The L(log L) ɛ endpoint estimate for maximal singular integral operators

The L(log L) ɛ endpoint estimate for maximal singular integral operators Tuomas Hytönen and Carlos Pérez J. Math. Anal. Appl. 428 (205), no., 605 626. Abstract We prove in this paper the following estimate for the maximal operator T associated to the singular integral operator T: T f L, (w) f (x) M ɛ L(log L) R n ɛ (w)(x) dx, w 0, 0 < ɛ. This follows from the sharp L p estimate T f L p (w) p ( δ )/p f L, < p <, w 0, 0 < δ. p (M L(log L) p +δ (w)) As as a consequence we deduce that T f L, (w) [w] A log(e + [w] A ) f w dx, R n extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate. Introduction and main results Very recently, the so called Muckenhoupt-Wheeden conecture has been disproved by Reguera-Thiele in [RT]. This conecture 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 H f L, (w) c f Mwdx. () where H is the Hilbert transform. The failure of the conecture was previously obtained by M.C. Reguera in [Re] for a special model operator T instead of H. This conecture was motivated by a similar inequality by C. Fefferman and E. Stein [FS] for the Hardy-Littlewood maximal function: M f L, (w) c f Mw dx. (2) R n 2000 Mathematics Subect Classification: 42B20, 42B25, 46E30. Key words and phrases: maximal operators, Calderón 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 MTM202-30748. R

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 L p Hardy-Littlewood maximal theorem: for every < p, q <, there is a finite constant c = c p,q such that ( (M f ) q) L q c ( f q) L q (3) p (R n ) p (R ). n This is a very deep theorem and has been used a lot in modern harmonic analysis explaining the central role of inequality (2). Inequality () was conectured by B. Muckenhoupt and R. Wheeden during the 70 s. That this conecture 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 M L(log L) ɛ, i.e., a maximal type operator that is ɛ-logarithmically bigger than M: T f L, (w) c ε R n f M L(log L) ε(w)dx w 0. where T is the Calderón-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 [P2] we find that c ε e ε. 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. A first main direction is to improve the exponential blow up e ε by a linear blow up ε. 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 M A = M A(L). We remark that the operator M L(log L) ε is pointwise smaller than M r = M L r, r >, which is an A weight and for which the result was known. Theorem.. Let T be a Calderón-Zygmund operator with maximal singular integral operator T. Then for any 0 < ɛ, T f L, (w) c T f (x) M L(log L) ɛ(w)(x) dx w 0 (4) ɛ R n If we formally optimize this inequality in ɛ we derive the following conecture: T f L, (w) c T R n f (x) M L log log L (w)(x) dx w 0, f L c (R n ). (5) To prove Theorem. we need first an L p version of this result, which is fully sharp, at least in the logarithmic case. The result will hold for all p (, ) but for proving Theorem. 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 establishes that for appropriate A and all γ (0, ), we have (M A f ) γ A with constant [(M A f ) γ ] A independent of A and f. The second property is that M Ā is a bounded operator on L p (R n ) where Ā is the complementary Young function of A. The main example is A(t) = t p ( + log + t) p +δ, p (, ), δ (0, ) since by (25). M Ā B(L p (R n )) p2 ( δ )/p 2

Theorem.2. Let < p < and let A be a Young function, then T f L p (w) c T p M Ā B(L p (R n )) f L p (M A (w /p ) p ) w 0. (6) In the particular case A(t) = t p ( + log + t) p +δ we have T f L p (w) c T p 2 p ( δ )/p f L p ( M L(log L) p +δ (w) ) w 0, 0 < δ. Another worthwhile example is given by M L(log L) p (log log L) p +δ instead of M L(log L) p +δ for which: T f L p (w) c T p 2 p ( δ )/p f L p ( M L(log L) p (log log L) p +δ(w) ) w 0, 0 < δ. There are some interesting consequences from Theorem., the first one is related to the one weight theory. We first recall the definition of the A constant considered in [HP] and where is shown it is the most suitable one. This definition was originally introduced by Fuii in [F] and rediscovered later by Wilson in [W]. Definition.3. [w] A := sup M(wχ ) dx. w() Observe that [w] A by the Lebesgue differentiation theorem. When specialized to weights w A or w A, Theorem. 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 adoint of T; the novelty in the corollary below consists of dealing with the maximal singular integral T. Corollary.4. T f L, (w) log(e + [w] A ) f Mw dx, (7) R n and hence T f L, (w) [w] A log(e + [w] A ) f w dx, (8) R n The key result that we need is the following optimal reverse Hölder s inequality obtained in [HP] (see also [HPR] for a better proof and [DMRO] for new characterizations of the A class of weights). Theorem.5. Let w A, then there exists a dimensional constant τ n such that ( w w) r /rw 2 w where r w = + τ n [w] A Proof of Corollary.4. To apply (4), we use log t tα α M L(log L) ɛ(w) α ɛ M L +ɛα(w) for t > and α > 0 to deduce that Hence, if w A we can choose α such that αɛ = τ n [w] A. Then, applying Theorem.5 ɛ M L(log L) ɛ(w) ɛ (ɛτ[w] A ) ɛ M L rw (w) ɛ [w]ɛ A M(w) and optimizing with ɛ / log(e + [w] A ) we obtain (7). 3

