Abstract
In 2019, Grafakos and Stockdale introduced an \(L^q\) 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 \(L^p\) boundedness for the “limited-range” instead of \(1< p < \infty \). However, in this paper, we show that the \(L^q\) mean Hörmander condition is actually enough to obtain the \(L^p\) boundedness for all \(1< p < \infty \) even in the worst case \(q=1\). 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 \(L^2\) boundedness for convolution type singular integral operators under the \(L^1\) mean Hörmander condition.
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1 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 \(L^{p_0}(\mathbb {R}^d)\) to \(L^{p_0, \infty }(\mathbb {R}^d)\) for some \(1< p_0 < \infty \) and its kernel K satisfies the Hörmander condition;
where the supremum \(\sup _{B \subset \mathbb {R}^d}\) is taken over all balls B in \(\mathbb {R}^d\), 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 \(L^1(\mathbb {R}^d)\) to \(L^{1, \infty }(\mathbb {R}^d)\). It follows that T is bounded on \(L^p(\mathbb {R}^d)\) 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 \(L^{p_0}(\mathbb {R}^d)\) to \(L^{p_0, \infty }(\mathbb {R}^d)\) for some \(1< p_0 < \infty \) and its kernel K satisfies the \(H_{q'}\) condition for some \(1 \le q < p_0\) where \(q'\) denotes the Hölder conjugate of q. Then T is bounded from \(L^{q}(\mathbb {R}^d)\) to \(L^{q, \infty }(\mathbb {R}^d)\). It follows that T is bounded on \(L^p(\mathbb {R}^d)\) for all \(q< p < p_0\).
Note that \([K]_{q_1} \le [K]_{q_2}\) if \(1 \le q_1 \le q_2 \le \infty \) and the \(H_\infty \) 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 \(L^1 \rightarrow L^{1, \infty }\) 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 \(L^{p_0}(\mathbb {R}^d)\) to \(L^{p_0, \infty }(\mathbb {R}^d)\) for some \(1< p_0 < \infty \) and its kernel K satisfies the \(H_{1}\) condition;
Then T is bounded from \(L^1(\mathbb {R}^d)\) to \(L^{1, \infty }(\mathbb {R}^d)\) with a constant proportional to \(\Vert T\Vert _{L^{p_0} \rightarrow L^{p_0, \infty }} + [K]_{H_1}\).
Our proof is motivated by Fefferman’s proof of the \(L^1 \rightarrow L^{1, \infty }\) 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 \(\widetilde{b}\).
Also we will give a criterion of the \(L^2\) boundedness for convolution type singular integral operators under the \(H_1\) condition.
Theorem 2
Let \(K \in \mathcal {S}'(\mathbb {R}^d) \cap { L^1_{\text {loc}} } (\mathbb {R}^d \setminus \{ 0 \})\) be such that
where \(V_d\) denotes the volume of the d dimensional unit ball, and define \(K_{\varepsilon , R} := K \chi _{ \{\varepsilon< |x| < R \} }\) for any \(0< \varepsilon< R < \infty \). Then \(K_{\varepsilon , R}\) satisfies
with a constant proportional to \(A + B + [K]_{H_1}\).
This is a natural generalization of the classical result stated by using the \(H_\infty \) 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 \([K]_{H_\infty } = \infty \) but \([K]_{H_1} < \infty \).
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 \(H^1 \rightarrow L^1\) boundedness under the assumption of Theorem 1.
2 Proof of Theorem 1
We use the following lemma:
Lemma A
[4, Decomposition Lemma] Let \(f \in L^1(\mathbb {R}^d)\) and \(\lambda > 0\). Then there exists a family of disjoint dyadic cubes \(\{Q_j\}_j\) such that
where
-
\(C_d\) denotes a constant which depends only on the dimension d,
-
\(B_j\) denotes the smallest ball circumscribing \(Q_j\),
-
\(\Omega := \bigcup _j Q_j\),
-
\(\Omega ^* := \bigcup _j 2B_j\).
Furthermore, if we define
then immediately it follows that
Lemma A is essentially the Whitney decomposition of \(\{ \, x \in \mathbb {R}^d \, : \, Mf(x) > \lambda \, \}\), where M is the Hardy–Littlewood maximal function with uncentered balls.Footnote 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 \(\int b_j = 0\). 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
Fix \(f \in L^1(\mathbb {R}^d) \cap L^\infty (\mathbb {R}^d)\), \(t, \lambda > 0\) and form the Calderón–Zygmund decomposition of f at height \(t \lambda \) (where t is given later to set appropriate estimates). In addition, fix \(\varphi \in C_c^\infty (\mathbb {R}^d)\) such that
and write \(\varphi _j(x) := s_j^{-d} \varphi (s_j^{-1} x)\) where \(r_j\) is the radius of \(B_j\) and \(s_j := r_j / 2\). We approximate \(b_j\) by \(\widetilde{b}_j := b_j * \varphi _j\) and b by \(\widetilde{b} := \sum _j \widetilde{b}_j\).
