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Conference on Harmonic Analysis and Related TopicsTitles and abstractsIn this page you will find the complete collection of titles and abstracts for the conference. Auscher, Pascal Abstract:
We present results obtained with J.-M. Martell on boundedness of
operators, commutators with BMO functions in absence of kernel estimates. Some
appplications will be described.
Franchi, Bruno Abstract: In
this talk we present some recent results obtained in collaboration with A.
Baldi, N. Tchou and M.C. Tesi: we prove a compensated compactness theorem for
differential forms of the intrinsic complex of a Carnot group. The proof relies
on a $L^s$--Hodge decomposition for these forms. Because of the lack of
homogeneity of the intrinsic exterior differential, Hodge decomposition is
proved using the parametrix of a suitable 0-order Laplacian on forms.
Grafakos, Loukas Abstract: We discuss recent advances on the following problem: Given an even, integrable with vanishing integral over $S^{d-1}$ homogeneous of degree $-d$ kernel on $R^d$, for what range of indices $1<p<\infty$, the Calderon-Zygmund singular integral operator associated with this kernel is bounded on $L^p(R^d)$? Certain results and surprising counterexamples shed some light in the study of this problem. Martell, José María Abstract: In joint work with Ana Jiménez del Toro we present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When measuring the oscillation in a given cube we replace the average by a semigroup scaled to that cube and whose whose kernel decays fast enough. We apply the method to obtain self-improving in the scale of Lebesgue spaces of Poincaré type inequalities with respect to a semigroup. In particular, we propose some expanded Poincaré estimates that take into account the lack of localization of the semigroup. Mateu, Joan Abstract: Let $T$ be a smooth homogeneous Calderon-Zygmund singular integral operator in $R^n$. The most basic form of control of the maximal singular T^*f $ by $Tf$ one may consider is the estimate of the $L^2(R^n)$ norm of $T*f$ by a constant times the $L^2(R^n)$ of $Tf$. It turns that if $T$ is an even higher order Riesz transform, the one has the stronger pointwise inequality $T^*f(x)\leCM(Tf)(x)$, where $C$ is a constant and $M$ is the Hardy-Litlewood maximal operator. In fact the $L^2$ estimate of $T*$ by $T$ is equivalent, for even smooth homogeneous Calderon-Zygmund operators, to the pointwise inequality between $T*$ and $M(T)$. Our main result characterizes the $L^2$ and pointwise inequality in terms of the kernel $\frac{\Omega(x)}{|x|^n} of $T$, where $T$ is an even homogeneous function of degree 0, of class $C^{\infty}(S^{n-1})$ and with zero integral on the unit sphere $S^{n-1}$. Similar results are obtained for odd higher order Riesz transforms, but now the pointwise inequality is $T^*f(x)\leCM^2(Tf)(x), where $M^2=M\circM$. Mauceri, Giancarlo Abstract: In this talk I shall present an
atomic $H^1-BMO$ theory on a class of nondoubling metric measure spaces, that
are locally doubling and satisfy two additional ``geometric" properties, called
\emph{approximate midpoint} (AM) property and \emph{isoperimetric} (I) property.
Roughly speaking, a space satisfies (AM) if its points do not become too sparse
at infinity and satisfies (I) if a fixed ratio of the measure of any bounded set
is concentrated near the boundary.\par Examples of spaces satisfying these
properties are Gauss space and noncompact complete Riemannian manifolds with
bounded geometry and spectral gap, e.g. noncompact Riemannian symmetric spaces.
We prove that $BMO$ is the dual of $H^1$ and that an inequality of John~-
Nirenberg type holds for functions in $BMO$. Furthermore, we show that the $L^p$
spaces are intermediate spaces between $H^1$ and $BMO$. Finally, we show that
our results yield endpoint estimates for spectral multipliers of the
Laplace-Beltrami operator on Riemannian manifolds and for Riesz transforms for
the Ornstein-Uhlenbeck operator on Gauss space. These results are joint work
with A. Carbonaro, S. Meda and M. Vallarino.
