)\right|}^p}{{\left|x-y\right|}^{l+\mathit{sp}}}⟩}_y⟩}_x,$$E4

which is defined over suitable Sobolev space.

Let us assume $u\in W_s^p\left(R^l\right)$ is a positive weak solution to (3) then inequality

$${sup}_{K}u\left(t-\tau ,\cdot \right)\le C{inf}_{K}u\left(t,\cdot \right)$$E5

holds for any compact set $K\subset \mathrm{\Omega }$ and a positive constant $C$ depends only on $K,\tau ,t,s,p$.

The fractional Laplace operator is intrinsically connected with the fractional Sobolev spaces $W_s^p\left(R^l\right)$ or, more specifically, $W_s^p\left(R^l\right)$ can be defined by using the fundamental solution of the fractional Laplace operator. So, for any $s\in \left(0,1\right)$ and any $p\in \left[1,\infty \right)$, the set $W_s^p$ of all functions $u$ such that

$$W_s^p=\left\{u\in L^p\left(R^l\right):\frac{\left|u\left(x\right)-u\left(y\right)\right|}{{\left|x-y\right|}^{\frac{l}{p}+s}}\in L^p\left(R^l\times R^l\right)\right\}$$E6

is called the fractional Sobolev space, which can be equipped with its natural norm

$${‖u‖}_{W_s^p\left(R^l\right)}={\left(⟨{\left|u\right|}^p⟩+⟨⟨\frac{{\left|u\left(x\right)-u\left(y\right)\right|}^p}{{\left|x-y\right|}^{l+\mathit{sp}}}⟩⟩\right)}^{\frac{1}{p}}.$$E7

Employing the perturbation theory, we can consider the fractional Laplace operator ${\left(-\mathrm{\Delta }\right)}_s^p$ with the perturbation $b$, in the form

$$\mathrm{\Lambda }\equiv {\left(-\mathrm{\Delta }\right)}_s^p+b\cdot \nabla ,$$E8

where $b\left(x\right)=c{\left|x\right|}^{-2s}x$. Since the vector $b$ at $x=0$ and at $x=\infty$ indicates a stronger singular attitude than permitted by the definition of the Kato classes, this vector does not belong to any Kato class, and the standard upper estimation of its heat kernel $e^{-t\mathrm{\Lambda }}\left(x,y\right)$ via the heat kernel of the correspondent Laplacian operator is not valid in this case. However, the weighted estimations are still holding.

The integral inequality

$$⟨\frac{{\left|\nabla u\right|}^p}{{\left|x\right|}^p}⟩\le \mathit{\text{const}}\left(l,p\right)⟨{\left|\mathrm{\Delta }u\right|}^p⟩$$E9

holds for any $u\in C_0^{\infty }\left(R^l\right),l>2$, $p\ge 2$, and the constant in the right part depends only on the dimension of the Euclidian space and on the convexity of the functional space. This estimation can be generalized to abstract spaces with convex norms. If in (9) we take $p=2$, then (9) becomes a well-established Hardy-Relich inequality with a sharp constant in the form

$$⟨\frac{{\left|\nabla u\right|}^2}{{\left|x\right|}^2}⟩\le c\left(l,p\right)⟨{\left|\mathrm{\Delta }u\right|}^2⟩,$$E10

which can be proven by the methods developed in [4, 5].

Applying arguments of the perturbation theory, the operator $\mathrm{\Lambda }\equiv {\left(-\mathrm{\Delta }\right)}_s^p+b\cdot \nabla$ can be considered a perturbation of the fractional Laplace operator ${\left(-\mathrm{\Delta }\right)}_s^p$ and utilizing the Duhamel formula the upper and lower bounds can be easily proven. There is a limiting constant $\tilde{k}>0$ such that the contraction semigroup $exp\left(-t\mathrm{\Lambda }\right)\left(x,y\right)$ exists.

