Reference concerning weak-type (1,1) operator in $\mathbb{R}^d$

Question

I wish someone give me some reference on weak-type (1,1) operator in the d-dimensional Euclidean space. Thanks for any kind help.

Answer 1

Let $(X, \mathcal{X}, \mu)$ and $(Y, \mathcal{Y}, \nu)$ be measure spaces, and suppose that $T$ is an operator mapping measurable functions defined on $X$ to measurable functions defined on $Y$. Let $1 \le p, q \le \infty$. We say that $T$ is of strong-type (p,q) if there is an absolute constant $C > 0$ such that $||Tf||_q \le C ||f||_p$, for all $f$ for which $T$ is defined. In applications, we frequently begin with $T$ defined on some dense subclass of $L^p(X)$ (say $X$ and $Y$ are Euclidean spaces with Lebesgue measure and $T$ is a sublinear operator defined on the Schwartz class or for continuous functions of compact support). In this case, an a priori estimate of this form guarantees that $T$ may be uniquely and meaningfully extended to all of $L^p$.

Define the distribution function of f by $\lambda_f(t) := \mu(\{x \in X: |f(x)| > t\})$. Then we have $\displaystyle ||f||_p^p = \int_X |f|^p \ge \int _{|f| > t} |f|^p \ge t^p \lambda_f(t) $, which is Chebyshev's inequality.

That is, $\lambda_f(t) \le \frac{||f||_p^p}{t^p}$. So, if $T$ satisfies the strong (p,q) bound, we get from Chebyshev that $\lambda_{Tf}(t) \le C \frac{||f||_p^q}{t^q}$. We say that $T$ is of weak type (p,q) if it satisfies this weaker bound. So a strong type (p,q) operator is of weak type (p,q), but this implication cannot be reversed in general.

One reason we care about weak type bounds is because there are a series of results (such as the Marcinkiewicz interpolation result, which you can read about in this blog post by Terry Tao http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/) which essentially say that, under some mild conditions, two weak type bounds $(p_1, q_1), (p_2, q_2)$ imply intermediary strong type bounds.