CoSy seminar with Natasha Samko

Everyone is welcome and the first 40 people to register will be treated to a free lunch sandwich. If you do not want lunch, you are still welcome to join.

Abstract:

In this presentation we focus on the contribution of non-separable spaces to solving the evolution equation. As related to that study we also concentrate on the study of weighted boundedness of some operators of harmonic analysis in the generalized Morrey spaces, a known example of non-separable spaces, with the emphasis on maximal and Hardy operators.

We discuss the Cauchy problem for the diffusion equation, which is a typical example of evolution equation, with the emphasis on the case where the space of the initial data is non-separable. More precisely, we deal with weighted generalized Morrey spaces and show that the semigroup corresponding to the Cauchy problem for the diffusion equation remains uniformly bounded in such spaces, though non-separable, under the only assumption on the weight that the maximal operator is bounded in this space. As regards the convergence of this semigroup, it holds in a weaker norm which is caused by the non-separability of the space. In this relation, we provide some recent results on the boundedness of the maximal operator in the generalized Morrey spaces with radial weights. These results are obtained by means of pointwise estimates for Hardy operators, which is a key tool in the study of Hardy operators in Morrey space. Since Hardy operators play an important role in various problems of analysis and applications, we also present a new result on two weight boundedness of Hardy operators in Morrey spaces in the case of general weights.

References:

• A. Kufner, L.E. Persson, and N. Samko. Weighted Inequalities of Hardy Type. Second edition. World Scientific Publishing Co. Inc., River Edge, NY, 2017 (480pp.).

• P.J. Nicklasson, N. Samko, H. Singh, and D. Storni. More on Contributions of Non-Separable Spaces to Solving the Evolution Equation: The Case of Weighted Generalized Morrey Spaces. J Fourier Anal Appl 32, 8 (2026) (16pp.).

• N. Samko. Weighted boundedness of certain sublinear operators in generalized Morrey spaces on quasimetric measure spaces under the growth condition. J. Fourier Anal. Appl., 28(2): Paper No. 27, 2022 (27 pp.).

This is a lecture in the seminar series held by CIM (Centre for Interdisciplinary Mathematics).