Regularity of Solutions to an Integral Equation on a Half-Space nR *

n R In this paper, we discuss the integral equation on a half space       1 1 d , n  > 0, . p n x x R n n R u x u y y u x y x y                     

In this paper we consider the regularity of positive solution of the following integral equation in where . It relates closely to the higher-order PDEs with Navier boundary conditions in : Li ([3]) showed the equivalence between the integral Equation (1.1) and partial differential Equation (1.2).For more results concerning integral equations, see [4][5][6].Firstly, in this paper we have the boundedness for the positive solutions of (1.1) by using the contracting operators.
Theorem 1.1.Let u be a solution of (1.1).If n p n   , and In [2], the authors proved that Theorem 1.

 
Then we employ the brand new method which is the combinations of contracting and shrinking operators introduced by Ma-Chen-Li ( [1]) to derive the Lipschitz continuity of solutions.

Estimate by Contracting Operators
In this section, we obtain estimate for positive solutions to the equation (0.1) by using the contracting operators.To prove the Theorem 1.1, we need the following equivalent form of Hardy-Littlewood-Sobolev inequality.
The proof is divided into two steps.
Step 1.We first show that , Obviously,

B
, and vanishes outside the ball The Equation (0.1) can be rewritten as We will show that, for any , 1) T A is a contracting map from to for A large, and , we apply Hardy-Littlewood-Sobolev inequality and Hölder inequality to obtain , by the definition of one can choose a large number A, such that and hence arrives at , we apply Hardy-Littlewood-Sobolev inequality and Hölder inequality to obtain , we see that s can be arbitrary.Since and hence

If
, we are done.If , repeat the above process and after a few steps, we arrive at Step 2. In this step we will show that , and by the result in Step 1, , for For 1 I , we apply Hölder inequality

Lipschitz Continuity by Combinations of Contracting Operators and Shrinking Operators
In the previous section we showed that the solution of (0.1) is in .In this section, we will use the regularity lifting by combinations of contracting and shrinking operators to prove  , the space of Lipschitz continuous functions with norm To prove the Theorem 1.2, we need introduce the following definition, property and a more general Regularity Lifting Theorem on the combined use of contracting and shrinking operators.
Let V be a Hausdorff topological vector space.Suppose there are two extented norms (i.e. the norm of an element in V might be infinity) defined on V, Definition. ("XY-pair") Suppose X, Y are two normed subspaces described above, X and Y are called "XY-pair", Remark 2. The "XY-pair" are quite common, here we choose , and with the norm defined in (3.1).
Theorem 3.1.(Regularity Lifting Theorem) Suppose Banach spaces X, Y are an "XY-pair", and let and be closed subsets of X and Y respectively.Suppose is a contraction: is shrinking:

Moreover, assume that
Then there exists a solution u of equation and more importantly,  The proof and some applications of Theorem 3.1 can be found in [1,7,8].
, by elementary calculus one can verify that It follows that the solution of (0.1) only differs by a constant multiple from the solution of the following equation

 
Hence, for convenience of argument, we prove that every positive solution u of (3.2) is Lipschitz continuous.Let Then obviously, u is a solution of the equation We will show that for sufficiently small, 1) is a contracting operator from to X.

2)
is a shrinking operator from to Y.

t T f x T g x f y g y y f y g y y t t p y f y g y y y f y g y y
and consequently Here we applied the Mean Value Theorem with both Therefore  is a contracting operator from to X for such a small .Y , n , then for any

Tv x Tv z v y y v y y v y y v y y t t v y v y z x y v y v y z x y t
the last equality above is from the fact Again choosing  sufficiently small, we derive by Combining this with the estimate in 1), we arrive at sup 3) To show F is Lipschitz continuous, we split it into two parts: For the first part, we have For the second part, we use a different approach.Write x z Note that here we applied the Mean Value Theorem with both 1 fore, and 2 valued between 0 and    .
Combining (3.4) and (3.5), we have The same inequality holds for     In fact,  sufficiently small ut independent of v), w 1), 2) and 3), by the The 3.1 and Remark 2, we conclude that the solutio is Lipschitz continuous.This completes the pro Th .Usually, contracting operators are used to lift reg ties.For a linear operator, if it is "shrinking", then it is contracting.While for nonlinear problems, as were seen in Section 3, sometimes it is very difficult or even impossible to prove that it is contracting in a given functio space.However, one can show that it is "shrinking", a can still lift the regularity of solutions in many cases.The ge la

Acknowledgements
enxiong Chen for his discussions.neral Regu rity Lifting Theorem is applied for integral equations and system of integral equations associated with Bessel potentials and Wolff potentials (see [1] and [7]), and therefore arrive at higher regularity as Lipschitz continuity of solutions.
Most of this work was completed when the first author was visiting the Department of Mathematics, Yeshiva University, and she would like to thank the hospitality of the Department.Besides, the authors would like to express their gratitude to Professor W hospitality and many valuable