Now we have \(f = g - (\widetilde{b} - b) + \widetilde{b}\) and it suffices to show the following inequalities.
Proof of (2.9)
Since T is bounded from \(L^{p_0}(\mathbb {R}^d)\) to \(L^{p_0, \infty }(\mathbb {R}^d)\), it follows that
\(\square \)
Proof of (2.10)
Since
it follows that
We will estimate the second term by the \(H_1\) condition (1.3). For each j and \(x \in \mathbb {R}^d \setminus 2B_j\), we write
and
since T is a singular integral operator with a kernel K. Therefore, for each j, we have
It follows that
\(\square \)
Proof of (2.11)
By the same argument as in the proof of (2.9), we have
Since it is obvious that
it is enough to show that \(\Vert \widetilde{b}\Vert _\infty \lesssim \lambda \). For each j, we have
Therefore, it follows from the bounded overlap property (2.2),
Hence we conclude that
\(\square \)
Combining estimates above, we obtain
Finally, remember that t and \(\varphi \) are arbitrary. Since \(\inf _{\varphi satisfies (2.8) } \Vert \varphi \Vert _\infty = V_d^{-1}\), we conclude that
\(\square \)
3 The Proof of Theorem 2
We use the following lemma:
Lemma 1
If \(K \in { L^1_{\text {loc}} } (\mathbb {R}^d \setminus \{ 0 \})\) satisfies (1.5) and (1.6), then
Proof of Lemma 1
It is obvious that
and the first term is bounded by \([K]_{H_1}\). To estimate other terms, note that (1.5) implies
for any \(c>1\). Since we have
under the condition \(2|y| \le 2r \le |x|\), the second and fifth terms are bounded by 3B/2, the third and fourth terms are bounded by 2B. \(\square \)
Proof of Theorem 2
Fix \(0< \varepsilon< R < \infty \) and \(\xi \in \mathbb {R}^d\). Since it is obvious that
we assume \(\xi \ne 0\) and write \(s := |\xi |^{-1}\). If we decompose \(\widehat{K_{\varepsilon , R}} (\xi ) \) as
then we easily get
To estimate \(I_3\), fix a radial function \(\varphi \in C_c^\infty (\mathbb {R}^d)\) such that
and define \(\varphi _s(x) := s^{-d} \varphi (s^{-1} x)\). Moreover, rewrite
and introduce
We decompose \(I_3\) into \((I_3 - I_4) + (I_4 + I_5) - (I_5 - I_6) - I_6\). By (3.2), (3.3) and Lemma 1, we get
For \(I_5 - I_6\), use (3.5), (3.6) and the mean value theorem to obtain
For \(I_4 + I_5\) and \(I_6\), remark that \(\widehat{\varphi _s}(\xi ) = \widehat{\varphi }(s\xi ) = \widehat{\varphi }(1)\) because \(\varphi \) is radial and \(s = |\xi |^{-1}\). Then it follows immediately that
Now we have
for any \(\xi \in \mathbb {R}^d\) (it is still valid in the case \(\xi = 0\)). Finally, remember \(|\widehat{\varphi }(1)| < 1\) to conclude that
\(\square \)
4 Remark
We can also obtain the \(H^1 \rightarrow L^1\) 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 \(L^{p_0}(\mathbb {R}^d)\) to \(L^{p_0, \infty }(\mathbb {R}^d)\) for some \(1< p_0 < \infty \) and its kernel K satisfies the \(H_{1}\) condition. Then T is bounded from \(H^1(\mathbb {R}^d)\) to \(L^1(\mathbb {R}^d)\).
To see this, note that Theorem 1 implies that T is bounded on \(L^p(\mathbb {R}^d)\) for any \(1< p < p_0\). Hence we assume that T is bounded on \(L^{p_0}(\mathbb {R}^d)\) for some \(1< p_0 < \infty \) without loss of generality. Now we can show the \(H^1(\mathbb {R}^d) \rightarrow L^1(\mathbb {R}^d)\) 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]).
Notes
The constant \(C_d\) in Lemma A is the \(L^1(\mathbb {R}^d) \rightarrow L^{1, \infty }(\mathbb {R}^d)\) bound of M, hence it can be taken to be \(3^d\).
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Communicated by Rodolfo H.Torres.
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This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 20J21771.
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Suzuki, S. The Calderón–Zygmund Theorem with an \(L^1\) Mean Hörmander Condition. J Fourier Anal Appl 27, 10 (2021). https://doi.org/10.1007/s00041-021-09810-9
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DOI: https://doi.org/10.1007/s00041-021-09810-9