Commutators of Singular Integrals with Kernels Satisfying Generalized Hörmander Conditions and Extrapolation Results to the Variable Exponent Spaces

Potential Analysis - We obtain boundedness results for the higher order commutators of singular integral operators between weighted Lebesgue spaces, including Lp-BMO and Lp-Lipschitz estimates. The...     

A Remark on the Hörmander Inequality

A sufficient condition is given in order to ensure the Hörmander inequality for a (formally) self-adjoint transversally elliptic pseudodifferential operator P in the case of the symplectic form changing rank on the characteristic manifold.     

Littlewood–Paley Operators on Spaces with Variable Exponent on Homogeneous Groups

In this paper,on homogeneous groups,we study the Littlewood–Paley operators in variable exponent spaces.First,we prove that the weighted Littlewood–Paley operators are controlled by the weighted Hardy–Littlewood maximal function,and obtain the vector-valued inequalities of the Littlewood–Paley operators,including the Lusin function,Littlewood–Paley g function and gλ* function.Second,we prove the boundedness of multilinear Littlewood–Paley gψ,λ* function.     

Hörmander Multipliers on the Heisenberg Group

 Let be the Heisenberg group of dimension . Let be the partial sub-Laplacians on and T the central element of the Lie algebra of . For any we prove that the operator is bounded on the Hardy spaces , if the function m satisfies a Hrmander-type condition on which depends on . We also obtain analogous results for the operators and , where the function m satisfies analogous H?rmander-type conditions on and on , respectively. Here is the Kohn-Laplacian on . (Received 28 July 1999; in final form 6 March 2000)     

Boundedness for Higher Order Commutators of Fractional Integrals on Variable Exponent Herz–Morrey Spaces

In this paper, the boundedness for higher order commutators of fractional integrals is obtained on variable exponent Herz–Morrey spaces \( M\dot{K}_{p, q(\cdot )}^{\alpha (\cdot ), \lambda }(\mathbb {R}^{n})\) applying some properties of variable exponent and \(\mathrm {BMO}\) function, where \(\alpha (x)\in L^{\infty }(\mathbb {R}^{n})\) are log-Hölder continuous both at the origin and at infinity, and q(x) satisfies the logarithmic continuity condition.     

A Survey on Frölicher Spaces

Abstract

This survey paper highlights a series of results in recent research on topology, geometry and categorical properties of spaces provided with a new structure in the mathematical literature, called Frölicher spaces. Without any fear of contradiction, these smooth spaces are stated to generalize the theory of differentiable manifolds. More precisely, the present study will give the state of research on this topic which historically links a fundamental theorem of calculus in so-called Boman's theorem (and its generalization) to abstract spaces, whether they are normable or not.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

Fractional Integrals on Variable Hardy–Morrey Spaces

Vector-valued fractional maximal inequalities on variable Morrey spaces are proved. Applying atomic decomposition of variable Hardy–Morrey spaces, we obtain the boundedness of fractional integrals on variable Hardy–Morrey spaces, which extends the Taibleson–Weiss’s results for the boundedness of fractional integrals on Hardy spaces. The corresponding boundedness for the fractional type integrals is also considered.     

A Note on Spaces with Regular Gδ-Diagonals

这篇简短的注记使用Mysior的例子说明具有正则G     

Hörmander index in finite-dimensional case

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.