My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, for any $1\leq p<\infty$. On $L^2$ it is easy to compute the norm using Plancherel. But I am not sure how to compute the norm in general of this induced operator as a bounded operator from $L^1$ to $L^1_w$. The most common 'norm' I have seen (for instance in Stein) is that of $||\hat{K}||_{L^\infty} + C$, where the $\hat{K}$ denotes the distributional Fourier transform, and $C$ is the constant such that $$|K(x)|\leq\frac{C}{|x|^n}$$ I know that $K$ commutes with translations, but am wondering about general transformations. A norm I am interested in is the constant $A$ such that
$$\int\limits_{\mathbb{R}^n\setminus B_{2\delta}(y)} |K(x-y)-K(x-\bar{y}) |dx\leq A$$ whenever $\bar{y}\in B_\delta(y)$, some $\delta>1$.
How does one prove that, if $K$ satisfies the above decay and regularity estimates, and if $T$ is a general transformation matrix on $\mathbb{R}^n$, then
$$\int\limits_{\mathbb{R}^n\setminus B_{2\delta}(y)} |K(T(x-y))-K(T(x-\bar{y})) ||\det(T)|dx\leq p(1+||T||||T^{-1}||)A?$$ where $p(x)$ is some $L^1_{loc}$ function? Any help would be appreciated.
