A priori estimates are established for the two-phase boundary value problems with Venttsel interface conditions for linear nondivergent parabolic and elliptic equations. By these estimates, the existence and uniqueness theorems in Sobolev and Hölder spaces are proved.

Suppose that X is a vector field on a manifold M whose flow, exp tX, exists for all time. If μ is a measure on M for which the induced measures μ_t≡(exptX)_*μ are absolutely continuous with respect to μ, it is of interest to establish bounds on the L^p (μ) norm of the Radon-Nikodym derivative dμ_t/dμ. We establish such bounds in terms of the divergence of the vector field X. We then specilize M to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples on C^m and on the Riemann surface for z^1/n.

We prove a Melin-Hörmander inequality for a Banach algebra of pseudo-differential operators whose calculus was developed by Sjöstrand. The main new difficulties in the proof are settled by a stationary phase method tailored to the low-regularity of the symbols.

A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associated m-function near −∞. The proof given is based on relations between the classical transformation operators and the m-function.

It is known that, unlike the one dimensional case it is not possible to find an upper bound for the zeros of an entire map from Cⁿ to Cⁿ, n≥2, in terms of the growth of the map. However, if we only consider the “non-degenerate” zeros, that is, the zeros where the jacobian is not “too small”, it becomes possible. We give a new proof of this fact.

Let $\mathcal{H}^{1,1} (T^1 )$ denote the Hardy space of real-valued functions on the unit circle with weak derivatives in the usual real Hardy space $\mathcal{H}^1 (T^1 )$ . It is shown that when the weak derivative of a locally Lipschitz continuous function f has bounded variation on compact sets the Nemytskii operator F, defined by F(u)=f·u, maps $\mathcal{H}^{1,1} (T^1 )$ continuously into itself. A further condition sufficient for the continuous Fréchet differentiability of F is then added.

LetS denote the class of schlicht functions. D. Bertilsson proved recently that for f∈S, p<0 and 1<-N<-2|p|+1 the modulus of the Nth Taylor coefficient of (f′)^p takes its maximal value if f is the Koebe function. Here a short proof of a generalisation of this result is presented.