As a consequence of Theorem. we have, by using some variations of the ideas from [CP], the following: Corollary.6. Let u, σ be a pair of weights and let p (, ). We also let δ, δ, δ 2 (0, ]. Then (a) If then K = sup u /p L p (log L) p +δ, ( ) /p σ dx <, (9) T ( f σ) L p, (u) δ K ( δ )/p f L p (σ) (0) (The boundedness in the case δ = 0 is false as shown in [CP].) (b) As consequence, if ( ) /p K = sup u /p L σ dx p (log L) p +δ, ( /p + sup u dx) σ /p L p (log L) p +δ 2, <, () then T ( f σ) L p (u) K ( δ δ ) p + δ 2 ( δ 2 ) p f L p (σ). (2) The first qualitative result as in (0) was obtained in [CP], Theorem.2 and its extension Theorem 4.. We remark that this result holds for any operator T which satisfies estimate (4). We also remark that this corollary improves the main results from [CRV] (see also [ACM]) by providing very precise quantitative estimates. We refer to these papers for historical information about this problem. We don t know whether the factors δ i, i =, 2 can be removed or improved from the estimate (2). Perhaps our method is not so precise to prove the conecture formulated in Section 7. However, it is clear from our arguments that these factors are due to the appearance of the factor ɛ in (4). Acknowledgments We would like to thank the anonymous referee for detailed comments that improved the presentation. 2 Basic definitions and notation 2. Singular integrals In this section we collect some notation and recall some classical results. By a Calderón-Zygmund operator we mean a continuous linear operator T : C 0 (Rn ) D (R n ) that extends to a bounded operator on L 2 (R n ), and whose distributional kernel K coincides away from the diagonal x = y in R n R n with a function K satisfying the size estimate K(x, y) c x y n 4

and the regularity condition: for some ε > 0, x z ε K(x, y) K(z, y) + K(y, x) K(y, z) c x y n+ε, whenever 2 x z < x y, and so that T f (x) = K(x, y) f (y)dy, R n whenever f C0 (Rn ) and x supp( f ). Also we will denote by T the associated maximal singular integral: T f (x) = sup K(x, y) f (y) dy ε>0 f C 0 (Rn ) y x >ε More information can be found in many places as for instance in [G] or [Duo]. 2.2 Orlicz spaces and normalized measures We will also need some basic facts from the theory of Orlicz spaces that we state without proof. We refer to the book of Rao and Ren [RR] for the proofs and more information on Orlicz spaces. Another interesting recent book is [W2]. A Young function is a convex, increasing function A : [0, ) [0, ) with A(0) = 0, such that A(t) as t. Such a function is automatically continuous. From these properties it follows that A : [t 0, ) [0, ) is a strictly increasing biection, where t 0 = sup{t [0, ) : A(t) = 0}. Thus A (t) is well-defined (single-valued) for t > 0, but in general it may happen that A (0) = [0, t 0 ] is an interval. The properties of A easily imply that for 0 < ε < and t 0 A(ε t) ε A(t). (3) The A-norm of a function f over a set E with finite measure is defined by ( ) f (x) f A,E = f A(L),E = inf{λ > 0 : A dx } E λ where as usual we define the average of f over a set E, E f = E E f dx. In many situations the convexity does not play any role and basically the monotonicity is the fundamental property. The convexity is used for proving that A,E is a norm which is often not required. We will use the fact that f A,E if and only if A ( f (x) ) dx. (4) Associated with each Young function A, one can define a complementary function E Ā(s) = sup{st A(t)} s 0. (5) t>0 Then Ā is finite-valued if and only if lim t A(t)/t = sup t>0 A(t)/t =, which we henceforth assume; otherwise, Ā(s) = for all s > sup t>0 A(t)/t. Also, Ā is strictly increasing on [0, ) if and only if lim t 0 A(t)/t = inf t>0 A(t)/t = 0; otherwise Ā(s) = 0 for all s inf t>0 A(t)/t. 5

