ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS LOUKAS GRAFAKOS Abstract. It is shown that maximal truncations of nonconvolution L2 -bounded singular integral operators with kernels satisfying H¨ ormander’s condition are weak type (1, 1) and Lp bounded for 1 < p < ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.
1. Introduction Consider a function k(x) on Rn \ {0} which satisfies |k(x)| dx < ∞ sup R>0 R≤|x|≤2R |k(x − y) − k(x)| dx < ∞ sup (1) y∈Rn \{0} |x|≥2|y|
and
sup 00
|x−x0 | ≤ 2 T ∗ (g) 2L2 + ∪j Q∗j + x ∈ α 2 4 2 α 2 2 ∗ ∗ ∗ ≤ 2 Cn (A1 +A2 +B) g L2 + |Qj | + x ∈ / ∪j Qj : |T (b)(x)| > α 2 j √ 2n+2 (5 n)n α 2 2 ≤ / ∪j Q∗j : |T ∗ (b)(x)| > γ Cn (A1 +A2 +B) f L1 + f L1 + x ∈ α αγ 2 Choosing γ = (2n+5 (A1 + A2 + B))−1 and using Lemma 1 we obtain the required estimate f L1 |{x ∈ Rn : |T ∗ (f )(x)| > α}| ≤ Cn (A1 + A2 + B) α √ with Cn = 2−3 Cn2 + (5 n)n 2n+5 + 2n+8 . This concludes the proof of (10). It remains to prove Lemma 1; this will be done in the next section. 3. The proof of Lemma 1 We now turn our attention to Lemma 1. The claimed estimate in the lemma will be a consequence of the fact that for x ∈ (∪j Q∗j )c we have the key inequality (21)
T ∗ (b)(x) ≤ 4E1 (x) + 2n+2 αγE2 (x) + 2n+3 αγA1 ,
where E1 (x) = E2 (x) =
j
Qj
j
Qj
|K(x, y) − K(x, yj )| |bj (y)| dy, |K(x, y) − K(x, yj )| dy,
and yj is the center of Qj . If we had (21), then we could easily derive (12). Indeed, fix γ ≤ (2n+5 A1 )−1 . Then α we have 2n+3 αγA1 < and using (21) we obtain 3 α x ∈ (∪j Q∗j )c : |T ∗ (b)(x)| > 2
α α ∗ c ∗ c n+2 (22) ≤ x ∈ (∪j Qj ) : 4E1 (x) > 12 + x ∈ (∪j Qj ) : 2 αγE2 (x) > 12 48 n+6 E1 (x) dx + 2 γ E2 (x) dx, ≤ α (∪j Q∗j )c (∪j Q∗j )c
ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS
since
α α α α = + + . We have 2 3 12 12 E1 (x) dx ≤ |bj (y)| (∪j Q∗j )c
≤
(23)
j
Qj
j
Qj
≤A2
(Q∗j )c
|K(x, y) − K(x, yj )| dx dy
|bj (y)|
j
9
|x−yj |≥2|y−yj |
|bj (y)| dy = A2
Qj
|K(x, y) − K(x, yj )| dx dy
bj L1 ≤ A2 2n+1 f L1 ,
j
1 5√ where we used the fact that if x ∈ (Q∗j )c then |x − yj | ≥ l(Q∗j ) = n l(Qj ). But 2 2 √ n l(Qj ) this implies that |x − yj | ≥ 2|y − yj |. Here we used the since |y − yj | ≤ 2 √ fact that the diameter of a cube is equal to n times its sidelength. Likewise we can obtain that f L1 (24) . E2 (x) dx ≤ A2 |Qj | ≤ A2 αγ (∪j Q∗j )c j Combining (23) and (24) with (22) yields (12). Therefore the main task in the proof of (12) is to show (21). Recall that b = j bj and to estimate T ∗ (b) it suffices to estimate each |T ε,N (bj )| uniformly in ε and N . To achieve this we will use that |T ε,N (bj )| ≤ |T ε,∞ (bj )| + |T N,∞ (bj )| .
