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In this paper, we will give characterizations for weighted Bergman spaces with admissible weights. The first criterion is established by the invariant gradient on the unit ball of $${\mathbb {C}}^N$$. The other criterion is given by the radial derivative of holomorphic functions in the unit ball.

Let $$\mathbb{B}$$ be the unit ball of a complex Banach space X. In this paper, we generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $$\mathbb{B}$$ by using the radial derivative. Next, we define an extended Cesaro operator Tφ with holomorphic symbol φ and characterize those φ for which Tφ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those φ for which Tφ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol φ. When $$\mathbb{B}$$ is the open unit ball of a finite dimensional complex Banach space X, this additional assumption is automatically satisfied.

Motivated and inspired by the improved Hardy inequalities studied in their well-known works by Brezis and Vazquez (Rev Mat Univ Complut Madrid 10:443–469, 1997) and Brezis and Marcus (Ann Scuola Norm Sup Pisa Cl Sci 25(1–2):217–237, 1997), we establish in this paper several identities that imply many sharpened forms of the Hardy type inequalities on upper half spaces $$\left\{ x_{N}>0\right\} $$. We set up these results for the distance to the origin, the distance to the boundary of any strip $$ \mathbb {R} ^{N-1}\times \left( 0,R\right) $$ and the distance to the hyperplane $$\left\{ x_{N}=0\right\} $$, using both the usual full gradient and radial derivative (in the case of distance to the origin) or only the partial derivative $$\frac{\partial u}{\partial x_{N}}$$ (in the case of distance to the boundary of the strip or hyperplane). One of the applications of our main results is that when $$\Omega $$ is the strip $$\mathbb {R}^{N-1}\times \left( 0,2R\right) $$, the bound $$\lambda \left( \Omega \right) $$ given by Brezis and Marcus in Brezis and Marcus (1997) can be improved to $$\frac{z_{0}^{2}}{R^{2}}$$, where $$z_{0} =2.4048 \ldots $$ is the first zero of the Bessel function $$J_{0}\left( z\right) $$. Our approach makes use of the notion of Bessel pairs introduced by Ghoussoub and Moradifam (Math Ann 349(1):1–57, 2011) and (Functional inequalities: new perspectives and new applications. Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2013) and the method of factorizations of differential operators. In particular, our identities and inequalities offer sharpened and more precise estimates of the second remainder term in the existing Hardy type inequalities on upper half spaces in the literature, including the Hardy-Sobolev-Maz’ya type inequalities.

Possibilities for defining the radial derivative of the delta distribution $\delta(\underline{x})$ in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar to but different from the standard distributions. The radial derivative of $\delta(\underline{x})$ then belongs to that new class of so-called signumdistributions. It is shown that these signumdistributions obey easy-to-handle calculus rules which are in accordance with those for the standard distributions in $\mathbb{R}^m$., Comment: 18 pages

In this paper, we obtain complete characterizations of the boundedness and compactness of the products of the multiplication and the radial derivative operator MuR from mixed norm spaces \({H(p, q, \phi)}\) to Zygmund-type spaces on the unit ball.

Least-squares collocation may be used for the estimation of spherical harmonic coefficients and their error and error correlations from GOCE data. Due to the extremely large number of data, this requires the use of the so-called method of Fast Spherical Collocation (FSC) which requires that data is gridded equidistantly on each parallel and have the same uncorrelated noise on the parallel. A consequence of this is that error-covariances will be zero except between coefficients of the same signed order (i.e., the same order and the same coefficient type C–C or S–S). If the data distribution and the characteristics of the data noise are symmetric with respect to the equator, then, within a given order and coefficient type, the error-covariances amongst coefficients whose degrees are of different parity also vanish. The deviation from this “ideal” pattern has been studied using data-sets of second order radial derivatives of the anomalous potential. A total number of points below 17,000 were used having an equi-angular or an equal area distribution or being associated with points on a realistic GOCE orbit but close to the nodes of a grid. Also the data were considered having a correlated or an uncorrelated noise and three different signal covariance functions. Grids including data or not including data in the polar areas were used. Using the functionals associated with the data, error estimates of coefficients and error-correlations between coefficients were calculated up to a maximal degree and order equal to 90. As expected, for the data-distributions with no data in the polar areas the error-estimates were found to be larger than when the polar areas contained data. In all cases it was found that only the error-correlations between coefficients of the same order were significantly different from zero (up to 88%). Error-correlations were significantly larger when data had been regarded as having non-zero error-correlations. Also the error-correlations were largest when the covariance function with the largest signal covariance distance was used. The main finding of this study was that the correlated noise has more pronounced impact on gridded data than on data distributed on a realistic GOCE orbit. This is useful information for methods using gridded data, such as FSC.

The recent development in this laboratory of an automated data capture system, for refractometric optics on the analytical ultracentrifuge has removed the requirement for tedious and time consuming manual acquisition which had led to a decline in the use of schlieren optics. At the same time this system has increased the amount of data easily available from such an optical system with maintained or increased precision. From the advent of such a system has arisen the need for a package to facilitate the analysis of these data and to extend the range of analytical methods used. Using the improved data sets now available has also enabled us to successfully use methods which have lapsed in popularity over the last two decades. We have also been able to successfully apply radial derivative methods (Bridgman 1942) which have not routinely been applied to the analysis of sedimentation velocity experiments using schlieren optics. In this paper we describe the methods we have so far used to analyse data and present results for previously well defined molecules to demonstrate that the results obtained are reliable.

We answer a question posed by B. Sims in 1972, by exhibiting an example of a Banach spaceX such that the numerical radius attaining operators onX are not dense. Actually,X is an old example used by J. Lindenstrauss to solve the analogous problem for norm attaining operators, but the proof for the numerical radius seems to be much more difficult. Our result was conjectured by C. Cardassi in 1985.

The “radial” polynomiality criterion for entire functions of several complex variables is proved.