Such Ā is also a Young function and has the property that and also st A(t) + Ā(s), t, s 0. (6) t A (t)ā (t) 2 t, t > 0. (7) The main property is the following generalized Hölder s inequality f g dx 2 f A,E g E Ā,E. (8) E As we already mentioned, the following Young functions play a main role in the theory: A(t) = t p ( + log + t) p +δ t, δ > 0, p >. 2.3 General maximal functions and L p boundedness: precise versions of old results Given a Young function A or more generally any positive function A(t) we define the following maximal operator ([P],[P2]) M A(L) f (x) = M A f (x) = sup f A,. x This operator satisfies the following distributional type estimate: there are finite dimensional constants c n, d n such that ( ) {x R n f : M A f (x) > t} c n A d n dx f 0, t > 0 (9) R n t This follows from standard methods and we refer to [CMP, Remark A.3] for details. A first consequence of this estimate is the following L p estimate of the operator, which is nothing more than a more precise version of one the main results from [P]. A second application will be used in the proof of Lemma 4.2. Lemma 2.. Let A be a Young function, then M A B(L p (R n )) c n α p (A) (20) where α p (A) is the following tail condition that plays a central role in the sequel ( α p (A) = A(t) t p ) /p dt <. (2) t Examples of functions satisfying condition (2) are A(t) = t q, q < p. More interesting examples are given by A(t) = t p ( + log + t) +δ A(t) t p log(t) log log(t) (+δ), p >, δ > 0. Often we need to consider instead of the function A in (2) the complementary Ā. We also record a basic estimate between a Young function and its derivative: which holds for any t (0, ) such that A (t) does exist. A(t) ta (t) (22) 6

There is the following useful alternative estimate of (20) that will be used in the sequel. Although variants of this lemma are well known in the literature (cf. [CMP], Proposition 5.0), we would like to stress the fact that we avoid the doubling condition on the Young functions B and B, which is important in view of the quantitative applications to follow: even if our typical Young functions are actually doubling, we want to avoid the appearance of their (large) doubling constants in our estimates. Lemma 2.2. Let B a Young function. Then where Proof. We first prove that for a > 0 B (a) M B B(L p (R n )) c n β p (B) (23) ( β p (B) = B (a) db(t) t p B() ( t ) pd /p B(t)) B(t) B (a) ( t ) pd B(t). (24) B(t) We discretize the integrals with a sequence a k := η k a, where η > and eventually we pass to the limit η. Then db(t) B (a k+ ) db(t) B (a k+ ) t p = t p B (a k ) p db(t) = B (a k ) p (a k+ a k ). Similarly, B (a) k= B (a k ) ( t ) pd B(t) = B(t) k=0 k=0 B (a k+ ) B (a k ) k= ( t ) pd B(t) B(t) ( B (a k+ ) ) p B (a k+ ) B( B (a k+ )) B (a k ) B (a k ) d B(t) = k=0 k= ( B (a k+ ) a k+ ) p(ak+ a k ), where we used the fact that t B(t)/t is increasing, so its reciprocal is decreasing. Moreover, and hence B (a k+ ) B (a k ) B (a k ) (7) a k+ a k+ B (a k ) B (a) db(t) t p a k a k+ B (a k ) = ηb (a k ) ( η p t ) pd B(t). B (a) B(t) Since this is valid for any η >, we obtain (24). Now, let t = max(, t 0 ), where t 0 = max{t : B(t) = 0}. Using B(t)dt/t db(t) and applying (24) with a = B(t + ɛ) > 0 ( α p (B) = lim ɛ 0 (24) ( lim ɛ 0 B(t) t +ɛ t p B (B(t +ɛ)) dt ) /p ( lim t ɛ 0 B (B(t +ɛ)) ( t ) pd B(t) ) /p ( B(t) where in the last step we used (7) with t = B(t + ɛ) to conclude that since B(t)/t is increasing and t. B (B(t + ɛ)) B(t + ɛ) t + ɛ 7 B(t ) t db(t) ) /p t p B() ( t ) pd B(t) ) /p, B(t) B(),

In this paper we will consider B so that B(t) = A(t) = t p ( + log + t) p +δ, δ > 0. Then, for 0 < δ A (t) 2p A(t) t > t and Thus, by the lemma Ā() = sup (t t p ) = (t t p ) t (0,) M Ā B(L p (R n )) c n Similarly for the smaller functional: (p )p p ( t ) p A(t) t=p /(p ) = (p )p p. /p ( A (t) dt c n p 2 δ ) /p B(t) = A(t) = t p ( + log + t) p ( + log + ( + log + t)) p +δ δ > 0. (25) Then, using that A (t) 3 p A(t) t t >, when 0 < δ and hence by the lemma ) /p M Ā B(L p (R n )) c n p 2 ( δ 2.4 The iteration lemma We will need the following variation of the Rubio de Francia algorithm. Lemma 2.3. Let < s < and let v be a weight. Then there exists a nonnegative sublinear operator R satisfying the following properties: (a) h R(h) (b) R(h) L s (v) 2 h L s (v) (c) R(h)v /s A with [R(h)v /s ] A cs Proof. We consider the operator Since M L s s, we have Now, define the Rubio de Francia operator R by S ( f ) = M( f v/s ) v /s S ( f ) L s (v) cs f L s (v). R(h) = k=0 2 k S k (h) ( S L s (v)) k. It is very simple to check that R satisfies the required properties. 8

