A Singular Integral Equation On The Upper Half Space

In this paper, we studied the characters of a singular integral equation on the upper half space, using the method of moving plane in integral form. Unlike the traditional methods of moving plane for partial differential equations, the method of moving plane in integral form doesn't need the restrictive conditions of some local properties. Let R+n be the n-dimensional upper half Euclidean space Let a be any real number satisfying 0<α<n, assume 0<β<α, andγ= (n+α-2β)/(n-α) We considered regularity, symmetry, and monotonicity of the solutions for the integral equation where x*=(x1,…,xn-1,-xn) is the reflection of the point x about the plane xn=0. And we also considered the relation between integral equation (0-1) and the well-known partial differential equationOur proofs depend on the regularity lifting theorem, the extremum principle of integral inequalities, the Holder inequlilty, the Hardy-Little-Sobolev inequality and so on.Our main results areTheorem 1 Let u∈Lq0{R+m} be a solution of (0-1) for some q0>n/(n-α). Assume thatThen u(x) is in Lq(R+n)∩L∞(R+n), for any n/(n-α)<q<∞. Hence u is continuous. Theorem 2 Let u∈Lq(R+m) be a solution of (0-1) for some q>n/(n-α). Assume thatThen u is rotationally symmetric about some line parallel to xn-axis.Theorem 3 Let u(x) be the smooth solution of (0-1), then u(x) satisfies (0-2).