Results 1  10
of
29
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].
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.
Nahmod A., Boundedness of bilinear operators with nonsmooth symbols
, 2000
"... Abstract. We announce the L pboundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies eal ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We announce the L pboundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of CoifmanMeyer for smooth multipliers and ones, such the Bilinear Hilbert transform of LaceyThiele, where the multiplier is not smooth.
ON THE HÖRMANDER CLASSES OF BILINEAR PSEUDODIFFERENTIAL OPERATORS
, 2010
"... Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition a ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear CalderónZygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well.
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.
ON THE BOUNDEDNESS OF BILINEAR OPERATORS ON PRODUCTS OF BESOV AND LEBESGUE SPACES
"... Abstract. We prove mapping properties of the form T: B ˙ α1,q1 p1 × L p2 → B ˙ α2,q2 p3 and T: B ˙ α1,q1 p1 × ˙ B α2,q2 p2 → Lp3, for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear HörmanderMihlin multiplier or a molecular paraproduct. Applications to bilinear Little ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We prove mapping properties of the form T: B ˙ α1,q1 p1 × L p2 → B ˙ α2,q2 p3 and T: B ˙ α1,q1 p1 × ˙ B α2,q2 p2 → Lp3, for certain related indices p1, p2, p3, q1, q2, α1, α2 ∈ R, where T is a bilinear HörmanderMihlin multiplier or a molecular paraproduct. Applications to bilinear LittlewoodPaley theory are discussed. 1.
TRANSFERENCE OF BILINEAR MULTIPLIER OPERATORS On Lorentz Spaces
, 2007
"... Let m(ξ, η) be a bounded continuous function in IR × IR, 0 < pi, qi < ∞ for i = 1,2 and 0 < p3, q3 ≤ ∞ where 1/p1+1/p2 = 1/p3. It is shown that ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Let m(ξ, η) be a bounded continuous function in IR × IR, 0 < pi, qi < ∞ for i = 1,2 and 0 < p3, q3 ≤ ∞ where 1/p1+1/p2 = 1/p3. It is shown that
BILINEAR MULTIPLIERS AND TRANSFERENCE
, 2005
"... We give de Leeuwtype transference theorems for bilinear multipliers. In particular, it is shown that bilinear multipliers arising from regulated functions m(ξ,η)inR × R can be transferred to bilinear multipliers acting on T × T and Z × Z. The results follow from the description of bilinear multipli ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
We give de Leeuwtype transference theorems for bilinear multipliers. In particular, it is shown that bilinear multipliers arising from regulated functions m(ξ,η)inR × R can be transferred to bilinear multipliers acting on T × T and Z × Z. The results follow from the description of bilinear multipliers on the discrete real line acting on L pspaces.