Meda, Stefano Abstract: There are interesting operators on noncompact symmetric spaces that are neither of weak type $1$, nor bounded from the Hardy space $H^1$, recently introduced by A. Carbonaro, G. Mauceri and Meda, to $L^1$. Examples include higher order resolvents of the Laplacian and higher order Riesz transforms. We shall define a sequence $\{ X^k\}$ of new function spaces on a class of manifolds that includes the symmetric spaces of the noncompact type. These spaces are isometric images of $H^1$. We shall show that if $p$ is in $(1, 2)$, then $L^p$ may be obtained as a complex interpolation space between $X_k$ and $L^2$ for every $k$. Furthermore, many important operators, including higher order resolvents of the Laplacian and higher order Riesz transforms, are bounded from $X_k$ to $L^1$ for suitable $k$. Finally, we shall show that functions in $X_k$ admit an atomic decomposition in term of “special atoms”. This is joint work with G. Mauceri and M. Vallarino. Parcet, Javier Abstract: The study of Calder\'on-Zygmund
operators acting on matrix or operator valued functions goes back to Bourgain's
work in the 80's, as part of the vector-valued theory. Regarding these functions
as elements in a suitable von Neumann algebra, we can go further and ask about
the weak type $(1,1)$ boundedness of these objects. This has been overlooked in
the theory, mainly because the vector-valued methods are not strong enough. Our
main tools for its solution are two. First, the lack of certain classical
inequalities in the noncommutative setting forces us to have a deeper
understanding of how fasts a singular integral decreases --in the $L_2$ sense--
far away from the support of the function on which it acts. This gives rise to a
pseudo-localization principle of independent interest, even in the classical
theory. Second, we construct a noncommutative form of Calder{\'o}n-Zygmund
decomposition by means of the recent theory of noncommutative martingales. Our
methods improve (in fact optimize) Bourgain's $L_p$ constants and settle some
tools for a noncommutative Calder\'on-Zygmund theory. At the end of the talk, we
will comment two related open problems.
Petermichl, Stephanie Abstract: We
present an overview of estimates for classical operators in harmonic and complex
analysis in weighted spaces.
Pott, Sandra Abstract: The space of
functions of logarithmic mean oscillation (LMO) in one variable is important in
the study of multipliers on BMO and for the boundedness behaviours of
paraproducts, Hankel operators and other singular integral operators on BMO(R).
In the talk, we define a product version of LMO in several variables,
characterize boundedness of dyadic paraproducts on dyadic product BMO spaces and
give a sufficient condition for the boundedness of little Hankel operators on
the product $H^1$ space. This is joint work with Benoit Sehba (Glasgow).
Tao, Terence Abstract: Can a signal with N degrees of
freedom be recovered using only m linear measurements (such as m Fourier
coefficients) if m << N? The answer, of course, is no. However, if the
signal is _sparse_ (so that significantly fewer than m degrees are active at any
given time) then it again becomes possible to reconstruct the signal exactly,
and in fact it can be done by a simple linear programming algorithm (basis
pursuit). The key is to ensure that the measurement matrix obeys a "uniform
uncertainty principle", which ensures that it always captures a "proportional"
amount of the energy of any sparse signal. The proof of this principle uses
techniques related to those used in Bourgain's solution of the Lambda(p)
problem.
