What's new:
What was new in 2001?
What was new in 2000?
What was new in 1999?
Dec 30, 2002
Dec 17, 2002

Withdrawn: "On
a problem by Arens, Goldberg, and Luxemburg", joint with Raymond Redheffer.
We have discovered that our main result was proven (by different methods,
and for a different algebra) by Goldberg last year, and so we are withdrawing
it from publication, although I will keep it available from my web page.

Updated: The links for the Analysis Seminar and
the Participating Analysis Seminar,
now displaying the schedule for the Winter quarter. Note that the
Participating Analysis Seminar has moved to MS 6201.
Dec 13, 2002
Dec 11, 2002

Uploaded: "L^p
bounds for a maximal dyadic sum operator", joint with Loukas
Grafakos and Erin Terwilleger.
In this note we observe that the LaceyThiele method of using wave packets
and combinatorics of tiles in the phase plane to prove Carleson's theorem
on convergence of Fourier series for L^2 also works for any L^p, 1 <
p < infinity, after a standard CalderonZygmund decomposition.
Actually, we do not bound Carleson's operator directly but proceed (as
with the LaceyThiele paper) by first considering a discretized model sum
operator, from which Carleson's operator can be recovered by a standard
averaging argument.

Linked: Six java applets, written by Kim
Chi Tran under my supervision, for Math
115A (linear algebra).
Dec 9, 2002

Added: Short program on Analysis and Resolution of Singularities (CRM Montreal,
Canada, Aug 18Sep 5)
Dec 6, 2002

Uploaded: "Polynomial
growth and orbital instability bounds for $L^2$subcritical NLS below the
energy norm", joint with Jim
Colliander, Mark
Keel, Gigliola
Staffilani, and Hideo Takaoka, submitted to Communications
on Pure and Applied Analysis. This is the sequel to "Polynomial
upper bounds for the orbital instability of the 1D cubic NLS below the
energy norm", but now we can obtain global wellposedness and orbital
instability bounds for certain H^s, s < 1 for all L^2subcritical nonlinear
Schrodinger powers. The main new technical innovation in this paper
is the extension of the "Imethod" to nonalgebraic nonlinearities (in
which the multifrequency games involving X^{s,b} spaces do not seem to
be available). In particular, we require an "Ifractional chain rule"
and a KatoPonce type "Icommutator estimate".

Uploaded: "On a
problem by Arens, Goldberg, and Luxemburg", joint with Raymond Redheffer.
In this short paper we give an example of a normed algebra which is 2bounded
(i.e.  A^2  <=  A ^2 for all A) but not 3bounded (i.e.  A^3
 is not always bounded by  A ^3). The idea is to take the unit
ball of a Banach algebra which is only barely multiplicative, and strategically
shave small pieces off of it to disrupt the 3boundedness property but
not the 2boundedness property.

Updated: "L^p improving
estimates for averages along curves", joint with Jim
Wright. This is the final version that has been accepted to JAMS.
The main new development is that we have simplified the argument somewhat,
removing any need for the complexity theory of semialgebraic sets, and
instead using the much more elementary complexity theory of onedimensional
polynomials of degree n (and in particular, using the basic fact that the
sublevel sets have at most O(n) connected components).
Nov 25, 2002
Nov 22, 2002
Nov 20, 2002
Nov 15, 2002

Added: 22nd Annual Western States Mathematical Physics Meeting (Feb 1718,
Caltech)
Nov 12, 2002
Nov 8, 2002
Oct 30, 2002
Oct 27, 2002
Oct 25, 2002
Oct 11, 2002
Oct 9, 2002
Oct 7, 2002

Uploaded: "A sharp
bilinear restriction estimate for paraboloids", submitted to GAFA.
The purpose of this paper is to extend Wolff's sharp bilinear cone restriction
estimate to paraboloids, and as a byproduct gain some progress on the
restriction conjecture for paraboloids and spheres. (Presumably there
is some application to BochnerRiesz problems as well). The argument
follows closely that of Wolff, and has all the same basic ingredients:
induction on scales, wave packet decomposition (in the spirit of Fefferman
and Cordoba, and later of Bourgain), local L^2 estimates (following some
ideas of Mockenhaupt), and the geometric exploitation of transversality.
The reason why Wolff's argument was initially restricted to the cone was
that it relied on the following geometric fact: if one takes all the light
rays through a point (i.e. a light cone), then any other light ray can
only intersect this light cone transversally in at most one point.
In the paraboloid case, light rays must be replaced by lines in arbitrary
direction, and this geometric fact fails. However, this can be rectified
by strengthening the local L^2 portion of the argument, in the spirit of
Bourgain and of Moyua, Vargas, and Vega, noting that locally one does not
need to control the multiplicity of all lines through a point, but only
those through a hypersurface (which in the paraboloid case happens to be
a hyperplane).
Oct 2, 2002
Sep 25, 2002