Let us consider the following example:

$$\frac{\partial }{\partial t}u=\sum _{k,j=1,\dots ,l}{\nabla }_k{a}_{\mathit{kj}}\left(x\right){\nabla }_{j}u-\sum _{k=1,\dots ,l}{b}_k\left(t,x\right){\nabla }_{k}u$$

under the integral condition of perturbation

$$\begin{array}{l}{\int }_{R_{+}}⟨b\left(t,\cdot \right)\cdot a^{-1}\left(t,\cdot \right)\cdot b\left(t,\cdot \right){\left|\phi \left(t,\cdot \right)\right|}^2⟩\mathit{dt}\le \\ \le C{\int }_{R_{+}}⟨a\left(t,\cdot \right)\cdot \nabla \phi \left(t,\cdot \right),\nabla \phi \left(t,\cdot \right)⟩\mathit{dt}+M{\int }_{R_{+}}⟨\phi \left(t,\cdot \right),\phi \left(t,\cdot \right)⟩\mathit{dt},\end{array}$$

where $C<4$ and $M<\infty$. So, $b$ can be a function such that

$$\sum _{k=1,\dots ,l}{b}_k^2\left(t,x\right)\le {\nu }^{2}C{\left(\frac{l-2}{2}\right)}^2\frac{1}{{\left|x\right|}^2}+M\frac{1}{\left|t\right|}{\left(ln\left(e+\frac{1}{\left|t\right|}\right)\right)}^{-\frac{3}{2}}.$$

If the matrix $a_{\mathit{ij}}$ is diagonal, then the differential operator is $-\mathrm{\Delta }+b\cdot \nabla$, and $b=\frac{l-2}{2}\sqrt{\beta }\frac{x}{{\left|x\right|}^2}$ and for some $0<\beta <4$.

Let us consider the elliptic equation

$$a\circ d^{2}u\equiv \sum _{i,j=1}^l{a}_{\mathit{ij}}\frac{\partial }{\partial x_i}\frac{\partial }{\partial x_j}u=0,$$

where the matrix $a$ is $a_{\mathit{ij}}={\delta }_{\mathit{ij}}+b\frac{x_i{x}_j}{{\left|x\right|}^2},$ $b=-1+\frac{l-1}{1-\chi },\chi <1,l\ge 3$. We calculate matrices

$$\nabla a=b\left(l-1\right)\frac{x}{{\left|x\right|}^2},{\left(a_{\mathit{ij}}\right)}^{-1}={\delta }_{\mathit{ij}}-\frac{b}{b+1}\frac{x_i{x}_j}{{\left|x\right|}^2},$$

and we have for the multiplication of the gradients of the matrices

$$\nabla a\circ a^{-1}\circ \nabla a={\left(1+b\right)}^{-1}{\left(\frac{l-1}{\mid x\mid }\right)}^2$$

then we have

$$⟨\nabla \phi \circ a\circ \nabla \phi ⟩\ge \left(1+b\right)\frac{l-2}{2}{‖\frac{\phi }{\left|x\right|}‖}_2^2\forall \phi \in W_1^2\left(R^l\right),l\ge 3,$$

so, if $\beta =4{\left(1+\frac{\chi }{l-2}\right)}^2$ then $\nabla a\circ a^{-1}\circ \nabla a\in {\mathit{PK}}_{\beta }\left(A\right)$ with the constant $c\left(\beta \right)=0$, for $\beta <4$ it is necessary $\chi \in \left(-2\left(l-2\right),0\right).$.

Let us assume that $u\left(\left|x\right|=1\right)=1$. As the solutions, we can consider two functions: the first is $u\equiv 1$ - tautological constant and the second is $u={\left|x\right|}^{\chi }$. If parameter $\chi =-\frac{l-2}{s}$ then $\beta =4{\left(1+\frac{\chi }{l-2}\right)}^2$ and $\beta \le 4$ for $p>s$ in the ball $K_1\left(0\right)$ function $u={\left|x\right|}^{\chi }\in L^p\left(K_1\left(0\right)\right)$ on another hand must hold the following estimation

$${‖exp\left(-t{\mathrm{\Lambda }}_p\right)‖}_{p\to s}\le Cexp\left(\frac{c\left(\beta \right)t}{\sqrt{\beta }}\right)t^{\frac{-\left(s-p\right)l}{2\mathit{ps}}},\frac{2}{2-\sqrt{\beta }}<p<s\le \infty ,$$

