Abstract. We prove the existence of equivariant finite time blow-up solutions for the wave map problem from R 2+1 → S 2 of the form u(t, r) = Q(λ(t)r) + R(t, r) where u is the polar angle on the sphere, Q(r) = 2 arctan r is the ground state harmonic map, λ(t) = t −1−ν , and R(t, r) is a radiative error with local energy going to zero as t → 0. The number ν > 1 2 can be described arbitrarily. This is accomplished by first "renormalizing" the blow-up profile, followed by a perturbative analysis.

Abstract. Given ν > 1 2 and δ > 0 arbitrary, we prove the existence of energy solutions ofin R 3+1 that blow up exactly at r = t = 0 as t → 0−. These solutions are radial and of the form u = λ(t)2 is the stationary solution of (0.1), and η is a radiation term with ZOutside of the light-cone there is the energy bound Zfor all small t > 0. The regularity of u increases with ν. As in our accompanying paper on wave-maps [10], the argument is based on a renormalization method for the 'soliton profile' W (r).

|V (y)| |x − y| dy < 4π.We also establish the dispersive estimate with an ε-loss for large energies provided V K + V 2 < ∞. Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x| −2−ε in dimensions n ≥ 3, thus solving an open problem posed by Journé, Soffer, and Sogge.

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Classical and Multilinear Harmonic Analysis This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and is intended for graduates and researchers in pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. The first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transforms. Then the higher-dimensional Calderón-Zygmund and Littlewood-Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. This second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman-Meyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not been collected previously in book form.

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Our approach is very different from both [JSS] and [Yaj1]. Journé, Soffer, and Sogge use a timedependent method and expand the evolution repeatedly by means of Duhamel's formula. For large energies the smallness needed to control the evolution e itH appearing on the right-hand side of such an expansion is obtained from Kato's smoothing estimate. For small energies they use the expansion of the resolvent around zero energy from [JenKat]. Since their method relies on the integrability of t − d 2 at infinity, it can only be used in dimensions d ≥ 3 and it also requires more regularity of V (V ∈ L 1 is a natural assumption for their proof). Yajima [Yaj1] uses the stationary approach of Kato [Kato] to bound the wave operators on L p . While his result is more general (it yields many more corollaries than just dispersive estimates), our approach to (2) is direct and also requires less of V . The one-dimensional case was open until recently. Weder [Wed1] proved a version of Theorem 1 under the stronger assumption that, and also Artbazar, Yajima [ArtYaj] established corresponding theorems for the wave-operators. More precisely, they showed that the wave operators are bounded on L p (R) provided 1 < p < ∞ under similar assumptions on V . While our analysis is in some ways similar to Weder's [Wed1], it turns out that the high energy case can be treated more easily by means of a Born series expansion, whereas small energies fall under the framework of the Jost solutions as developed in the fundamental paper by Deift, Trubowitz [DeiTru]. The latter was also observed by Weder, but there is no need to impose any stronger condition on V other than the one used in [DeiTru], i.e., V ∈ L 1 1 (R). Dispersive estimates in two dimensions are unknown in this degree of generality. Yajima [Yaj2] established the L p (R 2 ) boundedness of the wave operators under suitable assumptions on the decay of V as well as the behavior of the Hamiltonian at zero energy. Since his result requires that 1 < p < ∞, it does not imply the L 1 (R 2 ) → L ∞ (R 2 ) decay bounds for e itH P ac , although it does imply the Strichartz estimates. We claim that our three-dimensional argument can be adapted to two dimensions as well, since it does not require integrability of t −1 at infinity (unlike, say, [JSS]). Generally speaking, we expect the argument to apply to any dimension (in d = 1, however, we use a different strategy which yields sharper results). For small energies we use expansions of the perturbed resolvent around zero energy. These were unknown in R 2 for some time, but were recently obtained by Jensen, Nenciu [JenNen], whereas dimensions three and higher were treated by Jensen, Kato [JenKat] and Jensen [Jen1], [Jen2]. We plan to present appropriate versions of Theorem 2 in dimensions two, or four and higher, elsewhere.An interesting issue in Theorems 1 and 2 is the question of optimality. The decay rate of (1 + |x|) −2−ε appears to be a natural threshold for the dispersive estimates, and Theorem 1 achieves this rate. But we do not know at t...

We consider radially symmetric, energy critical wave maps from (1 + 2)-dimensional Minkowski space into the unit sphere S m , m ≥ 1, and prove global regularity and scattering for classical smooth data of finite energy. In addition, we establish a priori bounds on a suitable scattering norm of the radial wave maps and exhibit concentration compactness properties of sequences of radial wave maps with uniformly bounded energies. This extends and complements the beautiful classical work of Christodoulou-Tahvildar-Zadeh [3, 4] and Struwe [31, 33] as well as of Nahas [22] on radial wave maps in the case of the unit sphere as the target. The proof is based upon the concentration compactness/rigidity method of Kenig-Merle [6, 7] and a "twisted" Bahouri-Gérard type profile decomposition [1], following the implementation of this strategy by the second author and Schlag [17] for energy critical wave maps into the hyperbolic plane as well as by the last two authors [16] for the energy critical Maxwell-Klein-Gordon equation.