First, we need to introduce fractional Sobolev spaces and their seminorms, we refer to  for details. For and , is the fractional Sobolev space of all functions with finite fractional seminorm , given by
Under these definitions, we have the following fractional Sobolev-type inequality: there exists a constant such that, for each with , we have
where (see ).
First, we consider a positive smooth function with the following properties:
With the aid of this function, we split a function u into two parts. We will denote the ‘smooth’ part of u as v and the remaining as w. We let
Sometimes, for simplicity in the notation and where the context is clear, we will write u, v, and w as functions depending only of x.
As a first property of this decomposition we find that each norm of the functions v and w is controlled by the corresponding norm of u.
Lemma 2.1Letvandwbe given by (2.4). For each, we have
Proof We start with v. Denoting the Hölder conjugate of r and using the definition of v, we have
The inequality for w easily follows immediately from the triangular inequality in . □
Now we state a key result to get the desired estimate on the decay rate.
Proposition 2.2Letand letbe a kernel satisfying (1.2), ψsatisfying (2.3), , and. Then there exists a constantsuch that, for allandv, wdefined in (2.4), we have
The constantCdepends onψ, β, r, andn.
Proof For the estimate concerning w, we have
Applying Hölder’s inequality, we get
Since ψ is supported in , we have for all , and, since J verifies for , there exists a constant C depending only on such that . Then
Now we deal with the term with v. We split the fractional seminorm as
and look at these integrals separately. For , using the definition of v we have
Now, we can think of the measure as a probability measure (because of (2.3)), and since the function is convex in ℝ, we can apply Jensen’s inequality on the dz-integral in right-hand side of the last expression to obtain
which, after an application of Fubini’s theorem, gives
Then, applying the change , in the integral and using (2.3), we conclude
Using this last expression, we obtain from the assumption (1.2) that
Now we deal with . In this case, using the definition of v, we can write
Note that by using (2.3), we have for all
Thus, using this equality with (2.7), we get
However, note that if in the dz integral, since necessarily . Then, due to the fact that ψ is supported in the unit ball, the contribution of the integrand when is null in the dz integral. Taking this into account, applying Hölder’s inequality into the dz-integral, we have
By Fubini’s theorem we can write
Using the regularity of ψ, we have
and since , we conclude that the last integral is convergent, obtaining
which leads to the following estimate for :
From this, it is easy to get
which, by the use of (1.2), let us conclude that
This last estimate together with (2.6) concludes the proof. □