Boundedness of classical operators of harmonic analysis, such as maximal functions, fractional integrals and singular integrals have been extensively investigated in various function spaces. For example, we know that Hardy-Littlewood maximal operator $M$ is bounded from Lebesgue space $L^p(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ for $p>1$ and from $L^1(\mathbb{R}^n)$ to weak Lebesgue space $WL^1(\mathbb{R}^n)$; Riesz potential $I_{\alpha},~0<\alpha<n$ is bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for $1<p<\frac{n}{\alpha}$ and $q=\frac{np}{n-\alpha p}$ and from $L^1(\mathbb{R}^n)$ to $WL^q(\mathbb{R}^n)$ for $q=\frac{n}{n-\alpha}$; Calderon-Zygmund operator $T$ is bounded from $L^p(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ for $p>1$ and from $L^1(\mathbb{R}^n)$ to $WL^1(\mathbb{R}^n)$. These results are classical and can be found many textbook. These results have been extended to many other function spaces such as Morrey space, Lorentz space, Orlicz space etc, which are generalizations of Lebesgue space.
My question is: Where do we use these results? What is the advantage of these results? What is the importance of these operators? Could you give me concrete examples?