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} (1-|z|²) f^#(z) < ¥, where f^#(z) = |f¢(z)| / ( 1+|f(z)|² ). For a > 1, a meromorphic function f is called a-normal if sup_{z Î D} (1-|z|²)^a f^#(z) < ¥. H. Allen and C. Belna [1] have proved that there is an analytic function f₁, defined in D, such that

References

[1]

H. Allen and C. Belna, Non-normal functions f(z) with \iint_{|z| < 1} |f¢(z)|  dx  dy < ¥. J. Math. Soc. Japan 24(1972), 128-132.

[2]

H. Wulan, A non-normal function related Q_p spaces and its applications. In: Progress in Analysis I, II, World Sci. Publ., River Edge, NJ, 2003, 229-234.

[3]

S. Yamashita, A non-normal function whose derivative has finite area integral of order 0 < p < 2. Ann. Acad. Sci. Fenn. Ser. Math. 4(1978/1979), 293-298.

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 one-space H¹. On the unit ball in Cⁿ, n ³ 2, there is an analogous result for any f smooth on the closed ball (not necessarily conjugate-holomorphic), which is reminiscent of the trace formula of Helton and Howe. The last result extends also to arbitrary smoothly bounded strictly pseudoconvex domains in Cⁿ, 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 Brunn-Minkowski inequality is established for the (n-1)-capacity of radial sums of star bodies in Rⁿ. This inequality is a counterpart for the p-capacity of Minkowski sums of convex bodies in Rⁿ 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 (n-1)-capacity has the same status of volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.

This is joint work with Richard Gardner.

Let L = -D_Hⁿ + V be a Schrödinger operator on the Heisenberg group Hⁿ, where D_Hⁿ is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B_[(Q)/2] and Q is the homogeneous dimension of Hⁿ. The Riesz transforms associated with the Schrödinger operator L are bounded from L¹(Hⁿ) to L^1,¥ (Hⁿ). The L¹ integrability of the Riesz transforms associated with L characterizes a certain Hardy type space denoted by H¹_L(Hⁿ) which is larger than the usual Hardy space H¹(Hⁿ). We define H¹_L(Hⁿ) in terms of the maximal function with respect to the semigroup {e^{-s L} : s > 0}, and give the atomic decomposition of H¹_L(Hⁿ). As an application of the atomic decomposition theorem, we prove that H¹_L(Hⁿ) 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 quasi-geostrophic (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 Triebel-Lizorkin 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.