Heat Kernel Estimates And Function Spaces On Fractals

This thesis consists of the studies of the following three questions.1)When is the union of two unit intervals on the line Uλa self-similar set sat-isfying the open set condition?We prove that Uλis a self-similar set satisfying theopen set condition,if and only if,λ∈Λ:=n≥1Λn,the definition ofΛn is at sec-tion 3.1.Furthermore,we characterize the structure of the setΛas follows:the setΛ∩[0,2]consists exactly of 0,1,2,and the spectra of the words 123132,12313424,and wn=w1...w2n(n=3,4,···)defined by w1=w3=1,w2n?2=w2n=n,andw2j ?2=w2 j+1=j for 2≤j≤n?1。The definitions of word and spectra are at thesection 3.1.We give also two examples and present some unsolved questions.We obtain the above result by using the techniques Fractal Geometry,Theoryof Geometrical Measure,classical analysis,moreover,we used also graph theory andnon-negative matrix.2)First,we consider the heat kernel estimates on effective resistance metricspaces.We prove the existence of heat kernel and obtain the two-sided estimates,andthen somea pplications ont he real linea nd onp .c.f.self-similar setsa reg iven.We needto assume that the existence of irreducible local regular Dirichlet form,and theγ-chaincondition of effective resistance,furthermore,we need the measure to beα-regular.The estimates of local heat kernel is obtained by using Green function.Then,we provethe on-diagonal estimates of heat kernel,and obtain the off-diagonal upper bounds byusing the locality of Dirichlet form and the lower bounds of local heat kernel.Finally,we prove the off-diagonal lower bounds of heat kernel through near diagonal lowerbounds by usingγ-chain condition.3)We prove that,if there exists a double measure in a metric space,then theHaj?asz-Sobolev type spaces Mpσ(μ),that depend on quasi-distances,satisfy the fol-lowing embedding theorem,Mpσ(μ)?Lip(σ,p,∞)(μ).Moreover,Lip(σ,p,∞)(μ)?Mpσ?θ(μ),for any 0<θ<σ.Furthermore,we show that,on the Sierpinski gasket in R2,the domain of p-energy D(E)=Lip(βp/p,p,∞)(μ),and then we prove the Haj?asz-Sobolev typespaces Mpσ(μ)is non-trivial for 1<σ<βp/p,whereβp characterizes the intrinsicproperty of the Sierpinski gasket.