Determine an operator is weak type (1,1) by its Fourier multiplier I have an interest on associated Fourier multipler $m$ for a given operator $T$ defined for $f\in L^p(R^d)$, $1\le p<\infty$ by 
$$
\hat{Tf}(\xi)=m(\xi)\tilde{f}(\xi),\ \ \xi\in R^d
$$ 
where $\hat{f}$ is the Fourier transform of $f$. There are some results concerning the form of Fourier multiplier to detemine whether $T$ is a strong type (p,p) operator, $1\le p<\infty$. For instance, the multiplier of the $j$-th Riesz transform $R_j$ is $\frac{\xi_j}{|\xi|}$,$1\le j\le d$, that is,
$$
(R_j f)\hat{\,}(\xi)=\frac{\xi_j}{|\xi|}\hat{f}(\xi),\ \ \xi\in R^d
$$
which is of course a strong type (p,p) operator for any $1< p<\infty$ (If d=1, $R_j$ is just a Hilbert tranform). 
Unfortuately, few books and reference (All that I can know) discuss the characterization or some sufficient conditions for $T$ of weak type (1,1) from its multipier. I wish someone can give me some relative materials on this topic (the case weak type (1,1)). Thanks in advance.
 A: First note that $R_j$ is not of strong type (1,1). The Hilbert transform of an $L^1$ function doesn`t even need to be locally integrable.
The question you are asking is one of the main concerns of classical singular integral theory. A central notion in this context is that of a Calderón-Zygmund operator. Recall that a Fourier transform translates multiplication into convolution. Therefore a multiplier operator $\widehat{Tf}=m\widehat{f}$ can also be written $Tf=K*f$ with $\widehat{K}=m$ ($K$ may be only a distribution). 
Definition. $T$ is called a Calderón-Zygmund operator if 


*

*$K\in L^1_{loc}(\mathbb{R}^d\backslash{0})$

*$m\in L^\infty$

*There exists $C>0$ such that for all $y\not=0$ we have
$$\int_{|x|>2|y|} |K(x-y)-K(x)| dx\le C$$
The second condition guarantees boundedness of $T$ in $L^2$ (Plancherel's theorem). The last condition can be seen as a bound on the derivative of $K$ but is more general as it doesn't require $K$ to be differentiable.
Using a technique called Calderón-Zygmund decomposition one can show that under these assumptions, $T$ satisfies a weak $(1,1)$ bound. One then usually uses Marcinkiewicz interpolation and duality to obtain strong $L^p$ bounds for $1<p<\infty$. 
The weak (1,1) boundedness is often not stated as a separate result but lies at the heart of the proof of the strong bounds.
You can read all this in any standard reference on Euclidean harmonic analysis, including but not limited to the following:


*

*E.M. Stein. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press. 1993.

*E.M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press. 1970.

*Duoandikoetxea. Fourier Analysis. American Mathematical Society. 2001.
The first one is the standard reference which is an excellent book but not recommended for somebody seeing the material for the first time.  The second one is a bit more accessible. The last one is written for beginners in the subject and I would start there if I were you.