Uploaded: "A singularity
removal theorem for YangMills fields in higher dimensions", joint
with Gang Tian, submitted
to J. Amer. Math. Soc.. In
this paper we generalize a fourdimensional singularity removal theorem
of Uhlenbeck to higher dimensions. In four dimensions the theorem
states that any weakly YangMills field (the weak limit of smooth YangMills
fields) with sufficiently small energy (defined as the L^2 norm of the
curvature) must be smooth (after gauge transformation). (In three
and fewer dimensions the problem is subcritical, and smoothness is automatic;
the small energy condition is necessary due to instanton examples).
The fourdimensional case is critical  the energy is scale invariant 
and so the argument does not extend directly to the supercritical higherdimensional
cases. Nevertheless, we can still obtain regularity if the energy
(in a unit cube, say) is sufficiently small. There are several ingredients.
The first is Price's monotonicity formula, which bootstraps the L^2 curvature
bound to a scaleinvariant Morreynorm bound. The second (and surprisingly
nontrivial) step is to approximate the singular YangMills connection
as a limit of smooth connections, in such a way that the approximating
connections also have small curvature; this is done by a new iterative,
continuous gauge gluing procedure. Finally, we use some elliptic
theory to place these smooth connections in a Coulomb gauge, exploiting
the small Morrey norm; this is based on a similar lemma of Uhlenbeck for
Lebesgue spaces, and the Morrey extension has also been independently obtained
recently by Riviere and Meyer. Taking limits we finally obtain the
desired regularity.

Added: Fields
Institute Thematic Program on PDE (20032004)
Sep 23, 2002

Created: Math 115A
home page

Added: PDEs in Applied Mathematics (U. Nice, France, Feb 67)
Sep 12, 2002
Sep 6, 2002
Aug 12, 2002
Aug 5, 2002
Aug 2, 2002
July 25, 2002

Uploaded: "BochnerRiesz
summability for analytic functions on the mcomplex unit sphere and for
cylindrically symmetric functions on R^{n1} \times R", joint with
Adam
Sikora, submitted to Comm.
Anal. Geom.. In this paper we prove the optimal BochnerRiesz
estimates
for two special classes of functions: analytic functions on the unit sphere
{ z_1^2 + ... + z_m^2 = 1 } on C^m, and on cylindrically symmetric
functions on R^{n1} x R (i.e. functions which are symmetric with respect
to rotations preserving {0} x R). In the first case the problem simplifies
substantially because the Laplacian on analytic functions is in fact equivalent
to the operator iZ(iZ+m), where Z is the null vector field iz_1 dz_1 +
... + iz_m dz_m. In the second case we can use a Radon transform
to project the problem back down to a (weighted) BochnerRiesz problem
on R^2, which can be obtained from the standard BochnerRiesz result by
some interpolation and a weighted estimate involving Hardy's inequality.
July 14, 2002

Téléchargé: "Existence
globale et diffusion pour l'équation de Schrödinger nonlinéaire
répulsive cubique sur R^3 en dessous l'espace d'énergie",
écrit avec Jim Colliander,
Mark
Keel, Gigliola
Staffilani, et Hideo Takaoka, pour les actes de une conférence
à Forgesleseaux. (Le titre et l'abstrait sont en français,
mais le texte principal est en anglais  notre capacité d'écrire
en français est pas si bon!). In this expository paper we
present most (but not all) of the arguments leading to global wellposedness
and scattering of the threedimensional cubic nonlinear Schrodinger equation
below the energy norm (more precisely, in H^s for s > 4/5). This
improves upon work of GinibreVelo, who handled the energy class, and of
Bourgain, who handled the radial case (for s > 5/7) and also obtained global
wellposedness (for s > 11/13). We use the Imethod (which should
be no surprise by now), but also more crucially we employ a new ``interaction''
variant of the Morawetz inequality; this variant gives an a priori L^4_{t,x}
spacetime bound on the solution, with no weights on the lefthand side
to localize the estimate near the spatial origin; this inequality seems
of independent interest, and should have wider applications. One
consequence of our approach is that we can substantially simplify the energy
scattering theory (for instance, our L^p spacetime bounds are now polynomial
in the energy rather than exponential), and the bound is also well suited
for going below the energy norm. In a later paper we will provide fuller
details.

