Boundary Behavior Of Harmonic Functions On Metric Measure Spaces

This thesis is concerned with the boundary behavior of harmonic functions on the upper half-space.The Dirichlet problem of elliptic equations is one of the basic problems in PDEs.On the other hand,the study of harmonic extension of a function is one of the basic tools in harmonic analysis.For instance,the harmonic extension of a bounded mean oscillation(BMO)function plays a key role in Fefferman-Stein duality of Hardy and BMO spaces.Fefferman-Fabes et al proved a harmonic function u(x,t)defined on Rn ×(0,∞)satisfies the Carleson condition (?) if and only if,u can be represented as the Poisson integral of a BMO function f.Moreover,the norm of f in BMO space can be compared with the LHS of(*).In this thesis,we consider some more general underlying space X(than the Euclidean space)satisfying a doubling condition and admitting a Poincare inequality,and more general differential operator L(than the Laplace operator)generalized by a Dirichlet form.We show that the traces of all harmonic functions defined on X ×(0,∞),i.e.,(?),are functions of BMO space if and only if they satisfy the Carleson condition(*).Furthermore,similar problem for the Schrodinger operator (?),is also considered,where the non-negative potential V is in the Muckenhoupt weight class and satisfies a certain reverse H(?)lder inequality.