2.5 Two weight maximal function Our main new result is intimately related to a sharp two weight estimate for M. Theorem 2.4. Given a pair of weights u, σ and p, < p <, suppose that ( /p K = sup u(y) dy) σ /p X, <. (26) where X is a Banach function space such that its corresponding associate space X satisfies M X : L p (R n ) L p (R n ). Then M( f σ) L p (u) K M X B(L p (R n )) f L p (σ) (27) In particular if X = L B with B(t) = t p ( + log + t) p +δ, δ > 0, then by (25) where the last is valid for δ. M X B(L p (R n )) = M B B(L p (R n )) (p ) 2 ( δ )/p. This result together with some improvements can be found in [PR]. 3 Dyadic theory In this section we define an important class of dyadic model operators and recall a general result by which norm inequalities for maximal singular integral operators can be reduced to these dyadic operators. The result is due to Lerner [Le2], and comes from his approach to prove the A 2 theorem proved by the first author [H]. We say that a dyadic grid, denoted D, is a collection of cubes in R n with the following properties: ) each D satisfies = 2 nk for some k Z; 2) if, P D then P =, P, or ; 3) for each k Z, the family D k = { D : = 2 nk } forms a partition of R n. We say that a family of dyadic cubes S D is sparse if for each S, 2. Given a sparse family, S, if we define S E() := \ S then ) the family {E()} S is pairwise disoint 2) E(), and 3) 2 E(). If S D is a sparse family we define the sparse Calderón-Zygmund operator associated to S as T S f := f dx χ. S As already mentioned the key idea is to transplant the continuous case to the discrete version by means of the following theorem. 9,

Theorem 3.. Suppose that X is a quasi-banach function space on R n and T is a Calderón-Zygmund operator. Then there exists a constant c T T B(X) c T sup T S B(X). S D For Banach function spaces (without quasi- ), this theorem is due to Lerner [Le2]. The stated generalization was obtained independently by Lerner and Nazarov [LN] on the one hand, and by Conde-Alonso and Rey [CAR] on the other hand. As a matter of fact, the last two papers only explicitly deal with the Calderón Zygmund operator T rather than the maximal truncation T, but the version above follows immediately from the same considerations, say, by combining [HLP, Theorem 2.] and [CAR, Theorem A]. We will not prove this theorem, we will simply mention that a key tool is the decomposition formula for functions found previously by Lerner [Le] using the median. The main idea of this decomposition goes back to the work of Fuii [F2] where the standard average is used instead. 4 Proof of Theorem.2 4. Two lemmas Following the notion of dyadic singular integral operator mentioned in the section above we have the following key Lemma. Lemma 4.. Let w A. Then for any sparse family S D S T S f L (w) 8[w] A M f L (w) (28) Proof. The left hand side equals for f 0 f dx w() inf M f (z) w() ( (M f ) /2 dw ) 2 w(). z S By the Carleson embedding theorem, applied to g = (M f ) /2, we have ( g dw ) 2 w() 4K g 2 L 2 (w) = 4K M f L (w) S S provided that the Carleson condition w() Kw(R) (29) S R is satisfied. To prove (29), we observe that w() w() = inf M( Rw)(z) 2 E() 2 z S R S R S R R M( R w)(z)dz 2[w] A w(r). This proves (29) with K = 2[w] A, and the lemma follows. 0

Actually, in the applications we have in mind we ust need this for w A q A for some fixed finite q. The second lemma is an extension of the well known Coifman-Rochberg Lemma: If γ (0, ) then M(µ) γ A with [M(µ) γ ] A c n γ Lemma 4.2. Let A be a Young function and u be a nonnegative function such that M A u(x) < a.e. For γ (0, ), there is a dimensional constant c n such that [(M A u) γ ] A c n c γ. (30) A statement of this type is contained in [CMP], Proposition 5.32, but there it is suggested that the bound may also depend on the Young function A, while our version shows that it does not. This is again important for the quantitative consequences. Proof. We claim now that for each cube and each u M A (uχ )(x) γ dx c n,γ u γ. (3) A, By homogeneity we may assume u A, =, and so, in particular, that A(u(x)) dx. Now, the proof of (3) is based on the distributional estimate (9). We split the integral at a level λ b n, yet to be chosen: M A (uχ )(x) γ dx = γ t γ {x : M A (uχ )(x) > t} dt 0 t λ γt γ dt + γt γ a n A ( u(x) ) dt b n dx 0 t λ t t λ γ + γt γ b n a n A( u(x) ) dxdt λ t t λ γ + a n b n γ t γ 2 dt = λ γ γ + a n b n γ λγ. With λ = a n b n, we arrive at M A (uχ )(x) γ dx (a nb n ) γ γ, which is (3), in view of our normalization that u A, =. We will use the following fact that can be also found in [CMP]: for every λ M A (u χ R n \3 )(x) sup u χ R n \3 A,P x (32) P where the constant in the direction is dimensional (actually 3 n ). (32) shows that M A ( f χ R n \3) is essentially constant on. Finally since A is a Young, the triangle inequality combined with (3) and (32) gives for every y, M A u(x) γ dx 3 n M A (uχ 3 )(x) γ dx + M A (uχ R n \3)(x) γ dx. 3 c n,γ u γ + ( ) A,3 3n sup u χ γ R n \3 A,P P c n,γ M A u(y) γ.

