
Let D be the unit disk {z : z < 1} in the complex plane. A function f, meromorphic in D, is normal, denoted by f Î N, if sup_{z Î D} (1z^{2}) f^{#}(z) < ¥, where f^{#}(z) = f¢(z) / ( 1+f(z)^{2} ). For a > 1, a meromorphic function f is called anormal if sup_{z Î D} (1z^{2})^{a} f^{#}(z) < ¥. H. Allen and C. Belna [1] have proved that there is an analytic function f_{1}, defined in D, such that

 (1) 
We prove existence of longtime smooth solutions to mean curvature flow of entire Lipschitz Lagrangian graphs. As an application of the estimates for the solution, we establish a Bernstein type result for translating solitons.
The results are from joint work with Albert Chau and Weiyong He.
We discuss the membership of (big) Hankel operators H_{f} on weighted Bergman spaces in the Dixmier class. For the unit disc and f holomorphic, H_{[`(f)]} is in this class if and only if f¢ belongs to the Hardy onespace H^{1}. On the unit ball in C^{n}, n ³ 2, there is an analogous result for any f smooth on the closed ball (not necessarily conjugateholomorphic), which is reminiscent of the trace formula of Helton and Howe. The last result extends also to arbitrary smoothly bounded strictly pseudoconvex domains in C^{n}, where the formula for the Dixmier trace turns out to involve the Levi form, thus exhibiting an interesting link with the geometry of the domain. The proofs involve analysis of pseudodifferential operators on the boundary of the domains.
The case of the disc is joint work with Richard Rochberg, while the case of the ball and of pseudoconvex domains are joint with Kunyu Guo and Genkai Zhang.
A dual capacity BrunnMinkowski inequality is established for the (n1)capacity of radial sums of star bodies in R^{n}. This inequality is a counterpart for the pcapacity of Minkowski sums of convex bodies in R^{n} for 1 £ p < n, proved by Borell, Colesanti and Salani. When n ³ 3, the dual inequality follows from an inequality of Bandle and Marcus, but our new proof allows us to establish an equality condition. In the n=2 case, we use different techniques to establish the inequality and a different equality condition. These results show that in a sense (n1)capacity has the same status of volume in that it plays the role of its own dual set function in the BrunnMinkowski and dual BrunnMinkowski theories.
This is joint work with Richard Gardner.
Let L = D_{Hn} + V be a Schrödinger operator on the Heisenberg group H^{n}, where D_{Hn} is the subLaplacian and the nonnegative potential V belongs to the reverse Hölder class B_{[(Q)/2]} and Q is the homogeneous dimension of H^{n}. The Riesz transforms associated with the Schrödinger operator L are bounded from L^{1}(H^{n}) to L^{1,¥} (H^{n}). The L^{1} integrability of the Riesz transforms associated with L characterizes a certain Hardy type space denoted by H^{1}_{L}(H^{n}) which is larger than the usual Hardy space H^{1}(H^{n}). We define H^{1}_{L}(H^{n}) in terms of the maximal function with respect to the semigroup {e^{s L} : s > 0}, and give the atomic decomposition of H^{1}_{L}(H^{n}). As an application of the atomic decomposition theorem, we prove that H^{1}_{L}(H^{n}) can be characterized by the Riesz transforms associated with L.
I will present some results of a recent joint paper with A. Cianchi. The purpose is to embed the famous Serrin's symmetry result (Arch. Rational Mech. Anal., 1971) in a general symmetry principle for solutions to overdetermined elliptic problems, where the relevant symmetry is not necessarily of spherical type. The underlying idea of our contribution is that a symmetry result holds for any overdetermined problem involving an elliptic operator (with quadratic growth) from a suitable class, provided that the additional boundary condition imposed on the gradient of the solution matches the structure of the differential operator. The resulting symmetry of the domain (and of the corresponding solution u) reflects, in turn, the symmetry of the operator, that is: a solution exists if and only if the domain W is a ball in an appropriate Finsler metric associated with the operator (moreover the level sets of u are homothetic to W).
The 2D surface quasigeostrophic (SQG) equation can be seen as a 2D model equation for the 3D incompressible Euler equations. In this talk we will discuss necessary and sufficient conditions, characterized by Besov and TriebelLizorkin type spaces, for the conservation of various conserved quantities of the 2D SQG equation. These conditions can help in the searching of rigorous mathematical framework modeling turbulent flows.
In contrast to the classical situation, it is known that many Laplacian operators on fractals have gaps in their spectra. This surprising fact means there can be no limit in the Weyl counting formula and it is a key ingredient in proving that the convergence of Fourier series on fractals can be better than in the classical setting. Recently, it was observed that the Laplacian on the Sierpinski gasket has the stronger property that there exist intervals which contain no ratios of eigenvalues. In this talk, we give general criteria for this phenomenon and show that Laplacians on many interesting classes of fractals satisfy our criteria.
This is a joint work with Kathryn Hare.