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L multipliers and their H-L estimates on the Heisenberg group.

Chin-Cheng Lin — 1995

Revista Matemática Iberoamericana

We give a Hörmander-type sufficient condition on an operator-valued function M that implies the L-boundedness result for the operator T defined by (Tf)^ = Mf^ on the (2n + 1)-dimensional Heisenberg group H. Here ^ denotes the Fourier transform on H defined in terms of the Fock representations. We also show the H-L boundedness of T, ||Tf|| ≤ C||f||, for H under the same hypotheses of L-boundedness.

Bilinear operators associated with Schrödinger operators

Chin-Cheng Lin;  Ying-Chieh Lin;  Heping Liu;  Yu Liu — 2011

Studia Mathematica

Let L = -Δ + V be a Schrödinger operator in ℝ d and H ¹ L ( ℝ d ) be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by T ± ( f , g ) ( x ) = ( T ₁ f ) ( x ) ( T ₂ g ) ( x ) ± ( T ₂ f ) ( x ) ( T ₁ g ) ( x ) , where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from L p ( ℝ d ) × L q ( ℝ d ) to H ¹ L ( ℝ d ) for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.

The continuity of pseudo-differential operators on weighted local Hardy spaces

Ming-Yi Lee;  Chin-Cheng Lin;  Ying-Chieh Lin — 2010

Studia Mathematica

We first show that a linear operator which is bounded on L ² w with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space h ¹ w if and only if this operator is uniformly bounded on all h ¹ w -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to h ¹ w .

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