Hua-Kelvin Transform On The Spaces Of Harmonic Functions

The dissertation studies the Hua-Kelvin transform and its operator theory on harmonic function spaces.Loo-keng Hua in his book "Starting with the unit circle" pointed out that the transform preserves harmonicity, where??is the real Mobius transform on the unit ball B in Rn. Due to its similarity to the Kelvin transform and in honor of Hua, we coin the name by the Hua-Kelvin transform.Hua-Kelvin transform is a concrete weighted composition operator. The the-ory of composition operators plays an important role in mathematics and physics, especially in dynamic systems. Although the composition operator theory on ana-lytic function spaces has become very mature, there still lacks of the corresponding theory for harmonic functions. In the article we initiate the study of composition operators in the setting of harmonic functions.To study composition operators for harmonic functions, the paramount problem we have to face is to pick out the composition operators with the harmonicity-preserving property. Thanks to the result by Taishun Liu that all harmonicity-preserving maps should be conformal, we can give the complete classification of the composition operators with the harmonicity-preserving property. They are in fact the composition of the following two operators:(?) Hua-Kelvin transform.(?) The composition operators are C??u?u??, where?=??x+b,??R, b?Rn,??O(n)and???+?b??1. This shows the core role of the Hua-Kelvin transform in the operator theory of harmonic functions.The Hua-Kelvin transform on the harmonic Hardy spaces and harmonic Bergman spaces are studied in the thesis. The contents are as follow. In Chapter 2, we establish an integral formula where x,??B,k=0,1,2,...??C. The formula extends the Forelli-Rudin type estimate and is a basic tool in the operator theory of harmonic functions.We show that in Chapter 3 the harmonicity-preserving maps are the composi-tion of Hua-Kelvin operator with translation, dilation, and rotation.In Chapter 4, we take the Hua-Kelvin transform as an operator on the harmonic Hardy space hp(B) and study its boundedness, compactness, essential-normality, and the radius of the spectra. Some of our main results are listed as follows.We find the exact norm of the Hua-Kelvin transform as an operator on the harmonic Hardy space hp(B) Notice that the norm of Hua-Kelvin transform on hp(B) is the piecewise defined function of indices p. The critical index is given by 2*= 2(n-1)/(n-2), which is also the critical Sobolev trace exponents of 2 introduced by Sobolev in Sobolev's imbedding theorem. Why there exists such an amazing relation between the crit-ical index from the norms of Hua-Kelvin transform and the critical Sobolev trace exponents is still unknown.In the operator theory of Banach spaces, an operator is called of having the kernel supremum property if its norm is the limit of its action on a sequence of normalized reproducing kernels. Bourdon and Retsek studied when the analytic composition operators have the kernel supremum property in the analytic Hardy space on the unit disk. In the article we show that the Hua-Kelvin transform, as an operator in the harmonic Hardy space on the real unit ball, has the kernel supremum property. In particular, where py is the normalized extended Poisson kernel on the harmonic Hardy space h2(B). To achieve the proof of the above formula is a subtle problem.Shapiro introduced the concept of the essential norm in the operator theory on Banach spaces. The essential norm of an operator describes its distance with the compact operators. Because the essential norm is less than or equal to the norm, the nature question arises when the equal can be achieved. The example is given in the article by the Hua-Kelvin transform, when the Hua-Kelvin transform is taken to be an operator in harmonic Hardy space of the unit ball.In Chapter 5, we take the Hua-Kelvin transform as an operator on the harmonic Bergman space and study the analogous properties as in Chapter 4.In Chapter 6, with the same approach as in the study of the Hua-Kelvin transform, we can study an integral operator associated to the weighted harmonic Bergman projection Its precise norm turns out to be which gives an affirmative answer to the open problem posted by Choe, Koo and Nam, which states that whether the norm of Ta grows at most like (?+1)-1 as??-1.Hua-Kelvin transform is a weighted composition operator arising from the real Mobius transform. In view of the close relationship of the Mobius transform with hyperbolic geometry, general relativity, and gyrogroup theory, we can expect wide applications of Hua-Kelvin transform in the future.