Results 1  10
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62
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 75 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Multilinear Calderón Zygmund theory
 ADV. IN MATH. 40
, 1996
"... A systematic treatment of multilinear CalderónZygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators. ..."
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Cited by 45 (16 self)
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A systematic treatment of multilinear CalderónZygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators.
Uniform bounds for the bilinear Hilbert transforms
 889–993. MR2113017 (2006e:42011), Zbl 1071.44004. Xiaochun Li
, 2004
"... Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying  α β − 1  ..."
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Cited by 28 (15 self)
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Abstract. We continue the investigation initiated in [8] of uniform Lp bounds � for the family of bilinear Hilbert transforms Hα,β(f, g)(x) = p.v. f(x − αt)g(x − βt) R dt t. In this work we show that Hα,β map Lp1 (R) × Lp2 (R) into Lp (R) uniformly in the real parameters α, β satisfying  α β − 1  ≥ c> 0 when 1 < p1, p2 < 2 and 2 p1p2 3 < p = < ∞. p1+p2 As a corollary we obtain Lp × L ∞ → Lp uniform bounds in the range 4/3 < p < 4 for the H1,α’s when α ∈ [0, 1). 1.
The Marcinkiewicz multiplier condition for bilinear operators
 Studia Math. 146 (2001), 115–156. LOUKAS GRAFAKOS
"... Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions ..."
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Cited by 23 (7 self)
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Abstract. This article is concerned with the question of whether Marcinkiewicz multipliers on R2n give rise to bilinear multipliers on Rn × Rn.We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces. 1.
Continuous wavelets and frames on stratified Lie groups I
 Journal of Fourier Analysis and Applications
, 2006
"... Let G be a stratified Lie group and L be the subLaplacian on G. Let 0 ̸ = f ∈ S(R +). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies ’ criterion, then Lf(L)δ generates a frame for any sufficiently ..."
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Cited by 10 (7 self)
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Let G be a stratified Lie group and L be the subLaplacian on G. Let 0 ̸ = f ∈ S(R +). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies ’ criterion, then Lf(L)δ generates a frame for any sufficiently fine lattice subgroup of G.
Discrete decompositions for bilinear operators and almost diagonal conditions
 TRANS. AMER. MATH. SOC
, 1998
"... Using discrete decomposition techniques,bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This ..."
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Cited by 10 (6 self)
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Using discrete decomposition techniques,bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of CalderónZygmund type. Applications include a reduced T 1 theorem for bilinear pseudodifferential operators and the extension of an L p multiplier result of Coifman and Meyer to the full range of H p spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal estimate of Fefferman and Stein.
Nearly Tight Frames and SpaceFrequency Analysis
 on Compact Manifolds, Math. Z
"... Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the LaplaceBeltrami operator on M. Say 0 ̸ = f ∈ S(R +), and that f(0) = 0. For t> 0, let Kt(x, y) denote the kernel of f(t 2 ∆). Suppose f satisfies Daubechies ’ criterion, and b> 0. For each j, write M as a disjoint union ..."
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Cited by 6 (4 self)
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Let M be a smooth compact oriented Riemannian manifold, and let ∆ be the LaplaceBeltrami operator on M. Say 0 ̸ = f ∈ S(R +), and that f(0) = 0. For t> 0, let Kt(x, y) denote the kernel of f(t 2 ∆). Suppose f satisfies Daubechies ’ criterion, and b> 0. For each j, write M as a disjoint union of measurable sets Ej,k with diameter at most ba j, and comparable to ba j if ba j is sufficiently small. Take xj,k ∈ Ej,k. We then show that the functions φj,k(x) = [µ(Ej,k)] 1/2 Kaj (xj,k, x) form a frame for (I − P)L 2 (M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how welllocalized a function F ∈ L 2 is in space and in frequency, we can describe which terms in the summation F ∼ SF = P j P k 〈F, φj,k〉φj,k are so small that they can be neglected. Finally we explain in what sense the kernel Kt(x, y) should itself be regarded as a continuous wavelet on M, and characterize the Hölder continuous functions on M by the size of their continuous wavelet transforms, for Hölder exponents strictly between 0 and 1.
Singular integrals on Sierpinski gaskets, Publ
 Mat
"... Abstract. We construct a class of singular integral operators associated with homogeneous CalderónZygmund standard kernels on ddimensional, d < 1, Sierpinski gaskets Ed. These operators are bounded in L2 (µd) and their principal values diverge µd almost everywhere, where µd is the natural (ddim ..."
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Cited by 6 (5 self)
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Abstract. We construct a class of singular integral operators associated with homogeneous CalderónZygmund standard kernels on ddimensional, d < 1, Sierpinski gaskets Ed. These operators are bounded in L2 (µd) and their principal values diverge µd almost everywhere, where µd is the natural (ddimensional) measure on Ed. 1.
VectorValued Riesz Potentials: Cartan Type Estimates and Related Capacities. ↑2
"... Abstract. Our aim is to give sharp upper bounds for the size of the set of points where the Riesz transform of a linear combination of N point masses is large. This size will be measured by the Hausdorff content with various gauge functions. Among other things, we shall characterize all gauge functi ..."
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Cited by 6 (2 self)
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Abstract. Our aim is to give sharp upper bounds for the size of the set of points where the Riesz transform of a linear combination of N point masses is large. This size will be measured by the Hausdorff content with various gauge functions. Among other things, we shall characterize all gauge functions for which the estimates do not blow up as N tends to infinity (in this case a routine limiting argument will allow us to extend our bounds to all finite Borel measures). We also show how our techniques can be applied to estimates for certain capacities. 1.
Linear bound for the dyadic paraproduct on weighted Lebesgue space L2
 w). J. Func. Anal
"... The dyadic paraproduct is bounded in weighted Lebesgue spaces Lp(w) if and only if the weight w belongs to the Muckenhoupt class A d p. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L2(w) case. In this paper we prove the linear bound on the norm o ..."
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Cited by 5 (1 self)
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The dyadic paraproduct is bounded in weighted Lebesgue spaces Lp(w) if and only if the weight w belongs to the Muckenhoupt class A d p. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L2(w) case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L2(w) using Bellman function techniques and extrapolate this result to the Lp(w) case. 1 1