Thiele, Cristopher Abstract: TBA Tolsa, Xavier Abstract: Let E be a subset from R^n with finite s-dimensional Hausdorff measure H^s and denote by $\mu$ the restriction of this measure to E. It turns out that the principal value of the signed s-dimensional Riesz transform of $\mu$ exists $\mu$-a.e. if and only if s is an integer and E is s-rectifiable. Recall that an analogous result holds if one replaces the existence of principal values for signed s-dimensional Riesz transforms by existence of s-dimensional densities. A combination of ideas from geometric measure theory and quasiorthogonality arguments from Calderón-Zygmund theory is required. In this talk I will survey this result and other related questions. Torres, Rodolfo Abstract: We will present recent results about a multi(sub)linear maximal operator smaller that the m-fold product of the Hardy-Littlewood maximal function. This operator can be used to obtain a precise control on multilinear singular integral operators of Calder\'onZygmund type. This allows us to build a theory of weights intrinsically adapted to multilinear operators. Also, a (log) variant of the operator can be employed to study commutators of multilinear Calder\'on-Zygmund operators with BMO functions. We obtain the optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the classical Muckenhoupt weights and the new multilinear ones for these commutators too. This is joint work with A. Lerner, S. Ombrosi, C. Perez and R. Trujillo-Gonzalez. Some further consequences of these new results and related work in progress will be presented too as time permits. Vargas, Ana Abstract: This is a joint work with Sanghyuk Lee and Keith Rogers. Null form estimates for the wave equation have been considered by several authors, as Foschi, Klainerman, Machedon, Selberg, Tao, Tataru. They are inequalities of the form \begin{align} \|D_0^{\beta_0}D_{+}^{\beta_+}D_{-}^{\beta_-}(\phi\psi)\|_{L_t^qL_x^r}\le C&(\|\phi(0)\|_{\dot{H}^{\alpha_1}}+\|\phi_t'(0)\|_{\dot{H}^{\alpha_1-1}}) \\ &\times (\|\psi(0)\|_{\dot{H}^{\alpha_2}}+\|\psi_t'(0)\|_{\dot{H}^{\alpha_2-1}}), \end{align} where $\phi$ and $\psi$ are solutions of the wave equation. We obtain sharp estimates in dimension $n\ge3,$ except for the endpoints. Vega, Luis Abstract: I will present some recent work done
in collaboration with Valeria Banica on a model that is an approximation of the
evolution of a vortex filament. The underlying PDE is the so called
1d-Schrodinger map. We are interested in the stability of selfsimilar solutions,
so that we have to consider data with infinite energy. We will see that some
classical and well known estimates for the extension operator from a parabola
play a crucial role.
Verbitsky, Igor Abstract: A new approach to the Hessian Sobolev inequality of X.-J. Wang and Hessian Poincare inequalities of Trudinger and Wang, which are of current interest to conformal geometers, will be presented. A characterization of the Hessian capacity and removable singularities, as well as existence theorems and estimates of solutions in terms of Wolff's potentials for a class of fully nonlinear elliptic PDE of Monge-Ampere type will be discussed. Verdera, Joan Abstract: The quasiconformal mapping $\Phi$ associated with a Beltrami coefficient $\mu$ has gradient locally in $L^p$ for some $p$ larger than $2$. A central theme in QC mappings has been determining the best p and understanding how $\Phi$ deforms sets. In this paper we give a condition on $\mu$ which entails that $\Phi$ is bilipschitz. By Stoilov's factorization this gives a sufficient condition for Lipschitz regularity of quasiregular functions and thus, for Lipschitz regularity for solutions of elliptic equations in divergence form (in the plane). The argument uses in an essential way the special cancellation properties of even Calderón-Zygmund operators, which are formulated in a new suggestive way. Volberg, Alexander Abstract: The
solution of certain Monge-Amp\'ere equations allows to construct
Bellman functions and use them as "energy functionals" in heat
flow estimates (and such). This is a universal method to estimate the
norms of singular operators such as Riesz transforms and certain
Fourier multipliers. The Riesz transforms can be considered with
respect to quite general Laplacians. The estimates are better than
those that can be obtained by other methods. They are always
dimension-free too.
Wright, Jim Abstract: We extend work of D. Oberlin on sharp bounds for averaging operators along general polynomial curves with respect to the affine arclength measure.
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