Uploaded: The expository note "The
weaktype (1,1) of the spherical maximal function". This is a
counterpart to an older expository note, "The
weaktype (1,1) of the parabolic maximal function". Both arguments
rely on CalderonZygmund theory and a stopping time argument, but the circular
maximal function argument is easier as it does not rely on Christ's twoparameter
stopping time argument. The newer expository note attempts to unify
both proofs.

Added: Quantification
et Analyse Microlocale (Colloque en l'honneur de Andrew Unterberger)
(Sep 912, Reims, France), Yamabe
Memorial Symposium (Sep 2022, U. Minnesota MN), CoifmanMeyer conference
(Orsay, June 1821 2003), PASI program in PDE and Inverse Problems (Jan
618, 2003, Santiago Chile), PDE meeting (Forges les Eaux, June 26 2003),
Numerical
Simulation of Gravitational Wave Sources (Caltech Visitors Program,
20022003),
Gravitational
Interaction of Compact Objects (May 12Jul 11 2003, KITP, UC Santa
Barbara). Thanks to Mark Keel for the last two links.
July 2, 2002

Updated: "Asymptotics,
frequency modulation, and low regularity illposedness for canonical defocussing
equations", joint with Michael
Christ and Jim Colliander.
After hearing back from the referee we made several changes to the exposition
and removed a number of typos, as well as clarified the nature of wellposedness
and illposedness that our results give.

Added: Emerging Applications
of the Nonlinear Schrodinger Equations (Feb 37 2003, IPAM (UCLA),
California), AMS/MAA
Joint Mathematics Meeting (Special
session in Wavelets, Frames, and Operator Theory) (Baltimore, MD Jan
1518 2003), The
24th Midwest Probability Colloquium (Oct 1819, Northwestern U., IL)
and New
developments in evolution equations and approaches to nonlinear phenomena
(Research Inst. of Math. Sci. Kyoto Univ., Kyoto Japan Oct 1618).
June 24, 2002
June 21, 2002

Uploaded: "Polynomial
upper bounds for the orbital instability of the 1D cubic NLS below the
energy norm", joint with Jim
Colliander, Mark
Keel, Gigliola
Staffilani, and Hideo Takaoka, submitted to Communications
on Pure and Applied Analysis. This is another "Imethod" paper,
but now we are trying to show that this method can be used to do more than
just global existence for rough data, and in fact give other sorts of quantitative
and qualitative control of rough solutions of nonlinear evolution equations.
Here we are interested in the question of stability of special solutions
(either the zero solution, or a soliton solution)  if one makes a small,
but rough, perturbation of a special solution, does one stay close to a
special solution for all time? For demonstration we pick the 1D cubic
defocusing NLS, which is a fairly easy equation to handle analytically.
If the perturbation is in the energy class, an old result of Weinstein
shows that the class of solitons (aka ground states) at a fixed energy
are stable (i.e. solitons are orbitally stable). (This class is actually
a twodimensional cylinder, reflecting the phase and translation invariances
of the equation). Here we consider H^s perturbations for 0 < s
< 1. Unfortunately we are not able to obtain H^s orbital stability,
but we can show that the H^sdistance to the ground state grows at most
polynomially, which is not ideal but considerably better than the exponential
bounds one would expect from iterative methods (which cannot detect anything
about Lyapunov exponents). In a similar vein we can show polynomial
growth bounds for the H^s norm for large H^s data, which is sort of a stability
bound for the zero solution of NLS. For this latter problem we introduce
an "upsidedown" Imethod, which pushes up from the L^2 norm rather than
pushing down from the H^1 norm. For the orbital stability result
we apply the Imethod to the Lyapunov functional (a variant of the Hamiltonian),
together with some standard techniques regarding how to approximate a solution
by an appropriately chosen ground state.
June 17, 2002
June 10, 2002

Updated: "Carleson
measures, trees, extrapolation, and T(b) theorems", joint with Pascal
Auscher, Steve
Hofman, Camil Muscalu,
and Christoph Thiele.
After some very helpful comments from Stephanie Molnar, Joan Verdera, and
the anonymous referee, we have fixed several typos, added more explanation
to the arguments (including an additional figure), and included substantially
more historical background and context to the paper.

