Lixin Yan

Duality of Hardy and BMO spaces associated with operators with heat kernel bounds

The introduction and development of Hardy and BMO spaces on Euclidean spaces R in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations. These spaces were studied extensively in [32], [22], [18], [19], [31] and many others. An L function f on R is in the Hardy spaceH(R) if the area integral function of the Poisson integral e−t √ f satisfies

Classes of Hardy spaces associated with operators, duality theorem and applications

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.

Multilinear operators with non-smooth kernels and commutators of singular integrals

We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calder6n-Zygmund kernels. As a consequence, we deduce endpoint L 1 × ··· x L 1 to weak L 1/m estimates for the mth-order commutator of Calderon. Our results reproduce known estimates for m = 1, 2 but are new for m > 3. We also explore connections between the 2nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.

Comparison of the classical BMO with the BMO spaces associated with operators and applications

Let L be a generator of a semigroup satisfying the Gaussian upper bounds. In this paper, we study further a new BMOL space associated with L which was intro- duced recently by Duong and Yan. We discuss applications of the new BMOL spaces in the theory of singular integration such as BMOL estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space BMOL might coincide with or might be essentially different from the classical BMO space.

Hardy spaces of spaces of homogeneous type

Let X be a space of homogeneous type, and L be the generator of a semigroup with Gaussian kernel bounds on L 2 (X). We define the Hardy spaces H p s(X) of X for a range of p, by means of area integral function associated with the Poisson semigroup of L, which is proved to coincide with the usual atomic Hardy spaces H p at(X) on spaces of homogeneous type.

Duality of hardy and BMO spaces associated with operators with heat Kernel bounds on product domains

Let L be the infinitesimal generator of an analytic semigroup on L2 (ℝ) with suitable upper bounds on its heat kernels, and L has a bounded holomorphic functional calculus on L2 (ℝ). In this article, we introduce new function spaces HL1 (ℝ × ℝ) and BMOL(ℝ × ℝ) (dual to the space HL*1(ℝ × ℝ) in which L* is the adjoint operator of L) associated with L, and they generalize the classical Hardy and BMO spaces on product domains. We obtain a molecular decomposition of function for HL1(ℝ × ℝ) by using the theory of tent spaces and establish a characterization of BMOL (ℝ × ℝ) in terms of Carleson conditions. We also show that the John-Nirenberg inequality holds for the space BMOL (ℝ × ℝ). Applications include large classes of differential operators such as the magnetic Schrödinger operators and second-order elliptic operators of divergence form or nondivergence form in one dimension.

Blocking analysis andT(1) theorem

TheT(1) theorem with a weak condition on the distribution kernel is proved by using a new method—blocking analysis. It improves a result of Meyer’s.

An atomic decomposition for hardy spaces associated to schrödinger operators

Let L = + V be a Schrodinger operator on R n where V is a nonnegative function in the space L 1 (R n ) of locally integrable functions on R n . In this paper we provide an atomic decomposition for the Hardy space H 1 (R n ) associated to L in terms of the maximal function characterization. We then adapt our argument to give an atomic decomposition for the Hardy space H 1(R n R n ) on product domains.

Wavelet frames on lipschitz curves and applications

The purpose of this paper is to present constructions of wavelet frames on a Lipschitz curve Γ. As applications, we obtain characterizations of the Besov and Triebel-Lizorkin spaces on Lipschitz curves, and the trace theorem on Γ of the Besov spaces onR2.

Multilinear Multiplier Theorems and Applications

We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calderón and Calderón–Coifman–Journé.