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The L(Log L) Endpoint Estimate for Maximal Singular Integral Operators

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The L(Log L) Endpoint Estimate for Maximal Singular Integral Operators The L(log L) endpoint estimate for maximal singular integral operators Tuomas Hytonen¨ and Carlos Perez´ J. Math. Anal. Appl. 428 (2015), no. 1, 605–626. Abstract We prove in this paper the following estimate for the maximal operator T ∗ associated to the singular integral operator T: Z ∗ 1 kT f k 1;1 j f x j M w x dx; w ≥ ; < ≤ : L (w) . ( ) L(log L) ( )( ) 0 0 1 Rn This follows from the sharp Lp estimate ∗ 0 1 1=p0 kT f k p p k f k ; < p < 1; w ≥ ; < δ ≤ : L (w) . ( ) Lp(M (w)) 1 0 0 1 δ L(log L)p−1+δ As as a consequence we deduce that Z ∗ 1;1 kT f kL (w) . [w]A1 log(e + [w]A1 ) j f j w dx; Rn extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate. 1 Introduction and main results Very recently, the so called Muckenhoupt-Wheeden conjecture has been disproved by Reguera-Thiele in [RT]. This conjecture claimed that there exists a constant c such that for any function f and any weight w (i.e., a nonnegative locally integrable function), there holds Z kH f kL1;1(w) ≤ c j f j Mwdx: (1) R where H is the Hilbert transform. The failure of the conjecture was previously obtained by M.C. Reguera in [Re] for a special model operator T instead of H. This conjecture was motivated by a similar inequality by C. Fefferman and E. Stein [FS] for the Hardy-Littlewood maximal function: Z kM f kL1;1(w) ≤ c j f j Mw dx: (2) Rn 2000 Mathematics Subject Classification: 42B20, 42B25, 46E30. Key words and phrases: maximal operators, Calderon–Zygmund´ operators, weighted estimates. The first author is supported by the European Union through the ERC Starting Grant “Analytic–probabilistic methods for borderline singular integrals”. He is part of the Finnish Centre of Excellence in Analysis and Dynamics Research. The second author was supported by the Spanish Ministry of Science and Innovation grant MTM2012-30748. 1 The importance of this result stems from the fact that it was a central piece in the approach by Fefferman-Stein to derive the following vector-valued extension of the classical Lp Hardy-Littlewood maximal theorem: for every 1 < p; q < 1, there is a finite constant c = cp;q such that X 1 X 1 q q q q (M f j) ≤ c j f jj : (3) p n p n j L (R ) j L (R ) This is a very deep theorem and has been used a lot in modern harmonic analysis explaining the central role of inequality (2). Inequality (1) was conjectured by B. Muckenhoupt and R. Wheeden during the 70’s. That this conjecture was believed to be false was already mentioned in [P2] where the best positive result in this direction so far can be found, and where M is replaced by ML(log L) , i.e., a maximal type operator that is “-logarithmically” bigger than M: Z kT f kL1;1(w) ≤ c" j f j ML(log L)" (w)dx w ≥ 0: Rn where T is the Calderon-Zygmund´ operator T. Until very recently the constant of the estimate did not play any essential role except, perhaps, for the fact that it blows up. If we check the computations in 1 [P2] we find that c" ≈ e " . It turns out that improving this constant would lead to understanding deep questions in the area. One of the main purposes of this paper is to improve this result in several ways. 1 " 1 A first main direction is to improve the exponential blow up e by a linear blow up " . The second improvement consists of replacing T by the maximal singular integral operator T ∗. The method in [P2] cannot be used directly since the linearity of T played a crucial role. We refer to Section 2.3 for the definition of the maximal function MA = MA(L). We remark that the operator ML(log L)" is pointwise smaller than Mr = MLr , r > 1, which is an A1 weight and for