Updated: "The weaktype
(1,1) of Fourier integral operators of order (n1)/2". After
hearing back from the referee I made some minor typographical corrections.
June 7, 2002
May 22, 2002
May 14, 2002

Uploaded: "Endpoint
mapping properties of spherical maximal operators", joint with Andreas
Seeger and Jim
Wright. In this paper we sharpen the known endpoint L^p properties
of spherical maximal operators, where the radius is restricted to lie in
a fixed set E. The critical index of p is known, and is determined
by what is essentially the Minkowski dimension of E. One can also
get restricted weak type estimates at this exponent fairly easily.
But weak type or more general Lorentz space estimates appear to be more
subtle; in the general case one cannot obtain these results using the standard
techniques of L^2 smoothing, CalderonZygmund decompositions, uncertainty
principle, naive estimates of exceptional sets, etc., as we have an example
of a similar (but somehow "flatter") maximal operator which fails to obey
the expected endpoint estimates. However, if the set E is in some
sense "regular" (in that the accumulation points of E have strictly lower
dimension than E) then one can recover the endpoint estimates by using
the uncertainty principle more strongly to approximate the maximal function
on E by the maximal function on the accumulation points.

Uploaded: "Pointwise
convergence of lacunary spherical means", joint with Andreas
Seeger and Jim
Wright. Here we focus on the lacunary spherical maximal function
(where the radius is restricted to a lacunary set such as the powers of
2). The conjecture here is that these operators are of weaktype
(1,1), but this seems out of reach of our techniques. However, we
are able to show weak type L log log L. (An old argument of Mike
Christ shows that H^1 maps to weak type L^1, thus we have weak type L log
L). The idea is best explained by considering how the circular maximal
function acts on the characteristic function of a rectangle. This
maximal function is supported on a set of concentric rings. The small
radius rings can be placed inside an "exceptional set" of small measure.
The large radius rings can be controlled by L^2 estimates. Then there
are a small number (log log of the eccentricity) of rings in the middle
which are not well estimated by either estimate, and so we estimate those
crudely in L^1. The idea now is to take a general function and try
to split it efficiently as the sum of characteristic functions of "rectangles"
 objects for whom their measure is roughly their "length" times their
"thickness". (The length  the 1dimensional Hausdorff content 
is what controls the L^2 estimate, while the thickness controls the size
of the exceptional set). This is done by a stopping time argument
which may have application to other problems.

Uploaded: "Singular
maximal functions and Radon transforms near L^1", joint with Andreas
Seeger and Jim
Wright. In this paper we generalize the argument of the
previous paper to maximal functions over arbitrary surfaces of finite
type, and arbitrary lacunary sets of (possibly non isotropic) dilations.
Also we consider the singular integral analogue of these maximal functions.
This paper is unfortunately considerably more technical than the previous
one. The main innovation needed is to have a separate notion of length
and thickness for each dyadic scale, and to work harder on the L^2 side
of the estimates in the case of the singular integral.
May 13, 2002

Uploaded: "A Carlesontype
theorem for a Cantor group model of the Scattering Transform", joint
with Camil Muscalu and Christoph
Thiele. The motivation for this paper is the following question: given
a onedimensional Dirac or Schrodinger operator with L^2 potential, is
it true that the eigenfunctions for this operator are bounded for almost
every energy? This is a more quantitative version of the result (recently
proved by Deift and Killip) that such operators have absolutely continuous
spectrum almost everywhere. This result would be a nonlinear analogue
of Carleson's theorem and is still open; the best result so far is due
to Christ and Kiselev, who proved the same result for L^p potentials, 1
<= p < 2. We have not been able to prove this result yet, but
we have managed to prove the corresponding discrete Walsh group (aka Cantor
group) analogue of this problem. This is as to the eigenfunction
problem as WalshFourier series expansions are to Fourier expansions; the
exponentials exp(ikx) are replaced by the Walsh characters e(k * x) (which
resemble Radamacher functions, i.e. we replace sine waves with square waves).
The argument is a nonlinear version of Carleson's argument; rather amazingly,
Carleson's original argument transplants quite easily into the nonlinear
setting. The one new ingredient is that the linear Bessel inequality
(which is a triviality in the Walsh case) must be replaced by a nonlinear,
"multiplicative" analogue, which we have only been able to prove by a Bellmanfunction
"swapping" argument. It is this step which relies crucially on the
exact dyadic structure of the problem, and so we have not been able to
extend this method to the continuous case. (On the other hand, it
works not just for the dyadic model but more generally for the dadic model
for any finite d, but unfortunately with constants which grow in d).

