Clarkson's inequalities for periodic Sobolev space

The paper is devoted to developing the proof of Clarkson's inequalities for periodic functions belonging to the Sobolev space. The norm of the space has not been considered earlier. The importance of the discussed issue rests with the need to develop fundamentals in research of error estimation using functional analysis techniques. Thus, the error of approximation is represented via a linear functional over the Banach space. The approach allows searching for new criteria of approximation quality and ways to optimize the numerical method. Parameters that are responsible for technique quality need to be previously investigated in respect of extreme values. Therefore, fundamental features, such as uniform convexity, should be proved for being further used in extremum problems solving.
The aim of the study is to prove the uniform convexity for the Sobolev space of periodic functions normed without pseudodifferential operators. The norm includes integrals over the fundamental cube. The integrands are the absolute values of derivatives of all orders raised to the $p$-th power. The exponent p generates the non-Hilbert space. The methods used in the study include application of inverse Minkowski inequalities stated either for sums or for integrals to periodic functions from the Sobolev space. Furthermore, functional analysis concepts and techniques are used. As a result, Clarkson's inequalities are proved for periodic functions from the Sobolev space.
The obtained results are important for solving extremum problems. The problems require using only uniformly convex functional spaces, for example, the problem of error estimation of numerical integration of functions from the Sobolev space with the above norm.