We show that the L(w) operator norm of the composition M◦TΩ, where M is the maximal operator and TΩ is a rough homogeneous singular integral with angular part Ω ∈ L∞(Sn−1), depends quadratically on [w]A2 , and this dependence is sharp.

We show that a first order perturbation A(x) · D + q(x) of the polyharmonic operator (−∆), m ≥ 2, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in R, n ≥ 3. Notice that the corresponding result does not hold in general when m = 1.

The operator Jm p (λ, )f (z) was studied by Srivastava et al. [3] and Aouf et al. [4]. From (1.2) and (1.3), we observe that J−m p (λ, )f (z) = Im p (λ, )f (z)(m > 0), so the operator Jm p (λ, )f (z) is well-defined for l ≥ 0, l ≥ 0, p Î N and m Î Z = {..., -2, -1,0,1,2,...}. El-Ashwah Journal of Inequalities and Applications 2012, 2012:35 http://www.journalofinequalitiesandapplications.com/con...

A completely bounded bilinear operator φ: M×M → M on a von Neumann algebra M is said to have a factorization in M if there exist completely bounded linear operators ψj , θj : M → M such that

We classify the connected pseudo-Riemannian manifolds of signature (p, q) with q ≥ 5 so that at each point of M the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of M .

The invariant subspace problem relative to a von Neumann algebra M ⊆ B(H) asks whether every operator T ∈ M has a proper, nontrivial invariant subspace H0 ⊆ H such that the orthogonal projection p onto H0 is an element of M; equivalently, it asks whether there is a projection p ∈ M, p / ∈ {0, 1}, such that Tp = pTp. Even when M is a II1–factor, this invariant subspace problem remains open. In t...

We give a condition on Riemannian submersions from a Riemannian manifold M to a Riemannian manifold N which will ensure that it induces a differential operator on N from the Laplace-Beltrami operator on M . Equivalently, this condition ensures that a Riemannian submersion maps Brownian motion to a diffusion. 2000 Mathematics Subject Classification. Primary 58J65, 53C21.

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...

Consider a flat symplectic manifold (M, ω), l ≥ 2, admitting a metaplectic structure. We prove that the symplectic twistor operator maps the eigenvectors of the symplectic Dirac operator, that are not symplectic Killing spinors, to the eigenvectors of the symplectic Rarita-Schwinger operator. If λ is an eigenvalue of the symplectic Dirac operator such that −ılλ is not a symplectic Killing numbe...

The authors consider the multilinear Riesz potential operator defined by Iα,m − → f x ∫ Rd m f1 y1 f2 y2 · · · fm ym /| x−y1, . . . , x−ym |mn−α dμ y1 · · ·dμ ym , where − → f denotes themtuple f1, f2, . . . , fm , m,n the nonnegative integers with n ≥ 2, m ≥ 1, 0 < α < mn, and μ is a nonnegative n-dimensional Borel measure. In this paper, the boundedness for the operator Iα,m on the product of...