INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).

ROUGH OPERATORS AND COMMUTATORS ON HOMOGENEOUS WEIGHTED HERZ SPACES

The authors establish the baundedness on homogeneous weighted Herz spaces for a large class of rough operators and their commutators with BMO functions. In particular, the Calderon-Zygmund singular integrals and the rough R. Fefferman singular integral operators and the rough Ricci-Stein oscillatory singular integrals and the corresponding commutators are considered.

ON THE CLASSIC NONHOLONOMIC DYNAMICS

For first-order nonlinear nonholonomic systems,the present paper proves that the d-δ operations are commutative and derives the equation of motion without making use of the addition al Appell-Chetaev condition.This equation of motion coincides with the equation of'Vacco dynam- ics'.

Commuting Toeplitz Operators on the Hardy Space of the Polydisk

In this paper, we show that two Toeplitz operators Tf and Tg on the Hardy space of the polydisk can commute if and only if the Berezin transform of the commutator [Tf, Tg] is n-harmonic.

Commuting Toeplitz Operators on Fock-Sobolev Spaces of Negative Orders

In the setting of Fock-Sobolev spaces of positive orders over the complex plane,Choe and Yang showed that if the one of the symbols of two commuting Toeplitz operators with bounded symbols is non-trivially radial,then the other must also be radial.In this paper,we extend this result to the Fock-Sobolev space of negative order using the Fock-type space with a confluent hyper geometric function.

Θ-type Calderón-Zygmund Operators with Non-doubling Measures

Let μ be a Radon measure on Rd which may be non-doubling. The only condition that μ must satisfy is μ（B（x,r）） ≤ Urn for all x∈Rd, r 〉 0 and for some fixed 0 〈 n 〈 d. In this paper, under this assumption, we prove that 0-type Calder6n-Zygmund operator which is bounded on L2 （μ） is also bounded from L^∞（μ） into RBMO （μ） and from Hb （μ） into L1（μ）. According to the interpolation theorem introduced by Tolsa, the LP（μ）-boundedness （1 〈 p 〈 ∞） is established for θ-type Calder6n-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type CMderθn-Zygmundoperator with RBMO （μ） function are bounded on LP（μ） （1 〈 p 〈 ∞）.

Weakly k-hyponormal and polynomially hyponormal commuting operator pairs

We introduce the notion of weak k-hyponormality and polynomial hyponormality for commuting operator pairs on a Hilbert space and investigate their relationship with k-hyponormality and subnormality.We provide examples of 2-variable weighted shifts which are weakly 1-hyponormal but not hyponormal.By relating the weak k-hyponormality and k-hyponormality of a commuting operator pair to positivity of restriction of some linear functionals to corresponding cones of functions,we prove that there is an operator pair that is polynomially hyponormal but not 2-hyponormal,generalizing Curto and Putinar’s result(1991,1993)to the two-variable case.

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R^n) functions is a compact operator from L^(p1)(R^n, w_1) × L^(p2)(R^n, w_2) to L^p(R^n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R^(2n)).

Commutation of Geometry-Grids and Fast Discrete PDE Eigen-Solver GPA

A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditioning,the authors apply the intrinsic geometric invariance,the Grid matrix G and the discrete PDE mass matrix B,stiff matrix A satisfies commutative operator BG=GB and AG=GA,where G satisfies G^(m)=I,m<<dim(G).A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.In this paper,the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle,square,tetrahedron,cube,and so on.They give the general form of pre-processing matrix,theory and numerical test of GPA.The conclusion that“the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron”is obtained through research,and it is further found that“commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.