The Calderón–Zygmund Theorem with an L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} Mean Hörmander Condition

In 2019, Grafakos and Stockdale introduced an Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q$$\end{document} mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} boundedness for the “limited-range” instead of 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< p < \infty $$\end{document}. However, in this paper, we show that the Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q$$\end{document} mean Hörmander condition is actually enough to obtain the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} boundedness for all 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< p < \infty $$\end{document} even in the worst case q=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=1$$\end{document}. We use a similar method to that used by Fefferman (Acta Math 124:9–36, 1970): form the Calderón–Zygmund decomposition with the bounded overlap property and approximate the bad part. Also we give a criterion of the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} boundedness for convolution type singular integral operators under the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} mean Hörmander condition.


Introduction
The Calderón-Zygmund theorem is a well-known tool to investigate the L p boundedness of singular integral operators. It was originally developed by Calderón and Zygmund [2] and later improved by Hörmander [6]. Today it is usually stated as follows: Theorem A Let T be a singular integral operator with a kernel K . Suppose that T is bounded from L p 0 (R d ) to L p 0 ,∞ (R d ) for some 1 < p 0 < ∞ and its kernel K satisfies the Hörmander condition; where the supremum sup B⊂R d is taken over all balls B in R d , c(B) is the center of B, 2B denotes the ball with the same center as B and whose radius is twice as long. Then T is bounded from L 1 (R d ) to L 1,∞ (R d ). It follows that T is bounded on L p (R d ) for all 1 < p < p 0 .
In 2019, Grafakos and Stockdale [5] introduced an L q mean Hörmander condition (H q condition for short); and proved the following: Theorem B [5] Let T be a singular integral with a kernel K . Suppose that T is bounded from L p 0 (R d ) to L p 0 ,∞ (R d ) for some 1 < p 0 < ∞ and its kernel K satisfies the H q condition for some 1 ≤ q < p 0 where q denotes the Hölder conjugate of q. Then T is bounded from L q (R d ) to L q,∞ (R d ). It follows that T is bounded on L p (R d ) for all q < p < p 0 .
Note that [K ] q 1 ≤ [K ] q 2 if 1 ≤ q 1 ≤ q 2 ≤ ∞ and the H ∞ condition is the same as the classical Hörmander condition (1.1). They named Theorem B 'limited-range Calderón-Zygmund theorem' because it implies the L p boundedness not for all 1 < p < p 0 but for the 'limited-range'; q < p < p 0 . However, as stated in [5], they did not find any operators that satisfy the assumption of Theorem B and not bounded on L q . In this sense, there is no evidence that it is truly a limited-range theorem. In this paper, we show that it is not actually limited-ranged. In fact, the H q condition is enough for the L 1 → L 1,∞ boundedness even in the worst case q = 1.

Theorem 1 Let T be a singular integral operator with a kernel K . Suppose that T is bounded from
for some 1 < p 0 < ∞ and its kernel K satisfies the H 1 condition; Our proof is motivated by Fefferman's proof of the L 1 → L 1,∞ boundedness of strongly singular integral operators (see [4,Theorem 2']). In the proof, we form the Calderón-Zygmund decomposition of f ; f = g + b, and approximate the bad part b by a certain function b. Also we will give a criterion of the L 2 boundedness for convolution type singular integral operators under the H 1 condition.
where V d denotes the volume of the d dimensional unit ball, and define K ε,R : This is a natural generalization of the classical result stated by using the H ∞ condition (see [1,Theorem 3], [3, Proposition 5.5]). Note that it remains an open question: is the H 1 condition actually weaker than the classical one? As of this writing, we have no examples of K such that This paper is organized as follows. We prove Theorem 1 in Sect. 2 and Theorem 2 in Sect. 3. In Sect. 4, we will remark on the H 1 → L 1 boundedness under the assumption of Theorem 1.

Proof of Theorem 1
We use the following lemma: where -C d denotes a constant which depends only on the dimension d, -B j denotes the smallest ball circumscribing Q j , - then immediately it follows that where M is the Hardy-Littlewood maximal function with uncentered balls. 1 Note that our good part g (2. 3) and bad part b (2.4) are different from usual ones. Ordinarily, they are defined by to guarantee the zero mean condition b j = 0. However, our proof does not require it, hence we use our simpler definition. Now we are going to give the proof of Theorem 1.

Proof of Theorem 1
, t, λ > 0 and form the Calderón-Zygmund decomposition of f at height tλ (where t is given later to set appropriate estimates). In addition, where r j is the radius of B j and s j := r j /2. We approximate b j by b j := b j * ϕ j and b by b := j b j .
Now we have f = g −( b −b)+ b and it suffices to show the following inequalities.

Proof of (2.10)
Since We will estimate the second term by the H 1 condition (1.3). For each j and x ∈ R d \ 2B j , we write since T is a singular integral operator with a kernel K . Therefore, for each j, we have It follows that Proof of (2.11) By the same argument as in the proof of (2.9), we have Since it is obvious that Therefore, it follows from the bounded overlap property (2.2), Hence we conclude that Combining estimates above, we obtain Finally, remember that t and ϕ are arbitrary. Since inf ϕsatisfies (2.8)

The Proof of Theorem 2
We use the following lemma:

Proof of Lemma 1 It is obvious that
under the condition 2|y| ≤ 2r ≤ |x|, the second and fifth terms are bounded by 3B/2, the third and fourth terms are bounded by 2B.

Remark
We can also obtain the H 1 → L 1 boundedness under the assumption of Theorem 1.

Theorem 3 Let T be a singular integral operator with a kernel K . Suppose that T is bounded from
To see this, note that Theorem 1 implies that T is bounded on L p (R d ) for any 1 < p < p 0 . Hence we assume that T is bounded on L p 0 (R d ) for some 1 < p 0 < ∞ without loss of generality. Now we can show the H 1 (R d ) → L 1 (R d ) boundedness. We do not prove it here because its proof is the almost same as that of the classical theorem (see [3, Proposition 6.2, Corollary 6.3]).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.