Weighted Hardy-Littlewood-Sobolev Inequality on the Unit Sphere

Abstract

One of the main aims in this thesis is to establish analogues of the classical Hardy-Littlewood-Sobolev (HLS) inequality for weighted orthogonal polynomial expansions (WOPEs) on the unit sphere, the unit ball and the simplex. An optimal condition for which this inequality holds is obtained. Classical proofs of the optimality of this inequality on the usual Euclidean spaces rely on the dilation operators and do not seem applicable in our setting, where dilations are not available. The crucial ingredients in our proofs in this thesis are a series of new sharp pointwise estimates for some important kernel functions that appear naturally in the WOPEs. These estimates are more difficult to establish, andwill be useful for some other problems in WOPEs. The HLS inequality for the first order fractional integral operator has been playing important roles in many applications. The second part in this thesis proves an equivalent version of the first order HLS inequality, which involves the tangent gradient and the difference operators. This equivalent version has the advantages that it is much simpler and much easier to deal with in applications. While the main tool for the proof of this equivalent version is the Calderon-Zygmund decomposition, the details are much more involved. Of particular importance in our proof is an elegant decomposition of the second order differential-difference operator associated with the WOPEs, discovered in this thesis. It turns out that this decomposition is very useful in several other problems, such as the uncertainty principle of the WOPEs. The main results of this thesis have many interesting applications in $N$-widths, embedding of function spaces and approximation theory.