Updated: the URL for the National
Research Symposium on Elliptic Operators, Geometric Analysis and related
topics (July 912, ANU, Australia)
May 2, 2002

Added: Progress
in Partial Differential Equations and Applications (May 2325, Washington
State U.), Tenth Meeting on
Real Analysis and Measure Theory (Jul 1519, Ischia, Italy), Spring
School on Nonlinear Analysis, Function Spaces and Applications 7 (July
1722, Prague, Czech Republic), and The
10th International Conference on Finite or Infinite Dimensional Complex
Analysis and Applications (ICFI DCAA) (July 29Aug 2, Pusan, Korea).
Also added a link to the Mathematics
Web and Project Euclid (both
online repositories of mathematical journals), and the mathematical
physics archive.
Apr 24, 2002

Uploaded: the expository note "The
nonlinear Fourier transform". This note is an expanded (and heavily
rewritten) version of my previous
notes on Dirac inverse scattering. Here we view the Dirac scattering
transform as a nonlinear analogue of the Fourier transform  basically,
it is a noncommutative, multiplicative version of the (additive) Fourier
transform, yet obeys many of the same properties (good behaviour with respect
to modulations, scaling, etc., as well as analogues of all the usual Fourier
theorems, e.g. Plancherel, PaleyWiener, RiemannLebesgue, HausdorffYoung,
...). This transform is of course linked to the spectral theory of
the Dirac operator, but can also be used to solve the NLS and mKdV equations
(via the inverse nonlinear Fourier transform, more commonly known as the
inverse scattering transform). It seems that the nonlinear Fourier
transform deserves more attention from harmonic analysts, as it has many
applications to spectral theory and integrable systems, and yet the theory
is far less developed than that of the linear Fourier transform.
Apr 19, 2002

Uploaded: "A new
bound for finite field Besicovitch sets in four dimensions".
This paper deals with the Kakeya problem in four dimensions, which is in
some sense the (current) borderline between the lowdimensional (geometric)
Kakeya theory and the highdimensional (arithmetic) Kakeya theory.
Here we treat the simplest of all the Kakeya problems, namely that of bounding
the cardinality of Besicovitch sets in a finite field geometry F^4.
A result of Wolff shows that such sets have cardinality at least F^3
(ignoring constants, logs, etc). Here we improve this slightly, to
F^{3+1/16}. (Izabella
and I were able to obtain a small
improvement also in the upper Minkowski Euclidean case, but that argument
doesn't extend to finite fields). Interestingly, the argument uses
a touch of algebraic geometry, in particular computing the dimension of
some algebraic varieties formed by considering the class of all lines which
intersect a given triple of reguli; this idea was inspired by some similar
work of Nets
Katz, which hopefully should appear soon. It contains no arithmetic
arguments, being completely geometrical. It seems that the argument
should extend to the other flavors of the Kakeya problem, and perhaps one
could start using algebraic geometry to attack the problem in other low
dimensions as well.
Apr 18, 2002

Uploaded: "Kakeya
and restriction phenomena for finite fields", joint with Gerd
Mockenhaupt. The Kakeya and restriction problems are usually
set in Euclidean space, but the finite field case is technically a lot
simpler while retaining many of the important features of these problems.
In particular, the Kakeya and restriction conjectures are still unsolved
even in the finite field case. In this paper we try to systematically
study these problems in finite fields, using the known Euclidean techniques
to obtain reasonably good restriction estimates for the paraboloid and
cone, and also reproduce the Euclidean Kakeya estimates in the finite field
setting. We also indicate how the Restriction<>Kakeya connection
works in finite fields. We have tried to make the paper accessible
to people in related fields (number theory, combinatorics, algebraic geometry)
in that no experience with the Euclidean theory is required (in fact we
use very little harmonic analysis, other than just the basic properties
of the (finite field) Fourier transform and other fundamental tools such
as the pigeonhole principle and CauchySchwartz). For people in those
fields who are interested in these problems, this paper might be the place
to start.
Apr 11, 2002

Reorganized: My preprints page is now subdivided
into six categories, organized by topic of paper (PDE, honeycombs, Kakeya,
etc.), as the original page was getting a little unwieldy to navigate.

Uploaded: "Lowregularity behavior
of the Kortewegde Vries and modified Kortewegde Vries equations: local
and global wellposedness, asymptotics, illposedness, and symplectic nonsqueezing".
This is the lecture notes for a series of three lectures I gave recently
at the University of Chicago on the KdV and mKdV equations. In the
first lecture I discuss the "Iteam" work on global wellposedness for
the KdV and mKdV equations, consisting of a
short KdV paper (with nonsharp results) and a
longer version (with sharp results, and also needing some
technical estimates in the periodic case). In the second lecture
I discuss the work with Michael
Christ and Jim Colliander
on Asymptotics,
frequency modulation, and low regularity illposedness for canonical defocussing
equations. In the third lecture I discuss more recent work with
the Iteam on symplectic nonsqueezing for KdV (paper currently in preparation).

