This article will explore the connections between ”Harmonic Analysis” and several fundamental estimates, including the ”Poincar´e-Sobolev”, ”Trudinger”, and ”John-Nirenberg inequalities”. Our central theme is the ”self-improving property” of generalized Poincar´e-type inequalities. Through this exploration, we’ll show how these connections eliminate the need to rely on ”Potential Operators”, offering a more flexible approach that yields more precise estimates, especially for singular measures. We’ll also outline how certain older results can be modernized, including fractional-type results that improve upon celebrated findings by Bourgain-Brezis-Mironescu.