This completes the proof of the lemma. 4.2 Proof of Theorem.2 We have to prove T f L p (w) c T p M Ā B(L p (R n )) f L p (M A (w /p ) p ) w 0. and if we use the notation A p (t) = A(t /p ) this becomes T f L p (w) c T p M Ā B(L p (R n )) f L p (M Ap (w)). By Theorem 3. everything is reduced to proving that T S f L p (w) p M Ā B(L p (R n )) f L p (M Ap (w)) S D. (33) Now, by duality we will prove the equivalent estimate T S ( f w) L p (M Ap (w) p ) p M Ā B(L p (R n )) f L p (w). because the adoint of T S (with respect to the Lebesgue measure) is itself. The main claim is the following: Lemma 4.3. T S (g) L p (M Ap (w) p ) p M(g) L p (M Ap (w) p ) S D g 0. (34) Proof. Now T S (g) L p (M Ap (w) p ) = T S (g) M Ap w L p (M Ap w) and by duality we have that for some nonnegative h with h L p (M Ap w) = T S (g) M Ap w L p (M Ap w) = T S (g) h dx R n Now, by Lemma 2.3 with s = p and v = M Ap w there exists an operator R such that (A) h R(h) (B) R(h) L p (M Ap w) 2 h L p (M Ap w) (C) [R(h)(M Ap w) /p ] A cp. Hence, T S (g) L p (M Ap (w) p ) R n T S (g) Rh dx. Next we plan to replace T S by M by using Lemma 4.. To do this we to estimate the A q constant of Rh, for a fixed q > (in fact, q = 3) using property (C) combining the following two facts. The first one is well known, it is the easy part of the factorization theorem: if w, w 2 A, then w = w w p 2 A p, and [w] Ap [w ] A [w 2 ] p A The second fact is Lemma 4.2 2

Now if we choose γ = 2 in Lemma 4.2, [R(h)] A [R(h)] A3 = [R(h)(M Ap w) p ( (M Ap w) 2p ) 3 ] A3 [R(h)(M Ap w) p ] A [(M Ap w) 2p ] 3 A c n p [M A (w /p ) 2 ] 3 A c n p by the lemma and since A p (t) = A(t /p ). Therefore, by Lemma 4. and by properties (A) and (B) together with Hölder, T S (g)h dx T S (g)r(h) dx [R(h)] A M(g)R(h) dx R n R n R n p M(g) Rh L M Ap w p (M Ap w) = c N p M(g) M Ap w L p (M Ap w) L p (M Ap w). This proves claim (34). With (34), the proof of Theorem.2 is reduced to showing that M( f w) L p (M Ap (w) p ) c M Ā B(L p (R n )) f L p (w) for which we can apply the two weight theorem for the maximal function (Theorem 2.4) to the couple of weights (M Ap (w) p, w) with exponent p. We need then to compute (26): (We reproduce this short calculation from [CMP], Theorem 6.4, for completeness.) ( ) /p M Ap (w) p dy w /p A, w /p A p, w /p A, = w /p A, w /p A, =, since A p (t) = A(t /p ). Hence concluding the proof of the theorem. M( f w) L p (M A (w) p ) c MĀ B(L p (R n )) f L p (w) 5 Proof of Theorem. To prove the Theorem we follow the basic scheme as in [P2] (see also [LOP], [HP]). Thanks to Theorem 3., it is enough to prove the following dyadic version: Proposition 5.. Let D be a dyadic grid and let S D be a sparse family. Then, there is a universal constant c independent of D and S such that for any 0 < ɛ T S f L, (w) c f (x) M L(log L) ɛ(w)(x) dx w 0 (35) ɛ R n Note that in order to deduce Theorem. from the Proposition above, we need the full strength of Theorem 3. with quasi-banach function space, because the space L, is not normable. It is also possible to prove Theorem. directly (without going through the dyadic model); this was our 3

original approach, since the quasi-banach version of Theorem 3. was not yet available at that point. However, we now present a proof via the dyadic model, which simplifies the argument. Recall that the sparse Calderón-Zygmund operator T S is defined by, T S f = f dx χ. By homogeneity on f it would be enough to prove w{x R n : T S f (x) > 2} c f (x) M L(log L) ɛ(w)(x) dx. ɛ R n S We consider the the CZ decomposition of f with respect to the grid D at level λ =. There is family of pairwise disoint cubes { } from D such that < f 2 n Let Ω = and Ω = 3. The good part is defined by g = f χ (x) + f (x)χ Ω c(x), and it satisfies g L 2 n by construction. The bad part b is b = f )χ (x). Then, f = g + b and we split the level set as b where b (x) = ( f (x) w{x R d : T S f (x) > 2} w( Ω) + w{x ( Ω) c : T S b(x) > } + w{x ( Ω) c : T S g(x) > } = I + II + III. As in [P2], the most singular term is III. We first deal with the easier terms I and II, which actually satisfy the better bound I + II c T f L (Mw). The first is simply the classical Fefferman-Stein inequality (2). To estimate II = w{x ( Ω) c : T S b(x) > } we argue as follows: w{x ( Ω) c : T S b(x) > } T S b(x) w(x)dx R n \ Ω T S (b )(x) w(x)dx R n \3 We fix one of these and estimate now T S (b )(x) for x 3 : T S (b )(x) = b dy χ (x) = S S, + R n \ Ω S, = since x. Now, this expression is equal to ( f (y) f ) dy χ (x) S, and this expresion is zero by the key cancellation: only left with the singular term III. 4 T S (b )(x) w(x)dx S, ( f (y) f ) dy = 0. Hence II = 0, and we are

