• The $X^s_\theta$ spaces and unique continuation for solutions to the semilinear wave equation, Comm. PDE, 21 (1996), no 5-6, 841-887 The aim of this article is to introduce the variable coefficient version of the $X^s_\theta$ spaces associated to a partial differential operator and to study their properties (interpolation, microlocalization, duality, cannonical representations). The Sobolev-type embeddings of the $X^s_\theta$ spaces into $L^p$ spaces are also obtained here. These embeddings are used to derive $L^p$ Carleman estimates for solutions to the wave equation, which in turn yield unique continuation across strongly pseudoconvex surfaces for the semilinear wave equation ( or for the wave equation with an unbounded potential)
  • Remark on Strichartz type inequalities; Appendix II ( article by S. Klainerman and M. Machedon, Appendix I by J. Bourgain) IMRN 96, 5, pp 201-220 This article contains three different proofs of some sharp multiplicative estimates involving the $X^s_\theta$ spaces associated to the constant coefficient wave equation.
  • On the optimal local regularity for Yang-Mills equations in $R^{4+1}$ (joint with S. Klainerman),  J. Amer. Math. Soc. 12 (1999), no. 1, 93--116.  It is proved here that the Yang-Mills equations in 4+1 dimensions are locally well-posed in $H^{1+\epsilon}$ for any $\epsilon > 0$. Note that in 4+1 dimensions the scale-invariant space for the initial data is $H^1$, which is also the natural energy space. The proof of the result involves an unusual modification of the $X^s_\theta$ spaces. The key ingredient, though, is a sharp new bilinear estimate of Strichartz type for solutions to the wave equation; this is presented in the Appendix. 
  • Strichartz estimates for the wave equation in the hyperbolic space and global existence for the semilinear wave equation ,    Trans. Amer. Math. Soc.  353  (2001),  no. 2, 795--807 The first goal of the article is to obtain the fixed time Strichartz estimates for the wave equation in the hyperbolic space $H^n$. The corresponding space-time estimates follow, with the added bonus that one is allowed to use certain exponential weights in the estimates. These results can be reinterpreted in the context of the Minkovski space $R^{n+1}$ by introducing hyperbolic coordinates in the forward cone $t > |x|$. The corresponding estimates in the Minkovski space have Lorentz invariant weights, i.e. powers of $t^2-x^2$. As a consequence of these results one obtains a simpler proof of Strauss's conjecture on the existence of global solutions to the semilinear wave equation for small compactly supported initial data (proved by Gheorgiev-Lindblad-Sogge, 1996).
  • Local and global results for wave maps I , Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781--1793. It is proved here that the wave-maps equation in $R^{n+1}$ is locally well-posed for initial data in the Besov space $B^{2,1}_{n/2} \times B^{2,1}_{n/2-1}$, $n \geq 4$. Since this Besov space is at the critical level $n/2$, the local result yields a global result for small data in the same space. Furthermore, any additional regularity of the initial data is preserved in time (again, if the initial data is small in the above Besov space).
  • On global existence and scattering for the wave maps equation Amer. J. Math. 123(2001) no. 1, 37-77. This article extends the result in the above one to dimensions 2 and 3, but is considerably more technical. Thanks to Kenji Nakanishi, who found the error in the first manuscript.  The function spaces are now built using frequency localized classical solutions to the inhomogeneous wave equation with respect to characteristic directions.

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  • Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation American Journal of Mathematics 122 (2000), no. 2, 349--376. The aim of this article is threefold. First we use the FBI transform to set up a calculus for partial differential operators with non-smooth coefficients. Secondly, we use this calculus to prove Strichartz type estimates for the wave equation with nonsmooth coefficients. Here we do this in the case when the coefficients are $C^s$ for $0 \leq s \leq 1$. Finally, we use a version of these Strichartz estimates to improve the local theory for second order nonlinear hyperbolic equations.

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  • On the equation $\Box u = |\nabla u|^2$ in $5+1$ dimensions. Math. Res. Letters 6 (1999), no. 5-6, 469--485. The aim of this article is to prove sharp local well-posedness results for the the equation in the title. This problem is more difficult in dimension 5+1 and higher. Then well-posedness should hold up to the scaling level, and the Strichartz estimates and the $X^{s,\theta}$ spaces are insufficient. So, if you are curious about more complicated function spaces related to the wave operator, then this is the place. Availlable in dvi-letter (without pictures) and ps-letter format.

