Abstract. We prove old and new L bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that can be expected from known bounds in the degenerate and non-degenerate cases. For the new estimates with exponents p close to 1 the argument relies on ...

C. Muscalu, J. Pipher, T. Tao and C. Thiele proved in [27] that the standard bilinear and bi-parameter Hilbert transform does not satisfy any L estimates. They also raised a question asking if a bilinear and bi-parameter multiplier operator defined by Tm(f1, f2)(x) := ∫ R m(ξ, η)f̂1(ξ1, η1)f̂2(ξ2, η2)e 1122dξdη satisfies any L estimates, where the symbol m satisfies |∂ ξ ∂ ηm(ξ, η)| . 1 dist(ξ,Γ1...

Let L = −∆+ V be a Schrödinger operator on R, d ≥ 3, where V is a nonnegative function, V 6= 0, and belongs to the reverse Hölder class RHd/2. The purpose of this paper is three-fold. First, we prove a version of the classical theorem of Jones and Journé on weak∗-convergence in H L(R ). Secondly, we give a bilinear decomposition for the product space H L(R )×BMOL(R). Finally, we study the commu...

The aim of this paper is to establish some a priori bounds for solutions of Landesman-Lazer problem. We show the application for the solution structure of the nonlinear diierential equation of the fourth order 1. The general theory Let X be a real Banach space with the norm k k and let D(L) X be the domain of the closed Fredholm operator L : D(L) ! X with index zero. We shall suppose that 0 is ...

In joint work with C. Thiele, the author has shown that A. Calderón’s bilinear Hilbert transform extends to a bounded operator on certain products of L spaces. This article illustrates the method of proof by giving a complete proof of an inequality which is slightly weaker than the original conjecture of Calderón concerning this transform. 1991 Mathematics Subject Classification: 42A50

The bilinear maximal operator de ned below maps L L into L provided 1 < p; q <1, 1=p+ 1=q = 1=r and 2=3 < r 1. Mfg(x) = sup t>0 1 2t Z t t jf(x+ y)g(x y)j dy In particular Mfg is integrable(!) if f and, g are square integrable, answering a conjecture posed by Alberto Calder on. 1 Principal Results In 1964 Alberto Calder on de ned the maximal operator Mfg(x) = sup t>0 1 2t Z t t jf(x y)g(x y)j d...

We present the main ideas of the proof of the following result: The characteristic function of the unit disc in R is the symbol of a bounded bilinear multiplier operator from L1(R) × L2(R) into L(R) when 2 ≤ p1, p2 < ∞ and 1 < p = p1p2 p1+p2 ≤ 2.