5. Estimate for part III We now consider the last term III, the singular part. We apply Chebyschev s inequality and then (33) with exponent p and functional A, that will be chosen soon: III = w{x ( Ω) c : T S g(x) > } T S g p L p (wχ ( Ω) c ) (p ) p M Ā p B(L p (R n )) (p ) p M Ā p B(L p (R n )) g p M Ap (wχ ( Ω) )dx R n c g M Ap (wχ ( Ω) )dx, R n c using the boundedness of g by 2 n, and denoting A p (t) = A(t /p ). Now, we will make use of (32) again: for an arbitrary Young function B, a nonnegative function w with M B w(x) < a.e., and a cube, we have M B (χ R n \3 w)(y) M B (χ R n \3 w)(z) (36) for each y, z with dimensional constants. Hence, combining (36) with the definition of g we have g M Ap (wχ ( Ω) )dx f (x) dx inf M c Ap (wχ Ω ( Ω) ) c f (x) M Ap w(x) dx, and of course Combining these, we have Ω g M Ap (wχ ( Ω) c) dx Ω c III (p ) p M Ā p B(L p (R n )) Ω c f M Ap w dx. R d f M Ap (w)dx. We optimize this estimate by choosing an appropriate A. To do this we apply now Lemma 2.2 and more particularly to the example considered in (25), namely B is so that B(t) = A(t) = t p ( + log + t) p +δ, δ > 0. Then M Ā B(L p (R n )) c n ( t ) p A(t) Then A p (t) = A(t /p ) t( + log + t) p +δ and we have Now if we choose p such that III (p ) p ( δ ) p /p A (t) dt p ( ) /p δ R d f M L(log L) p +δ(w)(x) dx. p = ɛ 2 = δ < then (p ) p ( δ )p ɛ if ɛ <. This concludes the proof of (35), and hence of Theorem.. 0 < δ 5

6 Proof of Corollary.6 We follow very closely the argument given in [CP], the essential difference is that we compute in a more precise way the constants involved. We consider the set Then by homogeneity it is enough to prove Ω = {x R n : T ( f σ)(x) > } u(ω) /p δ K ( δ )/p f L p (σ) (37) where we recall that K = sup u /p L p (log L) p +δ, ( ) /p σ dx < (38) Now, by duality, there exists a non-negative function h L p (R n ), h L p (R n ) =, such that u(ω) /p = u /p χ Ω L p (R n ) = u /p h dx = u /p h(ω) f M L(log L) ε(u /p h) σdx Ω ε R n ( ) /p ( f p σdx M L(log L) ε(u /p h) p ε R n R n ) /p σdx, where we have used inequality (4) from Theorem. and then Hölder s inequality. Therefore everything is reduced to understanding a two weight estimate for M L(log L) ε. We need the following Lemma that can be found in [P] or in [CMP] Appendix A, Proposition A. Lemma 6.. Given a Young function A, suppose f is a non-negative function such that f A, tends to zero as l() tends to infinity. Given a > 2 n+, for each k Z there exists a disoint collection of maximal dyadic cubes { k } such that for each, and a k < f A, k 2 n a k, (39) {x R n : M A f (x) > 4 n a k } 3 k. Further, let D k = k and Ek = k \ (k D k+). Then the E k s are pairwise disoint for all and k and there exists a constant α >, depending only on a, such that k α Ek. Fix a function h bounded with compact support. Fix a > 2 n+ ; for k Z let Then by Lemma 6., Ω k = {x R n : 4 n a k < M A f (x) 4 n a k+ }. Ω k 3 k, where f A, k > a k. We will use a generalization of Hölder s inequality due to O Neil [O]. (Also see Rao and Ren [RR, p. 64].) We include a proof for the reader s convenience. 6