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  • Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients II  Amer. J. Math. 123 (2001), no. 3, 385--423. This is a revised and greatly expanded spin-off of the original article. The main result is that the full Strichartz estimates hold for operators with $C^2$ coefficients. Partial estimates are then proved for operators with $C^s$ coefficients for $1 \leq s \leq 2$.

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  • Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III   J. Amer. Math. Soc.  15  (2002),  no. 2, 419--442 This article improves the results in the previous article. It is proved here that the full Strichartz estimates hold for second order hyperbolic operators whose coefficients have two derivatives in $L^1(L^\infty)$. Partial estimates are obtain if the coefficients have less than two derivatives in the same space. This result also leads to further improvement for the local theory for nonlinear hyperbolic equations. Availlable in dvi-letter and ps-letter format.

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  • Strichartz estimates for a Schrodinger operator with nonsmooth coefficients (joint with Gigliola Staffilani)   Comm. Partial Differential Equations  27  (2002),  no. 7-8, 1337--1372. Here we begin the study of dispersive estimates for variable coefficient Schroedinger operators.  For now we confine ourselves to compactly supported  $C^2$ perturbations of the identity subject to a nontrapping condition. An essential role in our approach is played by the local smoothing effect, which allows us to  properly localize the problem.  Availlable in dvi-letter and ps-letter format.


    • (March 2001, revised April 2002)

  • Null form estimates for second order hyperbolic  operators with rough coefficients  Here we prove certain $L^p_t L^q_x$ null form estimates for second order hyperbolic equations with rough coefficients. This extends some earlier L^p results of Wolff and Tao for the constant coefficient case, and also some L^2 null form estimates for operators with rough coefficients due to Smith and Sogge.  Also it proves most of (the non-endpoint part of) a conjecture of Foschi and Klainerman. Availlable in dvi-letter and ps-letter format.


    • (Oct. 2001, revised Feb. 2002)

  •  Sharp local well-posedness results for the nonlinear wave equation (joint with Hart Smith) We prove that the Cauchy problem  for generic nonlinear hyperbolic equations is locally well-posed for initial data in $H^s \times H^{s-1}$ provided that s> n/2+ 3/4  for n=2  respectively s > (n+1)/2 for  n > 2. This result is sharp in dimensions n=2,3. Availlable in dvi-letter and ps-letter format.


    • (Oct. 2001)

  • Sharp counterexamples for Strichartz estimates for low regularity metrics.(joint with Hart Smith) We show that the Strichartz estimates for second order hperbolic equations with rough coefficients  proved in the papers (II) (III) above are sharp. Availlable in dvi-letter and ps-letter format.


    • (April 2003)

  • Rough solutions for the Wave-Maps equation  We consider the wave maps equation  with values into a Riemannian manifold which is isometrically embedded in $\R^m$. The main result asserts that the Cauchy problem is globally well-posed for initial data which is small in the critical Sobolev spaces. This extends and completes recent work of Tao and other authors. Availlable in pdf  and ps-letter format.

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  •  The wave maps equation . These are some notes which I wrote last winter for the AMS meeting in Baltimore. If you found the previous paper too long and technical, then maybe this one is more useful. Availlable in pdf  and ps-letter format.

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  •   Dispersive estimates for wave equations with rough coefficients. (joint with Dan Geba ) In this article we obtain a multiscale wave packet representation for the fundamental solution of the wave equation with rough coefficients. This leads to pointwise and weighted Lp bounds on the fundamental solution and also to a proof of dispersive estimates for such operators. Availlable in pdf   and ps-letter format.

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  •   Parametrices and dispersive estimates for Schroedinger operators with variable coefficients This is about variable coefficient time dependent Schr\"odinger evolutions in $\R^n$. Using phase space methods we construct outgoing parametrices and to prove global in time Strichartz type estimates. This is done in the context of $C^2$ metrics which satisfy a weak asymptotic flatness condition at infinity. Availlable in pdf