where semigroup $exp\left(-t{\mathrm{\Lambda }}_p\right)$ is generated by a linear operator ${\mathrm{\Lambda }}_p=A+b\left(l-1\right)\frac{x}{{\left|x\right|}^2}•\nabla$. That means ${\left|x\right|}^{\chi }\in L^{\frac{\mathit{pl}}{l-2}}\left(K_1\left(0\right)\right)$ but it is impossible because ${\left|x\right|}^{\chi }\notin L_{\mathit{loc}}^{\frac{\mathit{pl}}{l-2}}{K}_1\left(0\right)$ so the function ${\left|x\right|}^{\chi }$ cannot be a solution and there is only one trivial solution. If $\beta >4$, then the equation $a\circ d^{2}u=0$ always has two bounded solutions. Parallel with this equation, we can consider a Cauchy problem for a parabolic equation with the same differential operator. Let us assume that the linear operator $-{\mathrm{\Lambda }}_p\supset \nabla a\nabla -b\nabla$ defines over $D\left(A_p\right)$ generates holomorph semigroup in $L^p\left(R^l,d^{l}x\right)$-space. Let $b\circ a^{-1}\circ b\in {\mathit{PK}}_{\beta }\left(A\right)$, we denote $b_n={\chi }_{n}b$, where ${\chi }_n$ is an indicator of $\left\{x\in R^l:\left(b\circ a^{-1}\circ b\right)\left(x\right)\le n\right\}$ and ${lim}_{n\to \infty }exp\left(-t{\mathrm{\Lambda }}_p\left(b_n\right)\right)=exp\left(-t{\mathrm{\Lambda }}_p\left(b\right)\right)$ uniformly at $t\in \left[0,1\right]$. If $\beta <1,p\in \left[\frac{2}{2-\sqrt{\beta }},\infty \right[$ then there is $C_0$ - contraction semigroup, which is generated by the operator $A+b\nabla$ and the estimates

$${‖exp\left(-t{\mathrm{\Lambda }}_p\right)‖}_{p\to p}\le exp\left(\frac{c\left(\beta \right)t}{p-1}\right),$$

then we estimate the operator norm by the exponential function

$${‖exp\left(-t{\mathrm{\Lambda }}_p\right)‖}_{p\to s}\le Cexp\left(\frac{c\left(\beta \right)t}{\sqrt{\beta }}\right)t^{\frac{-\left(s-p\right)l}{2\mathit{ps}}},\frac{2}{2-\sqrt{\beta }}<p<s\le \infty$$

hold for $1\le \beta <4,p<s\in \left[\frac{2}{2-\sqrt{\beta }},\infty \right[$, operator sum $A+b\nabla$ cannot be defined correctly, however, semigroup exists and can be defined as a limit $exp\left(-t{\mathrm{\Lambda }}_p\left(b\right)\right)\equiv {lim}_{n\to \infty }exp\left(-t{\mathrm{\Lambda }}_p\left(b_n\right)\right),t\ge 0$ in this case it is a definition of the semigroup [6].

Let us remark that for any smooth enough function $f\left(x\right),x\in R^l$, we can write the equations

$$\begin{array}{l}{\left(-\mathrm{\Delta }\right)}_{\frac{1-b}{2}}f\left(x\right)=C⟨\frac{f\left(x\right)-f\left(\cdot \right)}{{\left|x-\cdot \right|}^{l+1-b}}⟩=\\ =C{lim}_{t\to 0}⟨\frac{f\left(x\right)-f\left(\cdot \right)}{{\left({\left|x-\cdot \right|}^2+{\left(1-b\right)}^2{\left|t\right|}^{\frac{2}{1-b}}\right)}^{\frac{l+1-b}{2}}}⟩=\\ ={lim}_{t\to 0}\frac{1}{t}⟨\widehat{K}\left(t,\cdot -x\right)\left(f\left(x\right)-f\left(\cdot \right)\right)⟩\\ ={lim}_{t\to 0}\frac{u\left(t,x\right)-u\left(0,x\right)}{t}\equiv \frac{\partial }{\partial t}u\left(0,x\right)\equiv u_t\left(0,x\right),\end{array}$$E11

