We prove versions of James' weak compactness theorem for polynomials and symmetric multilinear forms of finite type. We also show that a Banach space X is reflexive if and only if it admits and equivalent norm such that there exists x₀≠0 in X and a weak-^*-open subset A of the dual space, satisfying that x^*⊗x₀ attains its numerical radius. for each x^* in A.

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.

LetM be a non-compact connected Riemann surface of a finite type, and R⋐M be a relatively compact domain such that H₁(M, Z)=H₁(R, Z). Let $\tilde R \to R$ be a covering. We study the algebra H^∞(U) of bounded holomorphic functions defined in certain subdomains $U \subset \tilde R$ . Our main result is a Forelli type theorem on projections in H^∞(D).

For any proper polynomial map f: C^k→C^k define the function α as $\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$ Let f=(P₁,..., P_k) be a proper polynomial map. We define a notion of s-regularity using the extension of f to P^k. When f is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 and d₁,..., d_k, where d_i:=degP_i. We then describe dynamically the sets {α≤d_i}. We give a concrete description of regular maps. If d_i >1, this allows us to construct the equilibrium measure μ associated with f as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π, s)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.

We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in R³ away from the nuclei.

We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by c₀- and l₁-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

A fundamental solution of the acoustical equation with a variable refraction coefficient is constructed. The solution satisfies the limiting absorption and radiation conditions. The optimal high frequency estimate is proved for square means of the solution. The source function for the diffusion equation is a by-product of this construction.

In this paper, we characterize, for 1≤p<∞, the multiple (p, 1)-summing multilinear operators on the product of C(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators on C(K) in terms of their representing measures and a new multilinear characterization of L_∞ spaces. We also solve a problem stated by M.S. Ramanujan and E. Schock, improve a result of H. P. Rosenthal and S. J. Szarek, and give new results about polymeasures.

The sine(hyperbolic)-Gordon hierarchy is shown to be the extension of the modified Korteweg-de Vries (MKdV) hierarchy in the integrodifferential algebra extending the standard differential algebra by means of one antiderivative. The characterization by vanishing residues of the MKdV hierarchy yields the same characterization of the sine(hyperbolic)-Gordon hierarchy in the integrodifferential algebra.