which the result was known. Theorem 1.1. Let T be a Calder´on-Zygmundoperator with maximal singular integral operator T ∗. Then for any 0 < ≤ 1, Z ∗ cT kT f kL1;1(w) . j f (x)j ML(log L) (w)(x) dx w ≥ 0 (4) Rn If we formally optimize this inequality in we derive the following conjecture: Z ∗ 1 n k k 1;1 ≤ j j ≥ 2 R T f L (w) cT f (x) ML log log L(w)(x) dx w 0; f Lc ( ): (5) Rn To prove Theorem 1.1 we need first an Lp version of this result, which is fully sharp, at least in the logarithmic case. The result will hold for all p 2 (1; 1) but for proving Theorem 1.1 we only need it when p is close to one. There are two relevant properties properties that will be used (see Lemma 4.2). The first one γ γ establishes that for appropriate A and all γ 2 (0; 1), we have (MA f ) 2 A1 with constant [(MA f ) ]A1 p0 n independent of A and f . The second property is that MA¯ is a bounded operator on L (R ) where A¯ is the complementary Young function of A. The main example is A(t) = tp(1 + log+ t)p−1+δ; p 2 (1; 1); δ 2 (0; 1) since 2 1 1=p0 kM ¯k p0 n . p ( ) A B(L (R )) δ by (25). 2 Theorem 1.2. Let 1 < p < 1 and let A be a Young function, then ∗ 0 0 kT f kLp(w) ≤ cT p kM ¯k p n k f k p 1=p p w ≥ 0: (6) A B(L (R )) L (MA(w ) ) In the particular case A(t) = tp(1 + log+ t)p−1+δ we have ∗ 2 0 1 1=p0 k k p ≤ k k ≥ ≤ T f L (w) cT p p ( ) f Lp M (w) w 0; 0 < δ 1: δ L(log L)p−1+δ Another worthwhile example is given by ML(log L)p−1(log log L)p−1+δ instead of ML(log L)p−1+δ for which: ∗ 2 0 1 1=p0 k k p ≤ k k ≥ ≤ T f L (w) cT p p ( ) f Lp M (w) w 0; 0 < δ 1: δ L(log L)p−1(log log L)p−1+δ There are some interesting consequences from Theorem 1.1, the first one is related to the one weight theory. We first recall the definition of the A1 constant considered in [HP] and where is shown it is the most suitable one. This definition was originally introduced by Fujii in [F1] and rediscovered later by Wilson in [W1]. Definition 1.3. 1 Z [w]A1 := sup M(wχQ) dx: Q w(Q) Q Observe that [w]A1 ≥ 1 by the Lebesgue differentiation theorem. When specialized to weights w 2 A1 or w 2 A1, Theorem 1.1 yields the following corollary. It was formerly known for the linear singular integral T [HP], and this was used in the proof, which proceeded via the adjoint of T; the novelty in the corollary below consists of dealing with the maximal singular integral T ∗. Corollary 1.4. Z ∗ kT f kL1;1(w) . log(e + [w]A1 ) j f j Mw dx; (7) Rn and hence Z ∗ 1 kT f kL1; (w) . [w]A1 log(e + [w]A1 ) j f j w dx; (8) Rn The key result that we need is the following optimal reverse Holder’s¨ inequality obtained in [HP] (see also [HPR] for a better proof and [DMRO] for new characterizations of the A1 class of weights). Theorem 1.5. Let w 2 A1, then there exists a dimensional constant τn such that Z Z 1=rw − wrw ≤ 2− w Q Q where 1 rw = 1 + τn[w]A1 tα Proof of Corollary 1.4. To apply (4), we use log t ≤ α for t > 1 and α > 0 to deduce that 1 M (w) . M 1+α (w) L(log L) α L 1 Hence, if w 2 A1 we can choose α such that α = . Then, applying Theorem 1.5 τn[w]A1 1 1 1 ML L (w) . (τ[w]A ) MLrw (w) . [w] M(w) (log ) 1 A1 and optimizing with ≈ 1= log(e + [w]A1 ) we obtain (7). 3 As a consequence of Theorem 1.1 we have, by using some variations of the ideas from [CP1], the following: Corollary 1.6. Let u; σ be a pair of weights and let p 2 (1; 1). We also let δ; δ1; δ2 2 (0; 1]. Then (a) If !1=p0 1 Z k 1=pk 1 K = sup u Lp(log L)p−1+δ;Q σ dx < ; (9) Q jQj Q then ∗ 1 1 1=p0 kT ( f σ)k p;1 . K ( ) k f k p (10) L (u) δ δ L (σ) (The boundedness in the case δ = 0 is false as shown in [CP1].) (b) As consequence, if !1=p0 1 Z K = sup ku1=pk σ dx Lp(log L)p−1+δ1 ;Q Q jQj Q (11) Z !1=p 1 1=p0 + sup u dx kσ k 0 0 < 1; Lp (log L)p −1+δ2 ;Q Q jQj Q then 0 1 1 1 ! p0 ! p ∗ B 1 1 1 1 C kT ( f σ)kLp(u) .
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