Results 1  10
of
36
Uniform estimates for multilinear operators with one dimensional modulation symmetry
, 2001
"... In a previous paper [20] in this series, we gave L p estimates for multilinear operators given by multipliers which are singular on a nondegenerate subspace of some dimension k. In this paper we give uniform estimates when the subspace approaches a degenerate region in the case k = 1, and when al ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
(Show Context)
In a previous paper [20] in this series, we gave L p estimates for multilinear operators given by multipliers which are singular on a nondegenerate subspace of some dimension k. In this paper we give uniform estimates when the subspace approaches a degenerate region in the case k = 1, and when all the exponents p are between 2 and ∞. In particular we recover the nonendpoint uniform estimates for the Bilinear Hilbert transform in [12].
L p estimates for the biest II. The Fourier case
"... Abstract. We prove L p estimates (Theorem 1.2) for the “biest”, a trilinear multiplier operator with singular symbol. The methods used are based on the treatment of the Walsh analogue of the biest in the prequel [16] of this paper, but with additional technicalities due to the fact that in the Fouri ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
(Show Context)
Abstract. We prove L p estimates (Theorem 1.2) for the “biest”, a trilinear multiplier operator with singular symbol. The methods used are based on the treatment of the Walsh analogue of the biest in the prequel [16] of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.
On multilinear singular integrals of CalderónZygmund type
, 2011
"... A variety of results regarding multilinear CalderónZygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discret ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
A variety of results regarding multilinear CalderónZygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur’s test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain weighted norm inequalities.
The disc as a bilinear multiplier
 Amer. J. Math
"... Abstract. A classical theorem of C. Fefferman [3] says that the characteristic function of the unit disc is not a Fourier multiplier on L p (R 2) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit di ..."
Abstract

Cited by 16 (10 self)
 Add to MetaCart
(Show Context)
Abstract. A classical theorem of C. Fefferman [3] says that the characteristic function of the unit disc is not a Fourier multiplier on L p (R 2) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit disc in R 2 is the Fourier multiplier of a bounded bilinear operator from L p1 p2 p p1p2 (R) × L (R) intoL(R), when 2 ≤ p1,p2 < ∞ and 1 <p = ≤ 2. The proof p1+p2 of this result is based on a new decomposition of the unit disc and delicate orthogonality and combinatorial arguments. This result implies norm convergence of bilinear Fourier series and strengthens the uniform boundedness of the bilinear Hilbert transforms, as it yields uniform vectorvalued bounds for families of bilinear Hilbert transforms. 1.
Maximal theorems for the directional Hilbert transform on the plane
 electronic). MR 2219012
"... Abstract. For a Schwartz function f on the plane and a nonzero v ∈ R2 define the Hilbert transform of f in the direction v to be Hv f(x) = p.v. f(x − vy) dy y Let ζ be a Schwartz function with frequency support in the annulus 1 ≤ ξ  ≤ 2, and ζf = ζ ∗ f. We prove that the maximal operator sup v ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
(Show Context)
Abstract. For a Schwartz function f on the plane and a nonzero v ∈ R2 define the Hilbert transform of f in the direction v to be Hv f(x) = p.v. f(x − vy) dy y Let ζ be a Schwartz function with frequency support in the annulus 1 ≤ ξ  ≤ 2, and ζf = ζ ∗ f. We prove that the maximal operator sup v=1Hv ζf  maps L 2 into weak L 2, and L p into L p for p> 2. The L 2 estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series. 1.
NEW UNIFORM BOUNDS FOR A WALSH MODEL OF THE BILINEAR HILBERT TRANSFORM
"... The notion of bilinear Hilbert transform usually refers to a member of a family of bilinear operators parameterized by a unit vector β perpendicular to (1, 1, 1). We will write the bilinear operators in this family more symmetrically as dual trilinear forms Λβ, acting on three ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
The notion of bilinear Hilbert transform usually refers to a member of a family of bilinear operators parameterized by a unit vector β perpendicular to (1, 1, 1). We will write the bilinear operators in this family more symmetrically as dual trilinear forms Λβ, acting on three
Local estimates and global continuities in Lebesgue spaces for bilinear operators
, 2008
"... In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of “offdiagonal ” d ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
In this paper, we first prove some local estimates for bilinear operators (closely related to the bilinear Hilbert transform and similar singular operators) with truncated symbol. Such estimates, in accordance with the Heisenberg uncertainty principle correspond to a description of “offdiagonal ” decay. In addition they allow us to prove global continuities in Lebesgue spaces for bilinear operators with spatial dependent symbol.
L p estimates for the biest I. The Walsh case
"... Abstract. We prove L p estimates (Theorem 1.8) for the Walsh model of the “biest”, a trilinear multiplier with singular symbol. The corresponding estimates for the Fourier model will be obtained in the sequel [15] of this paper. 1. introduction The bilinear Hilbert transform can be written (modulo m ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We prove L p estimates (Theorem 1.8) for the Walsh model of the “biest”, a trilinear multiplier with singular symbol. The corresponding estimates for the Fourier model will be obtained in the sequel [15] of this paper. 1. introduction The bilinear Hilbert transform can be written (modulo minor modifications) as
Modulation invariant bilinear T(1) theorem
 J. Anal. Math
"... Abstract. We prove a T(1) theorem for bilinear singular integral operators (trilinear forms) with a onedimensional modulation symmetry. 1. ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We prove a T(1) theorem for bilinear singular integral operators (trilinear forms) with a onedimensional modulation symmetry. 1.
BILINEAR MULTIPLIERS ON LORENTZ SPACES
, 710
"... Abstract. 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. 1. Introduction. The bilinear Hilbert transform with parameter α ∈ R is the oper ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract. 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. 1. Introduction. The bilinear Hilbert transform with parameter α ∈ R is the operator given by Hα(f, g)(x) = 1