where the function $\widehat{K}\left(t,x\right)=C\left(l,b\right)\frac{t^{1-b}}{{\left({\left|t\right|}^2+{\left|x\right|}^2\right)}^{\frac{l+1-b}{2}}}$ is the fundamental solution to the associated extension problem. More precisely, the results, which concern the fractional Laplace problems, can be applied to the extension problem in the following form. A function $u:\left[0,\infty \right)\times R^l\to R$ is a solution to the initial problem

$$\begin{array}{l}\mathit{Div}\left(t^b\nabla u\right)=0\text{in}{R}^l,\\ u\left(t,0\right)=f\left(x\right),x\in R^l,\end{array}$$E12

or in expanded form

$$\begin{array}{l}{\left(-\mathrm{\Delta }\right)}_{s}u+\frac{b}{t}{u}_t+u_{\mathit{tt}}=0\text{in}{R}^l,\\ u\left(t,0\right)=f\left(x\right),x\in R^l.\end{array}$$E13

Below, we are going to construct the semigroup of contraction, which generator coincides with the realization $\mathrm{\Lambda }$ of the operator ${\left(-\mathrm{\Delta }\right)}^s-b\cdot \nabla$ in $L^p\left(R^l\right),1\le p<\infty ,l>2$, where the vector $b\left(x\right)=c{\left|x\right|}^{-2s}x$ is singular; and prove the Harnack inequality for a weak solution to the boundary problem $\left({\left(-\mathrm{\Delta }\right)}_p^s-b\cdot \nabla \right)u=0$, outside boundary $u=f$, and presuming $u\in W_s^p\left(R^l\right)$ is nonnegative in a ball with the center in point $x_0$ with a radius $\rho$ for all $r,\rho$ such that $0<r<\rho$.

Let us denote

$${\left({\left|u\right|}^p\right)}_{B\left(x_0,\rho \right)}=\frac{1}{\mathit{mes}\left(B\left(x_0,\rho \right)\right)}{\int }_{B\left(x_0,\rho \right)}{\left|u\left(y\right)\right|}^p\mathit{dy}$$

then we can formulate the next theorem.

Theorem 1. Let $1\le p<\infty$ and $s\in \left(0,1\right)$, and let $u\in W_s^p$ be a weak solution to (2), where its fundamental solution satisfies condition (15).

Then, the function $u\in W_s^p$ is locally Holder continuous and oscillation of the function satisfies the estimation

$$\begin{array}{c}\underset{B\left(x_0,r\right)}{\mathit{osc}}u\le C{\delta }^{\frac{\mathit{sp}}{p-1}}\sqrt[p]{{\left({\left|u\right|}^p\right)}_{B\left(x_0,\rho \right)}}+\\ +C{\delta }^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\rho \right)}^{\frac{1}{p-1}}\end{array}$$

holds for $\delta \in \left(0,1\right)$ and for all $r,\rho$ such that $0<r<\rho$.

There exist extensive literature dedicated to the partial differential fractional Laplacian operator, general questions can be found in [4, 7, 8, 9], a wide review is presented in [4], an interesting approach to nonlinear heat equations in modulation spaces and Navier-Stokes equations can be found in [10]; some aspects of weights inequalities are described in [2, 11]; fractional Laplacian is considered in [2, 12], the list of selected works consists of 29 works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In the recent works [1, 2], authors proved sharp two-sided estimations on the heat kernel of the fractional Laplacian with the perturbation of drift having critical-order singularity, also authors show that the operator with the heat kernel of the fractional Laplacian can be expressed as a Feller generator so that the probability measures uniquely determined by the Feller semigroup admits description as weak solutions to the corresponding SDE.