Lemma 6.2. Let A, B and C be Young functions such that Then for all functions f and g and all cubes, B (t)c (t) κa (t), t > 0. (40) f g A, 2κ f B, g C,. (4) Proof. The assumption (40) says that if A(x) = B(y) = C(z), then yz κx. Let us derive a more applicable consequence: Let y, z [0, ), and assume without loss of generality (by symmetry) that B(y) C(z). Since Young functions are onto, we can find a y y and x [0, ) such that B(y ) = C(z) = A(x). Then (40) tells us that yz y z κx. Since A is increasing, it follows that A ( yz) A(x) = C(z) = max(b(y), C(z)) B(y) + C(z). (42) κ Let then s > f B and t > g C. Then, using (42), A ( f g ) B ( f ) + κst s C ( g ) +, t and hence A ( f g ) 2κst 2 A ( f g ). κst This proves that f g A 2κst, and taking the infimum over admissible s and t proves the claim. If A(t) = t( + log + t) ε, the goal is to break M A in an optimal way, with functions B and C so that one of them, for instance B, has to be B(t) = t p ( + log + t) p +δ coming from (38). We can therefore estimate M A using Lemma 6. as follows: (M A (u /p h)) p σ dx = (M A (u /p h)) p σ dx R n k Ω k c a kp σ(ω k ) k c a kp σ(3 k ),k c σ(3 k ) u/p h p. A, k,k c,k σ(3 k ) u/p p h p, B, k C, k by (4). Now since u /p B, k disoint, 3 n u /p B,3 k, we can apply condition (38), and since the E k s are c 3 k,k σ dx u/p p 3 k M C (h) p dx K p,k E k K p M C (h) p dx. R n K p M C p B(L p (R n )) R n h p dx, B,3 k h p E k C, k 7

if we choose C such that M C is bounded on L p (R n ), namely it must satisfy the tail condition (2). We are left with choosing the appropriate C. Now, < p < and δ > 0 are fixed from condition (38) but ε > 0 is free and will be chosen appropriately close to 0. To be more precise we need to choose 0 < ε < δ/p and let η = δ pε. Then where and A t (t) ( + log + t) ε t /p = ( + log + t) ε+(p +η)/p t/p ( + log + t) (p +η)/p =B (t)c (t), B(t) t p ( + log + t) (+ε)p +η = t p ( + log + t) p +δ C(t) t p ( + log + t) (p )η. These manipulations follow essentially O Neil [O2] but we need to be careful with the constants. It follows at once from Lemma 2. that M C B(L p (R n )) ( ) /p η = ( δ pε )/p, where we suppress the multiplicative dependence on p. Finally if we choose ε = δ 2p we get the desired result: This completes the proof of part (a) of Corollary 6. To prove part (b) we combine Lerner s theorem 3., u(ω) /p δ K ( δ )/p f L p (σ) (43) T f L p (u) c T sup T S f L p (u), S D with the characterization of the two-weight inequalities for T S from [LSU] by testing conditions: a combination of their characterizations for weak and strong norm inequalities shows in particular that T S (.σ) L p (σ) L p (u) T S (.σ) L p (σ) L p, (u) + T S (.u) L p (u) L p, (σ) Now, as it is mentioned after the statement of Corollary.6, since T S satisfies estimate (4) (see (35)) we can apply the same argument as the ust given to both summands and since that estimate has to be independent of the grid and we must take the two weight constant K over all cubes, not ust for those from the specific grid. This concludes the proof of the corollary. 7 Conectures A conecture related to Corollary.6 is as follows: Conecture 7.. Let T, p, u, σ as above. Let X is a Banach function space so that its corresponding associate space X satisfies M X : L p (R n ) L p (R n ). If K = sup ( ) /p u /p X, σ dx <, (44) 8

then T ( f σ) L p, (u) K M X B(L p (R n )) f L p (σ). (45) As a consequence, if Y is another Banach function space with M Y : L p (R n ) L p (R n ) and if K = sup ( ) /p ( /p u /p X, σ dx + u dx) σ /p Y, <, (46) then T ( f σ) L p (u) K ( M X B(L p (R n )) + M Y B(L p (R n ))) f L p (σ) (47) This is a generalization of the conecture stated in [CRV] which arises from the work [CP, CP2]. We also refer to the recent papers [La, TV] for further results in this direction. If we could prove this, we would get as corollary: Corollary 7.2. T B(L p (w)) c[w] /p A p ( [w] /p A + [σ] /p A ) This last result itself is known [HL] (see also [HLP] for a more general case), but not as a corollary of a general two-weight norm inequality. References (48) [ACM] [CAR] [CMP] [CRV] [CP] [CP2] [Duo] T. C. Anderson, D. Cruz-Uribe and K. Moen, Logarithmic bump conditions for Calderón Zygmund operators on spaces of homogeous type, to appear Publicaciones Mathematique. J. M. Conde-Alonso and G. Rey, A pointwise estimate for positive dyadic shifts and some applications, preprint, arxiv:409.435. D. Cruz-Uribe, J. M. Martell and C. Pérez, Weights, Extrapolation and the Theory of Rubio de Francia, Series: Operator Theory: Advances and Applications, Vol. 25, Birkauser, Basel. D. Cruz-Uribe, A. Reznikov and A. Volberg, Logarithmic bump conditions and the twoweight boundedness of Calderón Zygmund operators, Advances in Math. 255 (204), 706729. D. Cruz-Uribe and C. Pérez, Sharp Two-weight, weak-type norm inequalities for singular integral operators, Mathematical Research Letters 6 (999),. D. Cruz-Uribe and C.Pérez, Two-Weight, Weak-Type Norm Inequalities For Fractional Integrals, Calderón Zygmund Operators and Commutators Indiana University Mathematical Journal 49 (2000) 697-72. J. Duoandikoetxea, Fourier Analysis, American Math. Soc., Grad. Stud. Math. 29, Providence, RI, 2000. [DMRO] J. Duoandikoetxea, F. Martín-Reyes, and S. Ombrosi, On the A conditions for general bases, preprint. [FS] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math., 93 (97), 07 5. 9

