Real-variable Characterizations Of Variable Hardy Spaces On Domains And Their Applications

Let p（·）:Rⁿ→（0,∞）a variable exponent function satisfying the globally logHolder continuous condition and 0<p-≤p+<∞,where p-:= ess （?）p（x）and p+:=ess （?）p（x）.The main aim of this article is threefold.1.Let Ω be a proper open subset of Rⁿ.In Chapter 2,the author first introduce the variable Hardy space H^{p（·）}on Ω by the radial maximal function,and then obtain the grand maximal function and the atomic characterization of H^{p（·）}（Ω）.Moreover,the author introduce the variable H（?）lder space Λ^{p（·）,q,d}（Ω）on Ω.As applications of atomic characterization of H^{p（·）}（Ω）,the author show that Λ^{p（·）,q,d}（Ω）is the dual space of H^{p（·）}（Ω）.The above results extend the main results in Miyachi（Studia Math 95:205-228,1990）to the case of variable exponents.2.Let n≥ 2 and Ω（?）Rⁿ be a bounded non-tangentially accessible domain（for short,NTA domain）.Assume that LD is a second order divergence form elliptic operator having real-valued,bounded,and measurable coefficients on L2（Ω）with the Dirichlet boundary condition.In Chapter 3,the author first establish the molecular characterization for the variable Hardy space H_LD^{p（·）}（Ω）associated with LD.In particular,when LD is nonnegative and self-adjoint,the author further obtain the atomic and the maximal function characterization of H_LD^{p（·）}（Ω）.Secondly,the author introduce the "geometrical" variable Hardy space Hrp（·）（Ω）and its local version hp（·）（Ω）by restricting any element of the variable Hardy space H^{p（·）}（Rⁿ）and its local version hp（Rⁿ）,respectively,to Ω,and then show that,when p-∈(n/n+θ,1],H^{p（·）}（Ω）=Hrp（·）（Ω）=H_LD^{p（·）}（Ω）=hrp（·）（Ω）with equivalent quasi-norms,where θ ∈(0,1]is the critical index of Holder continuity for the heat kernels{pt,LD}t>0 generated by LD.As applications,the author prove the boundedness of the Riesz transform â–½LD-1/2 on the variable Lebesgue space L^{p（·）}（Ω）when 1<p-≤p+≤2,and from H_LD^{p（·）}（Ω）to L^{p（·）}（Ω）when 0<p-≤p+≤1,or to H^{p（·）}（Ω）when n/n+1<p-≤p+≤1.Meanwhile,the author also show the boundedness of the fractional integral LD-βfrom L^{p（·）}（Ω）to Lq（·）（Ω）,when 1<p（·）<q（·）<∞,and from H_LD^{p（·）}（Ω）to Lq（·）（Ω）,when 0<p（·）≤1<q（·）<∞,or to HLDq（·）（Ω）,when 0<p（·）<q（·）≤1,where β∈（0,n/2）and 1/p（·）-1/q（·）=2β/n.As a corollary,global gradient estimates,in both L^{p（·）}（Ω）when 1<p-≤p+≤2 and H^{p（·）}Ωwhen n/n+1<p-≤p+≤1,for the inhomogeneous Dirichlet problem of LD on Ω are obtained.Finally,a div-curl lemma in the space Hrp（·）（Ω）and BMO^{p（·）}（Ω）=BMOzp（·）（Ω）with equivalent norm are established,where BMOzp（·）（Ω）:={f∈BMO^{p（·）}（Rⁿ）:supp（f）（?）Ω),BMO^{p（·）}（Rⁿ）and BMO^{p（·）}（Ω）,respectively,denote the variable BMO space on Rⁿ and Ω,and Ω denotes the closure of Ω（?） Rⁿ.The above results are new even when p（·）≡p ∈（0,∞）or Ω（?）Rⁿ is a bounded Lipschitz domain.3.Let n≥3 and Ω be a strongly Lipschitz domain of Rⁿ.Assume that LΩ:=-Δ+V is a Schrodinger operator on L2（Ω）with the Dirichlet boundary condition,where Δ denotes the Laplace operator and the nonnegative potential V belongs to the reverse Holder class RHq0（Rⁿ）for some q0∈(n/2,∞].In Chapter 4,the author first introduce the variable Hardy space HLΩp（·）（Ω）associated with LΩ on Ω,via the Lusin area function associated with LΩ,and the "geometrical" variable Hardy space H_LRⁿ,r^{p（·）},（Ω）,via the variable Hardy space HLRⁿp（·）（Rⁿ）associated with the Schrodinger operator LRⁿ:=-Δ+V on Rⁿ,and then prove that HLΩp（·）（Ω）=H_LRⁿ,r^{p（·）},（Ω）with equivalent quasi-norms.As an application,the author show that,when Ω is a bounded,simply connected,and semiconvex domain of Rⁿ and the nonnegative potential V belongs to the reverse H（?）lder class RHq0（Rⁿ）for some q0∈(maxï½›n/2,2},∞],the operators VLΩ-1 and â–½2LΩ-1 are bounded from H_LRⁿ,r^{p（·）},（Ω）to the variable Lebesgue space L^{p（·）}（Ω）,or to itself.As a corollary,the second-order regularity for the inhomogeneous Dirichlet problems of the corresponding Schrodinger equations in the scale of variable Hardy spaces H_LRⁿ,r^{p（·）}（Ω）is obtained.