2. The semigroup generated by $\mathrm{\Lambda }\equiv {\left(-\mathrm{\Delta }\right)}^s-b\cdot \nabla$, $b\left(x\right)=c{\left|x\right|}^{-2s}x$ in $L^p,1\le p<\infty$

Let us introduce the following mollifiers:

$$b_{\epsilon }\left(x\right)=c{\left(\left|x\right|+\epsilon \right)}^{-2s}x$$

and

$${\mathrm{\Lambda }}_{\epsilon }\equiv {\left(-\mathrm{\Delta }\right)}^s-b_{\epsilon }\cdot \nabla ,$$

with the domain defined as

$$D\left({\mathrm{\Lambda }}_{\epsilon }\right)\equiv {\left(1+{\left(-\mathrm{\Delta }\right)}^s\right)}^{-1}{L}^p$$

for small positive numbers $\epsilon >0$.

Since the following inequality

$$\left|\nabla \left(\varsigma +{\left(-\mathrm{\Delta }\right)}^s\right)\right|\le M\left(l,s\right){\left(\varsigma +{\left(-\mathrm{\Delta }\right)}^s\right)}^{-\frac{2s-1}{2s}}$$

holds for all complex numbers such that

$$\mathit{Re\varsigma }>0,$$

the operator norm ${‖b_{\epsilon }\cdot \nabla {\left(\varsigma +{\left(-\mathrm{\Delta }\right)}^s\right)}^{-1}‖}_{L^p\to L^p}$ is bounded by

$$M\left({‖b_{\epsilon }‖}_{\infty }\right){\left(\mathit{Re\varsigma }\right)}^{-\frac{2s-1}{2s}}.$$

So, the Liouville-Neumann series for resolvents converge in $L^p$ and the inequality

$${‖{\left(\varsigma +{\left(-\mathrm{\Delta }\right)}^s-b_{\epsilon }\cdot \nabla \right)}^{-1}‖}_{L^p\to L^p}\le M\left(\epsilon \right){\left|\varsigma \right|}^{-1}$$

holds for $\mathit{Re\varsigma }>c\left(\epsilon \right)$. Thus, according to the Hille-Kato perturbation theorem, the operator $-{\mathrm{\Lambda }}_{\epsilon }$, $\epsilon >0$ generates a holomorphic semigroup.

The next step is to show that there is the strong $L^p$ limit

$$\underset{\epsilon \downarrow 0}{lim}{e}^{-t{\mathrm{\Lambda }}_{\epsilon }}{=}^{\mathit{def}}{e}^{-t\mathrm{\Lambda }},L^p,1\le p<\infty$$

which defines a contraction continuous semigroup in $L^p$.

First, let us establish that a discretization of the set ${\left\{e^{-t{\mathrm{\Lambda }}_{\epsilon }}\right\}}_{\epsilon }$ satisfies the Cauchy condition for at least one Lebesgue space. Let us compose the integral identity

$$\begin{array}{l}⟨{\left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right)}^2⟩+{\int }_0^T⟨{\left({\left(-\mathrm{\Delta }\right)}^{\frac{s}{2}}\left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right)\right)}^2⟩\mathit{dt}-\\ \mathit{Re}{\int }_0^T⟨b_{\epsilon \left(n\right)}\nabla \left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right),e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}⟩\mathit{dt}-\\ \mathit{Re}{\int }_0^T⟨\left(b_{\epsilon \left(n\right)}-b_{\epsilon \left(m\right)}\right)\nabla \left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right),e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}⟩\mathit{dt}=0,\end{array}$$

so that we obtain

$$\begin{array}{l}⟨{\left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right)}^2⟩+{\int }_0^T⟨{\left({\left(-\mathrm{\Delta }\right)}^{\frac{s}{2}}\left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right)\right)}^2⟩\mathit{dt}+\\ c\frac{l-2s}{2}{\int }_0^T⟨{\left(\left|x\right|+\epsilon \right)}^{-2s},{\left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right)}^2⟩\mathit{dt}\le \\ {\int }_0^T\left|⟨\left(b_{\epsilon \left(n\right)}-b_{\epsilon \left(m\right)}\right)\nabla \left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right),e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}⟩\right|\mathit{dt}.\end{array}$$

Since

$${\int }_0^T\left|⟨\left(b_{\epsilon \left(n\right)}-b_{\epsilon \left(m\right)}\right)\nabla \left(e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}\right),e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}⟩\right|\mathit{dt}\stackrel{m,n\to \infty }{\to }0$$

we have the following limit

$${‖e^{-T{\mathrm{\Lambda }}_{\epsilon \left(n\right)}}-e^{-T{\mathrm{\Lambda }}_{\epsilon \left(m\right)}}‖}_2^2\stackrel{m,n\to \infty }{\to }0$$

uniformly at $T\in \left[0,1\right]$.