Added: National Research Symposium on Elliptic Operators, Geometric Analysis
and related topics (ANU, Australia Jul 912), CBMS
Lecture series  Alex Volberg  Nonhomogeneous harmonic analysis (U.
North Carolina, May 1317), Primer Taller Internacional de Analisis Armonico
y Ecuaciones Diferenciales Parciales (Puerto Vallarta, Mexico, June 2327
2003). Also added the URL for the Conference
in Harmonic Analysis and PDE at U. Missouri (May 811).
Mar 21, 2002

Uploaded: "Edinburgh lecture
notes on the Kakeya problem". These are my lecture notes on the
Kakeya problem for the upcoming instructional
conference at Edinburgh. These are an expanded version of the
Kakeya half of the IPAM lecture notes;
they try to focus on the actual mechanics of the various arguments used
to attack the Kakeya problem, and to invite other mathematicians (particularly
from other fields such as combinatorics) to try their hand at the problem.
In light of my recent work with Gerd (which will be uploaded shortly!)
we have found it best to emphasize the finite field model of the Kakeya
problem as it is much simpler technically; this point of view is prevalent
throughout the notes. Incidentally, these notes are not intended
to be an exhaustive survey of the subject, and I apologize for any references
etc. which are not present in the notes.

Uploaded: "Almost
conservation laws and global rough solutions to a nonlinear Schrodinger
equation", joint with
Jim
Colliander,
Mark
Keel, Gigliola
Staffilani, and Hideo Takaoka. This started out as one of our
first "Imethod" papers, but for various reasons (in particular, an emphasis
towards solving the GWP problem for KdV and mKdV) it was delayed until
now. Here we take one of the simplest nonintegrable nonlinear dispersive
equations  the 2D defocusing cubic NLS  and show that this equation is
globally wellposed in H^s for s > 4/7, by using the "Imethod".
This allows for comparison with Bourgain's Fourier truncation method, which
works for s > 3/5 (but gives better control on the evolution, in that it
differs from the linear flow by an H^1 error). In a later paper we
shall use the "correction term" refinement of the Imethod to push this
further, to a little below 1/2. We also handle the 3D cubic NLS,
getting GWP for s > 5/6 (Bourgain's Fourier truncation method gives s >
11/13). This will also be improved in a later paper by use of Morawetz
estimates.

For readers seeking an introduction to the "Imethod", this is probably
the best place to start (excepting perhaps the short
version of the KdV paper; Gigliola
Staffilani also has some nice notes from the Mt. Holyoke conference
which should be uploaded soon). Basically, the idea is to control
the H^s norm of the solution u by replacing it by the energy of Iu, where
I is a smoothing operator of order 1s which behaves like the identity
for low and medium frequencies. To get the best results, one needs
the multiplier of I to be smooth, and one needs to adapt the local wellposedness
theory to this I operator in order to get the best "almost conservation
law" for the energy of Iu.
Mar 11, 2002
Mar 7, 2002
Mar 5, 2002

Uploaded: "Asymptotics,
frequency modulation, and low regularity illposedness for canonical defocussing
equations", joint with Michael
Christ and Jim Colliander.
In this paper we investigate asymptotics and (lack of) wellposedness for
the defocusing NLS, modified KdV (mKdV), and KdV equations on the line
and on the circle. This parallels a recent paper of Kenig, Ponce
and Vega addressing the focusing case, exploiting the explicit soliton
and breather solutions available in that setting. In the defocusing
case soliton solutions are not available, but (in the NLS case) precise
asymptotics were obtained using the pseudoconformal transformation and
energy estimates by Ozawa. Interestingly, solutions of defocusing
NLS do not approach solutions of the free Schrodinger equation, but instead
evolve by a slightly modulated version of this equation (with a phase correction
of roughly O(log t)). From these asymptotics and the invariances
of NLS we can show that NLS is illposed in H^s for s < 0 (complementing
an old result of Tsutsumi, which establishes global wellposedness for
s>=0). We then use a wellknown approximation of mKdV by NLS to transplant
these results to mKdV, showing that equation is illposed for s < 1/4
(on R) and s < 1/2 (on T). We also construct asymptotics for global
solutions of mKdV by a method similar in spirit to Ozawa's, but where the
pseudoconformal invariance is not available. Finally, we use the
Miura transform to transplant these results to KdV, and also observe that
a simple modification of the Miura transform allows us to also push forward
the endpoint wellposedness of mKdV on R at s=1/4 to obtain wellposedness
of KdV on R at s=3/4.

