I am trying to prove the following fact.
Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define $$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$ where P.V. means principle value. Then (I guess) $$||T(u)||_{L^2{(\mathbb{R}^1)}}\leq C||u||_{L^2(\mathbb{R}^1)}.$$
My attempt: Notice $|T(u)|\leq C\int_{\mathbb{R}^1}\frac{|u(y)|}{|x-y|^{1/2}}dy=XI_{\frac12}(|u|)$, here $I_{\alpha}$ is the Riesz potential. Using the boundedness of Riesz potential $I_{1/2}$ from $\mathcal{H}^1(\mathbb{R}^1)$ to $L^2(\mathbb{R}^1)$, one gets $$||T(u)||_{L^2(\mathbb{R}^1)}\leq C||u||_{\mathcal{H}^1(\mathbb{R}^1)}.$$
My approach does not give the bound of in $L^2\to L^2$. I think the reason is that I did not use the cancellation of the kernel. I am hoping to get some analogous results to the Calderon-Zygmund operator. However, my knowledge about that is limited to the classical theory of Calderon-Zygmund operator.
Does anyone know the references study this type of operator? Or any idea how to prove the fact. Or my guess is wrong? Any comment is welcome.