Thus, the discretization of set ${\left\{e^{-t{\mathrm{\Lambda }}_{\epsilon }}\right\}}_{\epsilon }$ is a Cauchy sequence in $L^{\infty }\left(\left[0,1\right],L^2\right)$, and applying the contraction of $e^{-t{\mathrm{\Lambda }}_{\epsilon }}$, we estimate

$${‖e^{-t\mathrm{\Lambda }}u‖}_2\le {‖u‖}_2,$$

besides, we have the equality

$$\underset{\epsilon \downarrow 0}{lim}{‖e^{-t{\mathrm{\Lambda }}_{\epsilon }}u-e^{-t\mathrm{\Lambda }}u‖}_2=0,u\in L^2,t\in \left[0,1\right]$$

and applying group property, we have a continuous semigroup of contraction for $t\ge 0.$ So, the continuous semigroup of the contraction is constructed in $L^2$.

Lemma 1. Let the set of functions $\left\{f_a\left(x\right)\right\}$ converges in measure to function $f$ then following estimation

$${⟨f⟩}_E\le {sup}_a{⟨f_a⟩}_E$$

holds.

Now, let us take $p\in \left[1,\infty \right)$, then, from the lemma follows estimation

$${‖e^{-t\mathrm{\Lambda }}u‖}_p\le {‖u‖}_p,t\ge 0$$

for any $u\in C_0^{\infty }$. Next, by continuity we extend the semigroup from $C_0^{\infty }$ over $L^p$ so the contraction semigroup can be defined as $L^p$- closure of $e^{-t\mathrm{\Lambda }}$, thus, there is a $L^p$- strong limit

$$e^{-t\mathrm{\Lambda }}=\mathit{\text{strong}}-L^p-{lim}_{\epsilon \downarrow 0}{e}^{-t{\mathrm{\Lambda }}_{\epsilon }},t\ge 0,$$

which defined continuous semigroup of the contraction in $L^p$.

Thus, the contraction semigroup $e^{-t\mathrm{\Lambda }},t\ge 0$ in $L^p$ can be defined as a strong limit of holomorphic semigroups $e^{-t{\mathrm{\Lambda }}_{\epsilon }}$. Under accepted assumptions, for all $1\le p\le q$, the semigroup $e^{-t\mathrm{\Lambda }}$ satisfies natural conditions on its growth

$${‖e^{-t\mathrm{\Lambda }}‖}_{p\to q}\le C\left(l\right)t^{-\frac{l}{2}\left(\frac{1}{p}-\frac{1}{q}\right)},t\ge 0.$$

This estimation can be deduced from the next inequality

$$\begin{array}{l}\frac{1}{p}⟨{\left(\nabla e^{-T{\mathrm{\Lambda }}_{\epsilon }}u\right)}^p⟩+\\ +\frac{4\left(p-1\right)}{p^2}{\int }_0^T\sum _i⟨{\left({\left(-\mathrm{\Delta }\right)}^{\frac{s}{2}}\left({\nabla }_i{e}^{-T{\mathrm{\Lambda }}_{\epsilon }}u{\left|\nabla e^{-T{\mathrm{\Lambda }}_{\epsilon }}u\right|}^{\frac{p-2}{2}}\right)\right)}^2⟩\mathit{dt}+\\ +c\frac{l-2s-p}{p}{\int }_0^T⟨{\left(\left|x\right|+\epsilon \right)}^{-2s}{\left|\nabla e^{-T{\mathrm{\Lambda }}_{\epsilon }}u\right|}^p⟩\mathit{dt}+\\ +2\mathit{sc}{\int }_0^T⟨{\left(\left|x\right|+\epsilon \right)}^{2s-2}{\left|x\nabla e^{-T{\mathrm{\Lambda }}_{\epsilon }}u\right|}^2{\left|\nabla e^{-T{\mathrm{\Lambda }}_{\epsilon }}u\right|}^{p-2}⟩\mathit{dt}\le \\ \le \frac{1}{p}⟨{\left|\nabla u\right|}^p⟩\end{array}$$

that holds for all $T>0$.

