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Bilinear multipliers on Lorentz spaces

Francisco Villarroya (2008)

Czechoslovak Mathematical Journal

We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform.

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.

Biorthogonal wavelets, MRA's and shift-invariant spaces

Marcin Bownik, Gustavo Garrigós (2004)

Studia Mathematica

We give a characterization of biorthogonal wavelets arising from MRA's of multiplicity D entirely in terms of the dimension function. This improves the previous characterization in [8] removing an unnecessary angle condition. Besides we characterize Riesz wavelets arising from MRA's, and present new proofs based on shift-invariant space theory, generalizing the 1-dimensional results appearing in [17].

BMO and commutators of martingale transforms

Svante Janson (1981)

Annales de l'institut Fourier

The commutator of multiplication by a function and a martingale transform of a certain type is a bounded operator on L p , 1 &lt; p &lt; ∞ , if and only if the function belongs to BMO. This is a martingale version of a result by Coifman, Rochberg and Weiss.

BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces.

Joan Mateu, Joan Verdera Melenchón (1988)

Revista Matemática Iberoamericana

The spectral synthesis theorem for Sobolev spaces of Hedberg and Wolff [7] has been applied in combination with duality, to problems of Lq approximation by analytic and harmonic functions. In fact, such applications were one of the main motivations to consider spectral synthesis problems in the Sobolev space setting. In this paper we go the opposite way in the context of the BMO-H1 duality: we prove a BMO approximation theorem by harmonic functions and then we apply the ideas in its proof to produce...

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