(25)
We work with T ε,∞ and we note that T N,∞ can be treated similarly. For fixed x∈ / ∪j Q∗j and ε > 0 we define J1 (x, ε) = {j : ∀y ∈ Qj we have |x − y| < ε}, J2 (x, ε) = {j : ∀y ∈ Qj we have |x − y| > ε}, J3 (x, ε) = {j : ∃y ∈ Qj we have |x − y| = ε}. Note that T ε,∞ (bj )(x) = 0 whenever x ∈ / ∪j Q∗j and j ∈ J1 (x, ε). Also note that / ∪j Q∗j , j ∈ J2 (x, ε) and y ∈ Qj . Therefore K ε,∞ (x, y) = K(x, y) whenever x ∈ (26) T (bj )(x) + sup T (bj χ|x− · |≥ε )(x) sup |T ε,∞ (b)(x)| ≤ sup ε>0
but since (27)
ε>0
ε>0
j∈J2 (x,ε)
j∈J3 (x,ε)
sup T (bj )(x) ≤ |T (bj )(x)| = E1 (x) , ε>0
j∈J2 (x,ε)
j
it suffices to estimate the second term on the right in (26).
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LOUKAS GRAFAKOS
Here we need to make some geometric observations. Fix ε > 0, x ∈ (∪j Q∗j )c and also fix a cube Qj with j ∈ J3 (x, ε). Then we have ε≥
(28)
1 √ √ 1 ∗ l(Qj ) − l(Qj ) = (5 n − 1)l(Qj ) ≥ 2 n l(Qj ) . 2 2
Since j ∈ J3 (x, ε) there exists a y0 ∈ Qj with |x − y0 | = ε. Using (28) we obtain that for any y ∈ Qj we have √ ε ≤ ε − n l(Qj ) ≤ |x − y0 | − |y − y0 | ≤ |x − y|, 2 √ 3ε |x − y| ≤ |x − y0 | + |y − y0 | ≤ ε + n l(Qj ) ≤ . 2 We have therefore proved that
Qj ⊂ B(x, 3ε ) \ B(x, 2ε ) . 2
j∈J3 (x,ε)
−1
We now let cj (ε) = |Qj |
bj (y)χ|x−y|≥ε (y) dy and we note that property (P5) of Qj
the Calder´on-Zygmund decomposition yields the estimate |cj (ε)| ≤ 2n+1 αγ. Then K(x, y)bj (y)χ|x−y|≥ε (y) dy sup ε>0
j∈J3 (x,ε)
Qj
j∈J3 (x,ε)
Qj
≤ sup ε>0
K(x, y) bj (y)χ|x−y|≥ε (y) − cj (ε) dy
≤ sup ε>0
j∈J3 (x,ε)
j∈J3 (x,ε)
Qj
cj (ε) + sup ε>0
K(x, y) − K(x, yj ) bj (y)χ|x−y|≥ε (y) − cj (ε) dy n+1
+2 ≤
j
Qj
K(x, y) dy
αγ sup B(x, 3ε )\B(x, 2ε ) 2
ε>0
|K(x, y)| dy
K(x, y) − K(x, yj ) |bj (y)| + 2n+1 αγ dy
Qj
n+1
+2
αγ sup ε>0
ε ≤|x−y|≤ 3ε 2 2
|K(x, y)| dy
≤E1 (x) + 2n+1 αγE2 (x) + 2n+1 αγ(2A1 ). The last estimate above together with (27) and combined with (25) and the analogous estimate for supN >0 |T N,∞ (b)(x)| (which can be obtained entirely similarly), yields (21). This finishes the proof of Lemma 1 and thus of Theorem 1.
ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS
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References [1] A. Benedek, A. Calder´ on, and R. Panzone, Convolution operators on Banach-space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. [2] M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana, 1 (1955), 105–167. [3] G. David and J.-L. Journ´e, A boundedness criterion for generalized Calder´ on-Zygmund operators, Ann. of Math. 120 (1984), 371–397. [4] J. Duoandikoetxea, Fourier Analysis, Grad. Studies in Math. 29 Amer. Math. Soc., Providence RI, 2000. [5] N. Riviere Singular integrals and multiplier operators, Arkiv f. Math. 9 (1971), 243–278. [6] E. M. Stein, Harmonic analysis: Real variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton NJ, 1993. [7] S. Yano, An extrapolation theorem, J. Math. Soc. Japan 3 (1951), 296–305. [8] A. Zygmund, Trigonometric series, Vol. II, 2nd edition, Cambridge University Press, Cambridge UK 1959. Loukas Grafakos, Department of Mathematics, University of Missouri, Columbia, MO 65211, USA E-mail address:
[email protected] 