3. The Harnack inequality

For any measurable set $E\subset R^l$ and any integrable function $f$, we denote the mean value

$${\left(f\right)}_E=\frac{1}{\mathit{mes}\left(E\right)}{\int }_{E}f\left(y\right)\mathit{dy}$$

where $\mathit{mes}\left(E\right)$ is the Lebesgue measure of the set $E\subset R^l$.

Let us assume that $\mathrm{\Omega }\subset R^l$ is a bounded open set. We are going to consider the Harnack inequality for a weak solution to the following differential problem with the bounded condition

$$\begin{array}{l}\mathrm{\Lambda }u=0\text{in}\mathrm{\Omega },\\ u=f\text{in}{R}^l\backslash \mathrm{\Omega },\end{array}$$E14

where $\mathrm{\Lambda }\equiv {\left(-\mathrm{\Delta }\right)}^s-b\cdot \nabla$ acts on $L^p,1\le p<\infty$ under the condition that its kernel satisfies the following inequality

$$\frac{\mu }{{\left|x-y\right|}^{l+\mathit{sp}}}\le K\left(x,y\right)\le \frac{\lambda }{{\left|x-y\right|}^{l+\mathit{sp}}}$$E15

for almost all $x,y\in R^l,\left|x-y\right|\le 1$ and some $\mu ,\lambda$ such that $0<\mu \le \lambda <\infty$.

The general information can be found in [3, 18]. By the standard method, the following statements (1–3) can be proven.

Statement 1. Assuming that $u\in W_s^p$ is a weak solution to (14) and nonnegative in a ball with the center in point $x_0$ with radius $\rho$. Then, the following inequality

$$\begin{array}{l}{r}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,r\right)}^{\frac{1}{p-1}}\\ \le C\underset{B\left(x_0,r\right)}{sup}u+C{\left(\frac{r}{\rho }\right)}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{-u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\rho \right)}^{\frac{1}{p-1}}\end{array}$$

holds for $0<r<\rho$.

Statement 2. Assuming that $u\in W_s^p$ is a weak solution to (14) and nonnegative in a ball with the center in point $x_0$ with radius $\rho$. Then, for all $r,\rho$ such that $0<r<\rho$, the following inequality

$$\sqrt[\delta ]{{\left(f^{\delta }\right)}_{B\left(x_0,r\right)}}\le C{inf}_{B\left(x_0,r\right)}u+$$

$$+C{\left(\frac{r}{\rho }\right)}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{-u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\rho \right)}^{\frac{1}{p-1}}$$

holds for any $\delta \in \left(0,1\right)$.

Statement 3. Assuming that $u\in W_s^p$ is a weak solution to (14) and nonnegative in a ball with the center in point $x_0$ with radius $\rho$. Then, for all $r,\rho$ such that $0<r<\rho$, the following inequality

$$\underset{B\left(x_0,r\right)}{sup}u\le C{\epsilon }^{-\frac{p-1}{{\mathit{sp}}^2}l}\sqrt[p]{{\left({\left(max\left\{u,0\right\}\right)}^p\right)}_{B\left(x_0,\rho \right)}}+$$

$$+\mathit{C\epsilon }{r}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,r\right)}^{\frac{1}{p-1}}$$

holds for $\delta \in \left(0,1\right)$.

Now, we can prove the next theorem.

Theorem 2. Let $u\in W_s^p\left(R^l\right)$ be a weak solution to the problem

$$\begin{array}{l}\left({\left(-\mathrm{\Delta }\right)}_p^s-b\cdot \nabla \right)u=0\text{in}\mathrm{\Omega },\\ u=f\text{in}{R}^l\backslash \mathrm{\Omega },\end{array}$$E16

where $1\le p<\infty$ and $s\in \left(0,1\right)$. Assuming $u\in W_s^p\left(R^l\right)$ is nonnegative in a ball with the center in point $x_0$ with the radius $\rho$, then

$$\underset{B\left(x_0,r\right)}{sup}u\le C{inf}_{B\left(x_0,r\right)}u+C{\left(\frac{r}{\rho }\right)}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{-u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\rho \right)}^{\frac{1}{p-1}}$$

for $r,\rho$ such that $0<r<\rho$.