All three equations are completely integrable, and our asymptotics are
consistent with those obtained by methods such as inverse scattering.
However, our techniques here are quite general (for instance, we believe
they apply to the nonlinear wave equation NLW), and in particular we seem
to have obtained a general scheme to show illposedness for defocussing
equations at any supercritical regularity.

Added: Colloque
du Groupement de Recherche "Analyse des Equations aux Dérivées
Partielles", (Forgesleseaux, France, June 37). Thanks to Fabrice
Planchon for this link. And from the TMR
 Harmonic Analysis page: Spaces
of analytic functions and their operators (CIRM, France, Sep 30  Oct
4), Noncommutative
Phenomena and Random Matrices (UBC, Canada, Aug 69), Measure
Transportation and Geometric Inequalities (UBC, Canada, Jul 812),
Workshop
in Linear Analysis and Probability (Texas A&M, Jun 24Jul 19),
GPOTS
2002 (UNC, May 2226),
4th
conference on function spaces (Southern University of Illinois at Edwardsville,
May 1419).
Feb 27, 2002
Feb 19, 2002

Uploaded: "Upper
and lower bounds for Dirichlet eigenfunctions", joint with Andrew
Hassell. In this paper we consider eigenfunctions Delta u = lambda
u on a compact Riemannian smooth manifold with boundary M, with the Dirichlet
condition that u vanishes on the boundary, lambda is a large number.
We normalize things so that u has L^2 norm 1, thus nabla u has L^2 norm
lambda^{1/2}. We are interested in the extent to which nabla u concentrates
or vanishes on the boundary, and more specifically what the L^2 norm of
nabla u on the boundary is. If M is a domain in R^n, then this quantity
is always bounded above and below by a multiple of lambda^{1/2} (i.e. there
is no significant concentration or vanishing at the boundary); this is
a classical result of Rellich. In general domains, Rellich's identity
shows that the L^2 norm of nabla u on the boundary is bounded above by
lambda, but the lower bound is more subtle. It turns out the lower
bound still obtains as long as M contains no trapped geodesics (including
geodesics which skim the boundary); on the other hand, we have examples
which show that the lower bound is violated (in some cases quite dramatically)
if M contains trapped geodesics. To prove this, one cannot proceed
entirely by Rellich identities based on commutation with firstorder differential
operators (we have a "giraffeshaped" counterexample for this), but one
can instead use commutator estimates with firstorder pseudodifferential
operators, or (if the manifold is extremely smooth) by sufficiently
highorder differential operators (following Morawetz, Ralston, and Strauss).
In order to make the pseudodifferential method work, we need to also control
u and its derivatives near the boundary as well as on it (because
pseudodifferential operators are not perfectly local), and this is done
by an "energy estimate" method. Specifically, we show that for any
manifold M (regardless of having trapped geodesics), the size of u(x) is
bounded (in an L^2 averaged sense) by lambda^{1/2} dist(x, partial M);
this bound may be of independent interest.

Updated: "Multilinear
estimates for periodic KdV equations, and applications", joint with
Jim
Colliander, Mark
Keel, Gigliola
Staffilani, and Hideo Takaoka. The substance of this paper hasn't
changed much, but the introduction has been improved and the exposition
reworked in a couple places.

Added: 2nd
WSEAS International Conference on Simulation, Modelling and Optimization
and 2nd
WSEAS International Conference on Signal, Speech, and Image Processing
(both on Koukounaries, Skiathos Island, Greece, Sep 2528), and Southern
California Analysis and Partial Differential Equations Seminar (SCAPDE)
(Apr 2021, UC San Diego, web link not yet available)
Feb 13, 2002

Uploaded: "The T(b) theorem and its
variants", a survey article for the conference proceedings of the recent
The
International Conference on Harmonic Analysis and Related Topics (in
honour of the 60th birthday of Alan McIntosh). This short paper is
based on the much larger "Carleson
measures, trees, extrapolation, and T(b) theorems", joint with Pascal
Auscher, Steve
Hofman, Camil Muscalu,
and Christoph Thiele.
Basically the survey article isolates the portion of the large paper which
deals with T(b) theorems, and gives a somewhat nonstandard proof of the
usual T(1) and T(b) theorems, based on pointwise estimates of wavelet coefficients.
This perspective allowed us, eventually, to prove a new local T(b) theorem,
similar to an older theorem of Christ, but with L^infty hypotheses replaced
by the more natural L^2 counterparts.

