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18
On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation
 preprint. TERENCE TAO Department of Mathematics, UCLA, Los Angeles CA 900951555 Email address: tao@math.ucla.edu
"... Abstract. We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation iut+∆u = −u  2 u in R 3, assuming globally bounded H 1 (R 3) norm (i.e. no blowup in the energy space). We show that as t → ±∞, these solutions split into the sum of three terms: a radiat ..."
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Cited by 25 (12 self)
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Abstract. We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation iut+∆u = −u  2 u in R 3, assuming globally bounded H 1 (R 3) norm (i.e. no blowup in the energy space). We show that as t → ±∞, these solutions split into the sum of three terms: a radiation term that evolves according to the linear Schrödinger equation, a smooth function localized near the origin, and an error that goes to zero in the ˙ H 1 (R 3) norm. Furthermore, the smooth function near the origin is either zero (in which case one has scattering to a free solution), or has mass and energy bounded strictly away from zero, and obeys an asymptotic Pohozaev identity. These results are consistent with the conjecture of soliton resolution. 1.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
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.
Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
 Commun. Pure Appl. Anal
"... Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move away f ..."
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Cited by 8 (3 self)
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Abstract. We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing nonlinear Schrödinger equations in the H s (R n) norm for 1 − ε < s < 1, for small ε. In the L 2subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the H 1subcritical case then we cannot show this, but for defocussing equations we obtain global wellposedness and polynomial growth of H s norms for s sufficiently close to 1. 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 (ddimens ..."
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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.
Haar Multipliers, Paraproducts and Weighted Inequalities
, 1998
"... In this paper we present a brief survey on Haar multipliers, dyadic paraproducts, and recent results on their applications to deduce scalar and vector valued weighted inequalities. We present a new proof of the boundedness of a Haar multiplier in L p (R). The proof is based on a stopping time argu ..."
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Cited by 4 (3 self)
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In this paper we present a brief survey on Haar multipliers, dyadic paraproducts, and recent results on their applications to deduce scalar and vector valued weighted inequalities. We present a new proof of the boundedness of a Haar multiplier in L p (R). The proof is based on a stopping time argument suggested by P. W. Jones for the case p = 2, that it is adapted to the case 1 ! p ! 1 using an new version of Cotlar's Lemma for L p . We then prove some weighted inequalities for simple dyadic operators. 1 Introduction A Haar multiplier is an operator of the form: Tf(x) = X I2D ! I (x)hf; h I ih I (x); where the sum runs over the dyadic intervals D = f(k2 \Gammaj ; (k+1)2 \Gammaj ] : k; j 2 Zg; h I is the Haar function associated to I ; h:; :i denotes the L 2 inner product; and finally the symbol ! I (x) is a function of both the variables x 2 R and I 2 D. These operators are formally similar to pseudodifferential operators, but the trigonometric functions have been repl...
IMAGINARY POWERS OF LAPLACE OPERATORS
"... Abstract. We show that if L is a secondorder uniformly elliptic operator in divergence form on R d, then C1(1 + α) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + α) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ ..."
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Cited by 4 (2 self)
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Abstract. We show that if L is a secondorder uniformly elliptic operator in divergence form on R d, then C1(1 + α) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + α) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ C ∞ (R d), aij = aji for 1 ≤ i, j ≤ d and that κI ≤ (aij) ≤ τI for some positive constants κ and τ. We define a positive selfadjoint operator L on L 2 (R d) by the formula (1) L = − ∑ ∂iaij∂j. We refer readers to [8] for the precise definition and basic properties of L. In particular, L
Extensions Of The Heisenberg Group By Dilations And Frames
, 1995
"... . Two standard tools for signal analysis are the shorttime Fourier transform and the continuous wavelet transform. These tools arise as matrix coefficients of square integrable representations of the Heisenberg and affine groups respectively, and discrete frame decompositions of L 2 arise from a ..."
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Cited by 4 (2 self)
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. Two standard tools for signal analysis are the shorttime Fourier transform and the continuous wavelet transform. These tools arise as matrix coefficients of square integrable representations of the Heisenberg and affine groups respectively, and discrete frame decompositions of L 2 arise from approximations of corresponding reproducing formulae. Here we study two groups, the socalled affine WeylHeisenberg and upper triangular groups, which contain both affine and Heisenberg subgroups. Generalized notions of squareintegrable group representations allow us to fashion frames for L 2 and other function spaces. Such frames combine advantages of the shorttime Fourier transform and wavelet transform and can be tailored to analyze specific types of signals. 1 Research supported by an Australian Research Council grant. 2 Research supported by NSF contract DMS9307655. Typeset by A M ST E X 1. INTRODUCTION. Wavelet and Gabor techniques have played a prominent role in both signa...
Ramified Optimal Transportation in Geodesic Metric Spaces
 Advances in Calculus of Variations
"... Abstract. An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a treeshaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the ..."
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Cited by 3 (2 self)
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Abstract. An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a treeshaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called “the dimensional distance”, on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure. The optimal transportation problem aims at finding an optimal way to transport a given measure into another with the same mass. In contrast to the wellknown MongeKantorovich problem (e.g. [1], [6], [7], [14], [15], [18], [20], [22]), the ramified optimal transportation problem aims at modeling a branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to favor transportation in groups via a nonlinear (typically concave) cost function on mass. Transport networks with branching structures are observable not only in nature as in trees, blood vessels, river channel networks, lightning, etc. but also in efficiently designed transport systems such as used in railway configurations and postage delivery networks. Several different approaches have been done on the ramified optimal transportation problem in Euclidean
On The Resolvent Of The Dyadic Paraproduct, And A Nonlinear Operation On RH p Weights
"... . The existence of a bounded inverse of (I \Gamma \Pi b ) on L p (\Pi b is the dyadic paraproduct) does not imply the same for (I \Gamma \Pi b ), \Gamma1 ! 1 (we present a counterexample); but it guarantees the existence of 1 ! po such that there exist a bounded inverse in L po for every \Gamma ..."
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Cited by 1 (0 self)
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. The existence of a bounded inverse of (I \Gamma \Pi b ) on L p (\Pi b is the dyadic paraproduct) does not imply the same for (I \Gamma \Pi b ), \Gamma1 ! 1 (we present a counterexample); but it guarantees the existence of 1 ! po such that there exist a bounded inverse in L po for every \Gamma1 1. This is equivalent to showing that the RH d p class of weights is not preserved under certain nonlinear operation involving , but if ! 2 RH d p then there exists 1 ! po such that the transformed weight ! 2 RH d po for all \Gamma1 1. 1. Introduction The necessary and sufficient conditions for the existence of a solution f 2 L p (R) of the equation: (I \Gamma \Pi b )f = g; kfk p Ckgk p ; (1) are known (see [8]). Here I is the identity operator, \Pi b is the dyadic paraproduct (see (3) for the precise definition), associated to the function b of bounded mean oscillation (BMO), and g is a function in L p (R). We are interested in studying this equation under a simple perturb...