Proof. From statement 3, we can write the estimation

$$\underset{B\left(x_0,\frac{r}{2}\right)}{sup}u\le C{\epsilon }^{-\frac{p-1}{{\mathit{sp}}^2}l}{\left({\left({\left(max\left\{u,0\right\}\right)}^p\right)}_{B\left(x_0,\rho \right)}\right)}^{\frac{1}{p}}+$$

$$+\tilde{C}\epsilon r^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\frac{r}{2}\right)}^{\frac{1}{p-1}},$$

and using statement 1, we have the following inequality

$$\begin{array}{l}{r}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\frac{r}{2}\right)}^{\frac{1}{p-1}}\\ \le C\underset{B\left(x_0,r\right)}{sup}u+\tilde{\tilde{C}}{r}^{\frac{\mathit{sp}}{p-1}}{\rho }^{-\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{-u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\rho \right)}^{\frac{1}{p-1}}.\end{array}$$

So, we obtain the estimation

$$\begin{array}{l}\underset{B\left(x_0,\frac{r}{2}\right)}{sup}u\le \mathit{C\epsilon }\underset{B\left(x_0,r\right)}{sup}u+C{\epsilon }^{-\frac{p-1}{{\mathit{sp}}^2}l}{\left({\left({\left(max\left\{u,0\right\}\right)}^p\right)}_{B\left(x_0,\rho \right)}\right)}^{\frac{1}{p}}\\ +\mathit{C\epsilon }{r}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,r\right)}^{\frac{1}{p-1}}.\end{array}$$

Choosing $\frac{1}{2}<\eta <\tilde{\eta }<1$ applying the standard argument and the Young inequality, we obtain

$$\begin{array}{l}\underset{B\left(x_0,r\tilde{\eta }\right)}{sup}u\le \frac{1}{2}\underset{B\left(x_0,\mathit{r\eta }\right)}{sup}u+\tilde{c}\sqrt[\delta ]{{\left(f^{\delta }\right)}_{B\left(x_0,r\right)}}+\\ +C{\left(\frac{r}{\rho }\right)}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{-u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\rho \right)}^{\frac{1}{p-1}},\end{array}$$

now, iterating this argument and applying statement 2, we have proven Theorem 2.

Using Theorem 2, we write

$$\underset{B\left(x_0,{\epsilon }^i\frac{r}{2}\right)}{\mathit{osc}}u\equiv \underset{B\left(x_0,{\epsilon }^i\frac{r}{2}\right)}{sup}u-\underset{B\left(x_0,{\epsilon }^i\frac{r}{2}\right)}{inf}u\le$$

$$\le C{\left({\epsilon }^i\frac{r}{\rho }\right)}^{\frac{\mathit{sp}}{p-1}}\sqrt[p]{{\left({\left|u\right|}^p\right)}_{B\left(x_0,r\right)}}+$$

$$+C{\left({\epsilon }^i\frac{r}{\rho }\right)}^{\frac{\mathit{sp}}{p-1}}{⟨{\left|max\left\{u,0\right\}\right|}^{p-1}\frac{1}{{\left|\cdot -x_0\right|}^{l+\mathit{sp}}}⟩}_{R^l\backslash B\left(x_0,\frac{r}{2}\right)}^{\frac{1}{p-1}}$$

for all $i\in N$, thus Theorem 2 is a consequence of Theorem 1.

4. Conclusion

This chapter is dedicated to studying of the fundamental solutions to the generalization of the fractional Laplaciane by the methods of the theory of perturbation. We establish the regularity properties of the solutions to the fractional Laplacian equation with perturbations of different types. The Harnack inequality of a weak solution in the Sobolev space to the fractional Laplacian problem is studied, and the oscillation of the solution to the fractional Laplacian is estimated.

Mathematics subject classification

46B70, 43A15, 43A22, 44A05, 44A10, 44A45