Added: Workshop in general relativity(Stanford,
Apr 1  June 7)
Jan 31, 2002
Jan 24, 2002

Uploaded: "The
weaktype (1,1) of Fourier integral operators of order (n1)/2".
When I was a graduate student at Princeton, the first thesis problem my
advisor, Elias Stein, gave me was to determine whether Fourier integral
operators (FIOs) of the critical order (n1)/2 were of weaktype (1,1).
(With Andreas Seeger and Chris Sogge, he had already established that they
map H^1 to L^1). To do this, he suggested that one factorize these
operators into the product of a CalderonZygmund operator and an operator
bounded on L^1. This worked well enough when the phase function of
the FIO was nondegenerate, but I couldn't figure out what to do in the
degenerate case (except in the n=2 finite type case, where the stationary
phase asymptotics were so detailed that I could make the factorization
work). Eventually I gave up and my advisor gave me some other problems
which I did eventually solve. There the matter lay for about six
years, when I started looking at the problem again late last year.
After a few weeks of reconstructing my earlier attacks, I finally realized
that one should not try to factorize the degenerate portion of the FIO,
but just tackle that part directly by L^1 estimates. With that one
observation, the problem which was my nemesis for much of my graduate student
days turned out to be actually rather embarrassingly simple. So,
this theorem is not the most exciting or deep result out there in harmonic
analysis, but it does give me a great deal of personal satisfaction to
see it off at last. :)

Updated: The URL for Conference
on Harmonic Analysis and PDE (Institut Henri Poincare, Apr 2226).
Note slight change of date.

Added: Colloque en
l'honneur de JeanMichel Bony (Ecole Polytechnique, June 2628).
Thanks to Fabrice Planchon for this link. And from the TMR
 Harmonic Analysis page: Eighteenth
annual South Eastern Analysis Meeting (UNC Chapel Hill, NC, Mar 1517),
Conference
on Illposed and Inverse problems (in honour of the 70th anniversary
of the birth of Prof. M.M. Lavrent'ev, Novisibirsk, Russia, Apr 59), the
ICM
satellites on Abstract
and Applied Analysis, Nonlinear
Analysis, Complex
Analysis, Harmonic
Analysis and Applications, Clifford
Analysis, and Geometric
Function Theory in Several Complex Variables; NavierStokes
equations and related topics (NSEC8, St. Petersburg, Russia, Sep 1118).
Jan 20, 2002
Jan 18, 2002

Uploaded: The expository note "Inverse
scattering for the Dirac equation". These are some notes (arising
from discussions with Christoph Thiele, Michael Christ, Jim Colliander,
Gigliola Staffilani, Tom Mrowka, Hideo Takaoka, Camil Muscalu, and Mark
Keel) on how the scattering transform and inverse scattering transform
work for Dirac operators (d/dx + F), and how these can be used to solve
the modified Korteweg de Vries (mKdV) equation. The scattering transform
is the nonlinear analogue of the Fourier transform (or more precisely,
a noncommutative multiplicative analogue of the Fourier transform), and
the inverse scattering transform is the analogue of the inverse Fourier
transform. Just as the Airy equation can be solved as a Fourier multiplier,
the mKdV equation can be solved by a "scattering multiplier". Some
connections with the more traditional (but a bit more complicated) scattering
transform for the Schrodinger operator and the KdV equation (via the Miura
transform) are also given, as are some basic facts about the nonlinear
Scattering transform (nonlinear HausdorffYoung, nonlinear Plancherel,
etc.) This is a work in progress and I expect to add more sections
to these notes as I learn more.
Jan 8, 2002

Uploaded: "On the
bicarleson operator I. The Walsh case", joint with
Camil
Muscalu, and Christoph Thiele.
The "biCarleson operator" is a subbilinear operator which is a natural
hybrid between the Bilinear Hilbert transform B(f,g) and the Carleson maximal
operator C(f). (Indeed, it looks vaguely like B(f,Cg) or C(B(f,g))
but cannot actually be factorized into either form). This operator
appears naturally in the expansion of Dirac eigenfunctions in one dimension
(except for a crucial minus sign, which we discuss in another
paper), and is also of similar complexity to the "biest" operator (which
is a trilinear operator looking vaguely like B(f,B(g,h)) or B(B(f,g),h),
but again not actually factorizable into either form). We studied
the Walsh and Fourier
version of the biest in earlier papers. In this paper we study the
Walsh version of the biCarleson operator, using similar techniques (phase
space decomposition, construction of size, mass, energy, combinatorial
sorting into trees). As a byproduct we give a selfcontained proof
of Carleson's theorem for the Walsh case  not too much different from
the other proofs in recent literature, but here we are trying to put this
operator (and the bilinear Hilbert transform) into a unified framework
which also includes the biest, biCarleson, and several other types of
multilinear operators.
Jan 4, 2002