[F] N. Fuii, Weighted bounded mean oscillation and singular integrals, Math. Japon. 22 (977/78), no. 5, 529 534. [F2] [G] [H] [HL] N. Fuii, A proof of the Fefferman-Stein-Strömberg inequality for the sharp maximal functions, Proc. Amer. Math. Soc., 06(2):37 377, 989. L. Grafakos, Modern Fourier Analysis, Springer-Verlag, Graduate Texts in Mathematics 250, Second Edition, (2008). T. Hytönen, The sharp weighted bound for general Calderón Zygmund operators, Ann. of Math. (2) 75 (202), no. 3, 473-506. T. Hytönen and M. Lacey, The A p A inequality for general Calderón Zygmund operators. Indiana Univ. Math. J. 6 (202), no. 6, 2042092. [HP] T. Hytönen and C. Pérez, Sharp weighted bounds involving A, Analysis and P.D.E. 6 (203), 777 88. DOI 0.240/apde.203.6.777. [HLP] [HPR] [La] [LSU] [Le] [Le2] T. Hytönen, M. Lacey and C. Pérez, Sharp weighted bounds for the q-variation of singular integrals, Bulletin London Math. Soc. 203; doi: 0.2/blms/bds4. T. Hytönen, C. Pérez and E. Rela, Sharp Reverse Hölder property for A weights on spaces of homogeneous type, Journal of Functional Analysis 263, (202) 3883 3899. M. T. Lacey, On the Separated Bumps Conecture for Calderón Zygmund Operators, to appear in Hokkaido Math J. M. T. Lacey, E. T. Sawyer and I. Uriarte-Tuero, Two weight inequalities for discrete positive operators, preprint, arxiv:09.3437. A. K. Lerner, A pointwise estimate for local sharp maximal function with applications to singular integrals, Bull. Lond. Math. Soc. 42 (200), no. 5, 843 856. A. Lerner, A simple proof of the A 2 conecture, International Mathematics Research Notices, rns45, 2 pages. doi:0.093/imrn/rns45. [LN] A. Lerner, F. Nazarov, Intuitive dyadic calculus: the basics, preprint, http://www.math.kent.edu/ zvavitch/lerner Nazarov Book.pdf (204). [LOP] A. Lerner, S. Ombrosi and C. Pérez, A bounds for Calderón Zygmund operators related to a problem of Muckenhoupt and Wheeden, Mathematical Research Letters (2009), 6, 49 56. [O] R. O Neil, Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc. 5 (965), 300-328. [O2] R. O Neil, Integral transforms and tensor products on Orlicz spaces and L p,q spaces, J. D Anal. Math. 2 (968), -276. [P] [P2] C. Pérez, On sufficient conditions for the boundedness of the Hardy Littlewood maximal operator between weighted L p spaces with different weights, Proc. of the London Math. Soc. (3) 7 (995), 35 57. C. Pérez, Weighted norm inequalities for singular integral operators, J. London Math. Soc. 49 (994), 296 308. 20

[PR] C. Pérez and E. Rela, A new quantitative two weight theorem for the Hardy-Littlewood maximal operator, Proc. Amer. Math. Soc., 43, February (205), 64-655. [RR] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 99. [Re] M. C. Reguera, On Muckenhoupt-Wheeden Conecture, Advances in Math. 227 (20), 436 450. [RT] [TV] M. C. Reguera and C. Thiele, The Hilbert transform does not map L (Mw) to L, (w), Math. Res. Lett. 9 (202), 7. S. Treil, A. Volberg, Entropy conditions in two weight inequalities for singular integral operators, preprint, arxiv:408.0385. [W] M. J. Wilson, Weighted inequalities for the dyadic square function without dyadic A, Duke Math. J., 55(), 9 50, 987. [W2] M. J. Wilson, Littlewood-Paley Theory and Exponential-Square Integrability, Lectures Notes in Math. 924, Springer-Verlag, Berlin 2008. Department of Mathematics and Statistics P.O. Box 68 (Gustaf Hällströmin katu 2b) FI-0004 University of Helsinki, Finland E-mail address: tuomas.hytonen@helsinki.fi Department of Mathematics University of the Basque Country UPV/EHU, Leioa, Spain, and IKERBASUE, Basque Foundation for Science, Bilbao, Spain E-mail address: carlos.perezmo@ehu.es 2