Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on Rn or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p > 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L)−1/2 on the Lp space. As an application, for 1 < p < 3+ ε, we establish the Lp boundedness...

Abstract. Let L = divA∇ be a real, symmetric second order elliptic operator with bounded measurable coefficients. Consider the elliptic equation Lu = 0 in a bounded Lipschitz domain Ω of R. We study the relationship between the solvability of the L Dirichlet problem (D)p with boundary data in L(∂Ω) and that of the L regularity problem (R)q with boundary data in W (∂Ω), where 1 < p, q < ∞. It is...

Abstract Let $$\mathcal {B}$$ B be a homogeneous differential operator of order $$l=1$$ l = 1 or $$l=2$$ 2 . We show that sequence functions the form $$(\mathcal {B}u_j)_j$$ ( u</mml:...

let (m,g ) be a compact immersed hypersurface of (rn+1,) , λ1 the first nonzeroeigenvalue, α the mean curvature, ρ the support function, a the shape operator, vol (m ) the volume of m,and s the scalar curvature of m. in this paper, we established some eigenvalue inequalities and proved theabove.1) 1 2 2 2 2m ma dv dvn∫ ρ ≥ ∫ α ρ ,2)( )2 2 1 2m 1 mdv s dvn nα ρ ≥ ρ∫ − ∫ ,3) if the scalar curvatu...

In this paper, we introduce 1-way and 2-way quantum nite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeter-ministic and probabilistic 1-way and 2-way nite state automata. We prove the following facts regarding 2qfa's. 1. For any > 0, there is a 2qfa M which recognizes the non-regular language L = fa m b m j m 1g with (one-sided) error bounded by , an...

We prove that there is no sparse hard set for P under logspace computable bounded truth-table reductions unless P=L. In case of reductions computable in NC, the collapse goes down to P=NC. We parameterize this result and obtain a generic theorem allowing us to vary the sparseness condition, the space bound and the number of queries of the truth-table reduction. Another instantiation yields that...