Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the HSrmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy Spaces for k≥ 3 using the multi- parameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k 3, and the method works for all the cases k ≥ 3： Tmf（x1,x2,x3） =1/（（2π）＋n1＋n2＋n3） ∫ R n1×R n2×R n3 m（ξ）f（ξ）e 2π ix.ξ dξ. where x = （x1,x2,x3） ∈ Rn1 × Rn2 × R n3 and ξ = （ξ1,ξ2,ξ3） ∈ R n1 × Rn2 ×R n3. One of our main results is the following： Assume that m（ξ） is a function on Rn1＋n2＋n3 satisfying sup j,k,l ∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3）〈∞ with si 〉 ni（1/p-1/2） for 1 ≤ i ≤ 3. Then Tm is bounded from HP（R n1 × R n2 ×R n3） to HP（R n1 ×R n2 × R n3） for all 0 〈 p ≤ 1 and ｜｜Tm｜｜ Hp→Hp≤ sup j,k,l∈Z ｜｜mj,k,l｜｜ W（s1,s2,s3） Moreover, the smoothness assumption on sl for 1 ≤ i ≤ 3 is optimal. Here we have used the notations mj,k,l （ξ）= m（2 j ξ1,2 k ξ2, 2 l ξ3） ψ（ξ1） ψ（ξ2） ψ（ξ3） and ψ（ξi） is a suitable cut-off function on R ni for 1 ≤ i ≤ 3, and W（s1,s2,s3） is a three-parameter Sobolev space on R n × R n2 × Rn 3. Because the Fefferman criterion breaks down in three parameters or more, we consider the Lp boundedness of the Littlewood-Paley square function of T mf to establish its boundedness on the multi-parameter Hardy spaces.

H^p Spaces Associated with Square Root Operator

We characterize Hp spaces associated with square root operator by using maximal functions.

Estimates of Some Integral Operators with Bounded Variable Kernels on the Hardy and Weak Hardy Spaces over R^n

In this paper,we first introduce Lσ1-（log L）σ2 conditions satisfied by the variable kernelsΩ（x,z） for 0≤σ1≤1 and σ2≥0.Under these new smoothness conditions,we will prove the boundedness properties of singular integral operators TΩ,fractional integrals TΩ,α and parametric Marcinkiewicz integrals μΩρ with variable kernels on the Hardy spaces Hp（Rn） and weak Hardy spaces WHP（Rn）.Moreover,by using the interpolation arguments,we can get some corresponding results for the above integral operators with variable kernels on Hardy-Lorentz spaces Hp,q（Rn） for all p 〈 q 〈 ∞.

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be a a-finite yon Neumann algebra and let 9i C M be a maximal subdiagonal algebra with respect to a faithful normal conditional expectation Ф. Based on the Haagerup＇s noncommutative Lp space LP（M） associated with M, we consider Toeplitz operators and the Hilbert transform associated with 8i. We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2（A4） is just the right analytic Toeplitz algebra. Furthermore, the Hilbert transform on noncommutative LP（Yt4） is shown to be bounded for 1 （ p （ ce. As an application, we consider a noncomnmtative analog of the space BMO and identify the dual space of noncommutative Hl（M） as a concrete space of operators.