LetM be a (n−1)-dimensional manifold in Rⁿ with non-vanishing Gaussian curvature. Using an estimate established in the early work of the author [4], we will improve the known result of Y. Domar on the weak spectral synthesis property by reducing the smoothness assumption upon the manifold M. Also as an application of the method, a uniqueness property for partial differential equations with constant coefficients will be proved, which for some specific cases recovers or improves Hörmander's general result.

We determine the smallest Schatten class containing all integral operators with kernels in L_p(L_{p', q})^symm, where 2 < p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the C^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces A²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spaces A²(C^N), N≧1.

A multidimensional extension of Bailey's transform is utilised to deduce two new generating relations of quite a general character. These expressions are then specialised to give more practical formulae in terms of Karlsson's generalised Kampé de Fériet functions which embody very many generating relations. A number of interesting special cases are given in an appendix which includes results involving Lauricella polynomials, generalised hypergeometric polynomials and the polynomials of Meixner, Charlier and Laguerre.

We consider singular integral operators of the form (a)Z₁L⁻¹Z₂, (b)Z₁Z₂L⁻¹, and (c)L⁻¹Z₁Z₂, where Z₁ and Z₂ are nonzero right-invariant vector fields, and L is the L²-closure of a canonical Laplacian. The operators (a) are shown to be bounded on L^p for all p∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for any p∈[1, ∞).

It is shown that the Bargmann-Fock spaces of entire functions, A^p (C), p≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces.

We define a partition of Z into intervals {I_j} and prove the Littlewood-Paley inequality ‖f‖_p≦C_p‖Sf‖_p, 2≦p<∞. Here f is a function on [o, 2π) and $Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} $ . This is a new example of a partition having the Littlewood-Paley property since the {I_j} are not of the type obtained by iterating lacunary partitions finitely many times.

We prove that there is a unique way to construct a geometric scale of Hilbert spaces interpolating between two given spaces. We investigate what properties of operators, other than boundedness, are preserved by interpolation. We show that self-adjointness is, but subnormality and Krein subnormality are not. On the way to this last result, we establish a representation theorem for cyclic Krein subnormal operators.

In this paper, we prove a good-λ inequality between the nontangential maximal function and the square area integral of a subharmonic function u in a bounded NTA domain D in Rⁿ. We achieve this by showing that a weighted Riesz measure of u is a Carleson measure, with the Carleson norm bounded by a constant independent of u. As consequences of the good-λ inequality, we obtain McConnell-Uchiyama's inequality and an analogue of Murai-Uchiyama's inequality for subharmonic functions in D.