Let Ω be a bounded convex domain in C_n, with smooth boundary of finite type m.

The equation $\bar \partial u = f$ is solved in Ω with sharp estimates: if f has bounded coefficients, the coefficients of our solution u are in the Lipschitz space Λ. Optimal estimates are also given when data have coefficients belonging to L^p(Ω), p≥1.

We solve the $\bar \partial $ -equation by means of integral operators whose kernels are not based on the choice of a “good” support function. Weighted kernels are used; in order to reflect the geometry of bΩ, we introduce a weight expressed in terms of the Bergman kernel of Ω.

Letf=g_t+h_t be the optimal decomposition for calculating the exact value of the K-functionalK(t, f; $\bar X$ ) of an element f with respect to a couple $\bar X$ =(X₀ , X₁) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spaces X₀ and X₁ is either L^∞ or L¹. Many examples are given of couples of lattices $\bar X$ for which |g_t| increases monotonically a.e. with respect to t. It is shown that this property implies a sharpened estimate from above for the Brudnyi-KrugljakK-divisibility constant γ( $\bar X$ ) for the couple. But it is also shown that certain couples $\bar X$ do not have this property. These also provide examples of couples of lattices for which γ( $\bar X$ ).

The distance between the second and first eigenvalues for the Dirichlet Laplacian of a domain is called its (spectral) gap. We show that the gap of a convex planar domain D symmetric about both the x and y axes is no smaller than the gap of an oriented rectangle which contains D.

It is proved in Benamara-Nikolski [1] that if the spectrum σ(T) of a contraction T with finite defects (rank(I−T^*T)=rank (I−TT^*)<∞) does not coincide with $\bar D$ , then the contraction is similar to a normal operator if and only if $C_1 (T) = \mathop {\sup }\limits_{\lambda \in C\backslash \sigma (T)} \parallel (T - \lambda )^{ - 1} \parallel dist(\lambda ,\sigma (T))< \infty .$

The examples of Kupin-Treil [9] show that the result is no longer true if we replace the condition rank (I−T^*T)<∞ by its weakened versionG, whereG denotes the class of nuclear operators.

We prove in this paper that, however, the following theorem holds

LetH^∞ be the algebra of bounded analytic functions in the unit disk D. Let I=I(f₁,..., f_N) be the ideal generated by f₁,..., f_N∈H^∞ and J=J(f₁,..., f_N) the ideal of the functions f∈H^∞ for which there exists a constant C=C(f) such that |f(z)|≤C(|f₁(z)|+...; +|f_N(z)|), z∈D. It is clear that $I \subseteq J$ , but an example due to J. Bourgain shows that J is not, in general, in the norm closure of I. Our first result asserts that J is included in the norm closure of I if I contains a Carleson-Newman Blaschke product, or equivalently, if there exists s>0 such that $\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $

Our second result says that there is no analogue of Bourgain's example in any Hardy space H^p, 1≤p<∞. More concretely, if g∈H^p and the nontangential maximal function of $|g(z)|/\sum\nolimits_{j = 1}^N {|f_j (z)|} $ belongs to L^p (T), then g is in the H^p-closure of the ideal I.

We prove a second fundamental theorem in the sense of Nevanlinna's theory of meromorphic functions replacing the constants a in $\bar N$ (r, f, a) by rational functions R with R(∞)=a. The key argument is Ahlfors's second fundamental theorem from his theory of covering surfaces.

Characteristic numbers of families of maps of nodal curves to P² are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.

We study different classes of Jensen measures for plurisubharmonic functions, in particular the relation between Jensen measures for continuous functions and Jensen measures for upper bounded functions. We prove an approximation theorem for plurisubharmonic functions in B-regular domain. This theorem implies that the two classes of Jensen measures coincide in B-regular domains. Conversely we show that if Jensen measures for continuous functions are the same as Jensen measures for upper bounded functions and the domain is hyperconvex, the domain satisfies the same approximation theorem as above.

The paper also contains a characterisation in terms of Jensen measures of those continuous functions that are boundary values of a continuous plurisubharmonic function.