Sobolev's embedding on time scales

Area of Science:

• Mathematics
• Analysis

Background:

• Sobolev spaces are fundamental in analysis, with embeddings crucial for understanding function properties.
• Previous work established Sobolev embeddings on time scales for specific cases (p=2) and generalized cases.

Purpose of the Study:

• To present embeddings of Sobolev spaces $W^{k,p}(\mathbb{T})$ for $k o \infty$ on time scales.
• To develop general Sobolev embeddings for $W^{k,p}(\mathbb{T})$ on time scales using the established embeddings.

Main Methods:

• Developing embeddings for Sobolev spaces $W^{k,p}(\mathbb{T})$ on time scales.
• Utilizing these embeddings to derive general Sobolev embeddings for higher-order Sobolev spaces on time scales.

Main Results:

• Established embeddings for Sobolev spaces $W^{k,p}(\mathbb{T})$ for $k \to \infty$ on time scales.
• Developed general Sobolev embeddings for $W^{k,p}(\mathbb{T})$ on time scales, where k is a non-negative integer.

Conclusions:

• The study successfully extends Sobolev embedding theory to higher-order Sobolev spaces on arbitrary time scales.
• The developed embeddings provide a valuable tool for analyzing functions and solving differential equations on time scales.