We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.

We solve, in two dimensions, the "square root problem of Kato". That is, for L ≡ -div (A(x)∇), where A(x) is a 2 x 2 accretive matrix of bounded measurable complex coefficients, we prove that L1/2: L12(R2) → L2(R2).[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].

We shall consider the Schrödinger operators on ${ℝ}^2$ with the magnetic field given by a nonnegative constant field plus random $\delta$ magnetic fields of the Anderson type or of the Poisson-Anderson type. We shall investigate the spectrum of these operators by the method of the admissible potentials by Kirsch-Martinelli. Moreover, we shall prove the lower Landau levels are infinitely degenerated eigenvalues when the constant field is sufficiently large, by estimating the growth order of the eigenfunctions...

Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1\<p\<\infty$ and $v\ge 0$, we show there exists $C\>0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\text{'}}\>0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\text{'}}}dx\le C^{\text{'}}{\int }_{Q}v\left(x\right)dx\<\infty$ for all dyadic cubes Q, where $p^{\text{'}}=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

Let $L$ be a non-negative self-adjoint operator acting on $L^2\left({ℝ}^n\right)$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_r$ weight on ${ℝ}^n\times {ℝ}^n$, $1<r<\infty$. In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L,w}^p({ℝ}^n\times {ℝ}^n)$, $0<p\le 1$ associated to $L$. Based on the atomic decomposition, we show the dual relationship between $H_{L,w}^1({ℝ}^n\times {ℝ}^n)$ and ${\mathrm{BMO}}_{L,w}({ℝ}^n\times {ℝ}^n)$.