Uniform convexity of the weighted Sobolev space

In this paper, uniform convexity of the non-Hilbert weighted Sobolev space of functions of several variables is established. The Sobolev space is normed without pseudodifferential operators. The norm contains partial derivatives of all intermediate orders from zero to the given one. The norm includes a weight function represented in this work in the most general form. The norm is expressed in terms of an improper integral over the entire space of several variables. The first and second Clarkson inequalities are established for such norms. Validity of the two inequalities may be expressed in geometrical terms. It means that the mid-point of a variable chord of a unit sphere of the space cannot approach the surface of the sphere unless the length of the chord tends to zero. Each of the two Clarkson inequalities is valid on a certain range of the summability index of the Banach space under consideration. The limit value of the range is p = 2, which corresponds to the Hilbert space. In this case, the two inequalities turn into a single identity. The proof is based on well-known numerical inequalities. The derivation involves reverse Minkowski inequalities for finite sums and integrals, as well as integration of the right- and left-hand sides of the inequalities.

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