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57
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 72 (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.
Painlevé’s problem and the semiadditivity of analytic capacity
, 2001
"... Abstract. Let γ(E) be the analytic capacity of a compact set E and let γ+(E) be the capacity of E originated by Cauchy transforms of positive measures. In this paper we prove that γ(E) ≈ γ+(E) with estimates independent of E. As a corollary, we characterize removable singularities for bounded analy ..."
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Cited by 21 (5 self)
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Abstract. Let γ(E) be the analytic capacity of a compact set E and let γ+(E) be the capacity of E originated by Cauchy transforms of positive measures. In this paper we prove that γ(E) ≈ γ+(E) with estimates independent of E. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that γ is semiadditive. 1.
Localized Hardy spaces H 1 related to admissible functions on RDspaces and applications to Schrödinger operators
"... Abstract. Let X be an RDspace, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of lo ..."
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Cited by 8 (6 self)
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Abstract. Let X be an RDspace, which means that X is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first introduce the notion of admissible functions ρ and then develop a theory of localized Hardy spaces H1 ρ(X) associated with ρ, which includes several maximal function characterizations of H1 ρ (X), the relations between H1 ρ (X) and the classical Hardy space H1 (X) via constructing a kernel function related to ρ, the atomic decomposition characterization of H1 ρ(X), and (X) via a finite atomic the boundedness of certain localized singular integrals on H1 ρ decomposition characterization of some dense subspace of H1 ρ (X). This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on Rn, or the subLaplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality. 1
ANALYTICITY OF LAYER POTENTIALS AND L 2 SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR DIVERGENCE FORM ELLIPTIC EQUATIONS WITH COMPLEX L ∞ COEFFICIENTS
, 705
"... Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and tindependent. We show that for such operators, boundedness and invertibility of the corresp ..."
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Cited by 8 (6 self)
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Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and tindependent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L2 (Rn)=L 2 (∂Rn+1 +), is stable under complex, L ∞ perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L2 (Rn) whenever A(x) is real and symmetric (and thus, by our stability result, also when A is complex,‖A − A0‖ ∞ is small enough and A0 is real, symmetric, L ∞ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L2 (resp. ˙L 2 1) data, for small complex perturbations of a real symmetric matrix. Previously, L2 solvability results for complex (or even real but nonsymmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A j,n+1 = 0=An+1, j, 1 ≤ j≤n, which corresponds to the Kato square root problem.
HARDY SPACES ASSOCIATED TO NONNEGATIVE SELFADJOINT OPERATORS SATISFYING DAVIESGAFFNEY ESTIMATES
"... Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characte ..."
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Cited by 7 (0 self)
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Abstract. Let X be a metric space with doubling measure, and L be a nonnegative, selfadjoint operator satisfying DaviesGaffney bounds on L 2 (X). In this article we develop a theory of Hardy and BMO spaces associated to L, including an atomic (or molecular) decomposition, square function characterization, duality of Hardy and BMO spaces. Further specializing to the case that L is a Schrödinger operator on R n with a nonnegative, locally integrable potential, we establish addition characterizations of such Hardy space in terms of maximal functions. Finally, (X) for p> 1, which may or may not coincide with the space L p (X), and show that they interpolate with H 1 L(X) spaces by the complex method. we define Hardy spaces H p L The authors gratefully acknowledge support from NSF as follows: S. Hofmann (DMS
The Riesz kernels do not give rise to higher dimensional analogues of the MengerMelnikov curvature
, 1998
"... Ever since the discovery of the connection between the MengerMelnikov curvature and the Cauchy kernel in the L 2 norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1sets was that any such (nontrivial, nonnegative) ex ..."
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Cited by 6 (1 self)
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Ever since the discovery of the connection between the MengerMelnikov curvature and the Cauchy kernel in the L 2 norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for msets in R n , even in any L k norm (k 2 N), will probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies rectifiability properties of the underlying sets or measures. Answering such questions would also have an impact on another important problem, namely whether totally unrectifiable msets are removable for Lipschitz harmonic functions in R m+1 : It has been generally believed that some such expressions should exist at least for some choices of m; k; or n; but the apparent complexity involved made the search rather difficult, even with the aid of computers. However, our rather surprising ...
Some recent works on multiparameter Hardy space theory and discrete LittlewoodPaley analysis
 In: “Trends in Partial Differential Equations”, ALM 10, High Education Press and International Press (2009), BeijingBoston
"... The main purpose of this paper is to briefly review the earlier works of multiparameter Hardy space theory and boundedness of singular integral operators on such spaces defined on product of Euclidean spaces, and to describe some recent developments in this direction. These recent works include disc ..."
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Cited by 6 (5 self)
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The main purpose of this paper is to briefly review the earlier works of multiparameter Hardy space theory and boundedness of singular integral operators on such spaces defined on product of Euclidean spaces, and to describe some recent developments in this direction. These recent works include discrete multiparameter Calderón reproducing formulas and LittlewoodPaley theory in the framework of product of two homogeneous spaces, product of CarnotCaretheodory spaces, multiparameter structures associated with flag singular integrals and the Zygmund dilation. Using these discrete multiparameter analysis, we are able to establish the theory of multiparameter Hardy spaces associated to the aforementioned multiparameter structures and prove the boundedness of singular integral operators on such Hardy H p spaces and from H p to L p for all 0 < p ≤ 1, and derive the dual spaces of the Hardy spaces. These Hardy spaces are canonical and intrinsic to the underlying structures since they satisfy CalderónZygmund decomposition for functions in such spaces and interpolation properties between them. Proving boundedness of singular integral operators on product Hardy spaces was an extremely difficult task two decades ago. Our method avoids the use of very difficult Journe’s geometric lemma and is a unified approach to the multiparameter theory of Hardy spaces in all aforementioned settings. 1
New properties of Besov and TriebelLizorkin spaces on RDspaces, arXiv: 0903.4583
"... Abstract An RDspace X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first give several equivalent characterizations of RDspaces and show that the definitions of spaces of test fu ..."
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Cited by 5 (5 self)
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Abstract An RDspace X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X. In this paper, the authors first give several equivalent characterizations of RDspaces and show that the definitions of spaces of test functions on X are independent of the choice of the regularity ǫ ∈ (0, 1); as a result of this, the Besov and TriebelLizorkin spaces on X are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and TriebelLizorkin spaces by the norms of homogeneous Besov and TriebelLizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and TriebelLizorkin spaces. 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 5 (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.
CARLESON MEASURES, TREES, EXTRAPOLATION, AND T(b) THEOREMS
, 2002
"... Abstract. The theory of Carleson measures, stopping time arguments, and atomic decompositions has been wellestablished in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The pur ..."
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Cited by 4 (0 self)
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Abstract. The theory of Carleson measures, stopping time arguments, and atomic decompositions has been wellestablished in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation for Carleson measures, as well as a twosided local dyadic T(b) theorem which generalizes earlier T(b) theorems of David, Journe, Semmes, and Christ.