In this paper, we shall discuss about Bessel-Riesz operators. Kurata et al. have investigated their boundedness on generalized Morrey spaces with weight. The boundedness of these operators on Lebesgue spaces and Morrey spaces will be reproved using a different approach. Moreover, we also find the norm of the operators are bounded by the norm of the kernels.

We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in [3,7]. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of p-summable sequences. Analogous results for weak type discrete Morrey spaces is also presented. In addition, we show that each of these inclusion relations is proper. Some connections between inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed. MSC (2010): 42B35, 46A45, 46B45.

In this paper, we prove the boundedness of Bessel-Riesz operators on generalized Morrey spaces. The proof uses the usual dyadic decomposition, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator.Our results reveal that the norm of the operators is dominated by the norm of the kernels.

We revisit the space p of p -summable sequences of real numbers. In particular, we show that this space is actually contained in a (weighted) inner product space. The relationship between p and the (weighted)inner product space that contains p is studied. For 2 p  we also obtain a result which describe how the weighted inner product space is associated to the weights.

In this paper, we wish to define a 2-inner product, non-standard, possibly with weights, on p . For this purpose, we aim to obtain a different 2-norm ., . 2, v , w , which is not equivalent to the usual 2-norm ., . p on p (except with the condition p = 2 ), satisfying the parallelogram law. We discuss the properties of the induced 2-norm ., . 2, v , w and its relationships with the usual 2-norm on p . We also find that the 2-inner product ., .|. v , w is actually defined on a larger space.