  The classical Gagliardo-Nirenberg inequality, known as an interpolation
inequality, involves Lebesgue norms of functions and their derivatives. We
established an interpolation lemma to connect Lebesgue and H\"older spaces,
thus extending the Gagliardo-Nirenberg inequality. This extension involved
substituting arbitrary Sobolev norms with appropriate H\"older norms, allowing
for a wider range of applicable parameters in the inequality.
