On the existence of principal values for the Cauchy integral on weighted Lebesgue spaces for non-doubling measures.

Transcription

1 On the existence of rincial values for the auchy integral on weighted Lebesgue saces for non-doubling measures. J.García-uerva and J.M. Martell Abstract Let T be a alderón-zygmund oerator in a non-homogeneous sace (, d, µ), where, in articular, the measure µ may be nondoubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil and A. Volberg and, indeendently by. Tolsa. In the resent work we study some weighted inequalities for T, which is the suremum of the truncated oerators associated with T. Secifically, for < <, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresonding L weighted inequality holds for T. The main tool to deal with this roblem is the theory of vector-valued inequalities for T and some related oerators. We discuss it first by showing how these oerators are connected to the general theory of vector-valued alderón-zygmund oerators in non-homogeneous saces, develoed in our revious aer [GM]. For the auchy integral oerator, which is the main examle, we aly the two-weight inequalities for to characterize the existence of rincial values for functions in weighted L. Date: January 2, (Revised version: October 8, 2000) Mathematics Subect lassification: 42B20, 42B25, 30E20. Key words and hrases: Non-doubling measures, alderón-zygmund oerators, vector-valued inequalities, weights, auchy integral, rincial value. Both authors are artially suorted by DGES Sain, under Grant PB We would like to exress our dee gratitude to J.L. Torrea for many valuable discussions and remarks.

2 Introduction. A non-homogeneous sace (, d) will be a searable metric sace endowed with a non-negative n-dimensional Borel measure µ, that is, µ(b(x, r)) r n, for all x, r > 0, where B(x, r) = {y : d(x, y) r} and n is a fixed ositive number (not necessarily an integer). Definition. A bounded linear oerator T on L 2 (µ) is said to be a alderón-zygmund oerator with n-dimensional kernel K if for every f L 2 (µ), T f(x) = K(x, y) f(y) dµ(y), for µ-almost every x \ su f, where, for some A > 0, δ > 0, K : satisfies (i) K(x, y) A, for all x y; d(x, y) n (ii) K(x, y) K(x, y), K(y, x) K(y, x ) A d(x, x ) δ x, y and d(x, y) 2 d(x, x ). whenever x, d(x, y) n+δ For such a T, we define the maximal oerator associated with it as follows: T f(x) = su T r f(x) = su K(x, y) f(y) dµ(y). r>0 r>0 \B(x,r) This kind of oerators have been introduced by [NTV2], where strong and weak tye estimates for them as well as for the corresonding maximal oerators have been obtained. The main examle is the auchy integral given by the -dimensional kernel K(z, ξ) = ; where the metric sace z ξ is endowed with some measure which has linear growth (that is, n = ) and such that this oerator is bounded in L 2 (µ). This kind of measures has been characterized in [To]. The existence of the rincial value is treated in [To2]. Many references about the develoment of the auchy integral and other toics related with it can be found in [Da2], [Da], [hr], [Mur], [MMV]. About the boundedness on L 2 (µ) of these alderón-zygmund oerators, the reader is also referred to [NTV] and for the secial case of the auchy integral to [Ver]. 2

3 In our revious work [GM], we were concerned with the following roblem: for < <, find conditions on 0 v < µ-a.e. (res. u > 0 µ-a.e.) such that T f(x) u(x) dµ(x) f(x) v(x) dµ(x), for f L (v) = L (v dµ), holds for some u > 0 µ-a.e. (res. 0 v < µ-a.e.). We introduced there the classes of weights D and Z, < <, which were defined by { } D = 0 w < µ-a.e. : w(x) ( + d(x, x 0 )) n dµ(x) < { } Z = w > 0 µ-a.e. : w(x) ( + d(x, x 0 )) n dµ(x) <. for some x 0. Note that these classes of weights do not deend on the oint x 0 and that this definition becomes simler for finite diameter saces. By using vector-valued methods we obtained in that work, that the answer to this roblem is v D (res. u Z ). For the case of auchy integral oerator, we even obtained the necessity of these classes. The aim of the resent work is to extend these results to the maximal oerator associated with T. We shall rove that these classes are sufficient to have the desired inequality when we relace T by T. As in [GM], the main tool will be the vector-valued theory develoed there. We shall see that, after some modifications, the maximal oerator fits into that theory. This will allow us to obtain some vector-valued inequalities which will be the key to coe with the weighted inequalities. For the case of the auchy integral oerator further consequences will be obtained. We shall be able to rove some results about the existence of rincial values on weighted Lebesgue saces. Namely, we obtain the equivalence of the existence of rincial values, the finiteness almost everywhere of the suremum of the truncated auchy integrals, some two-weight inequalities and the fact that one of the weights belongs to the corresonding class. The main result, which is roved in Section 4, is: Theorem.2 Let µ be a -dimensional measure in such that the auchy integral oerator is bounded on L 2 (µ). Take < < and let v be a µ-a.e. ositive measurable function such that v L loc (µ). Then the following statements are equivalent: (a) v D. 3

4 (b) There exists a µ-a.e. ositive measurable function u, such that, f(z) u(z) dµ(z) f(z) v(z) dµ(z), for any f L (v). (c) There exists a µ-a.e. ositive measurable function u, such that, u(z) dµ(z) f(z) v(z) dµ(z), λ {z : f(z) >λ} for any f L (v), λ > 0. (d) There exists a µ-a.e. ositive measurable function u, such that, ( f(z)) u(z) dµ(z) f(z) v(z) dµ(z), for any f L (v). (e) There exists a µ-a.e. ositive measurable function u, such that, u(z) dµ(z) f(z) v(z) dµ(z), λ {z : f(z)>λ} for any f L (v), λ > 0. (f) If f L (v), the rincial value exists for µ-a.e. z. lim ε 0 ε f(z) = lim ε 0 ε< z ξ < ε (g) If f L (v), then f < µ-a.e.. f(ξ) z ξ dµ(ξ) Note that we have introduced here a slightly different tye of truncated integrals ε and we have denoted by the suremum of them when 0 < ε <. The same equivalences for will be also obtained. The lan of the aer is the following. Section 2 is devoted to rove vector-valued inequalities for the maximal oerator. To do this we shall handle some smooth version of it which will be a vector-valued alderón- Zygmund oerator. In Section 3 we obtain the sufficiency of the classes of weights for the roblem we are dealing with. Finally, in Section 4 we rove the main result. 4

5 2 Vector-valued inequalities for maximal oerators. Let A, B be a coule of Banach saces, and denote by L(A, B) the set of bounded oerators from A to B. Roughly seaking, K : L(A, B) is a vector-valued n-dimensional alderón-zygmund kernel if it verifies (i) and (ii) (of Definition.) but with the L(A, B)-norm instead of the comlex modulus. In the same manner, T is a vector-valued alderón-zygmund oerator if it is given by one of these kernels away from the suort of the functions and if it extends to a bounded oerator between L 2 A (µ) and L2 B (µ). For the recise definition and more details, the reader is referred to [GM]. The classical theory for these oerators can be found in [BP], [RRT] and [GR]. In [GM], we roved that these oerators satisfy the same inequalities as the scalar ones treated in [NTV2]. That is, they are bounded from L A (µ) to L, B (µ) and from L A (µ) to L B (µ), for < <. Besides, that result had an immediate self-imrovement: with no extra hyothesis the oerator can be extended to sequence-valued functions. The concrete result we obtained there, is as follows: Theorem 2. Let T be a vector-valued alderón-zygmund oerator and take q, < q <. Then (i) T is bounded from L l q A { µ x : T f (x) q B (µ) to L, (µ), that is, l q B } } q > λ λ f (x) q A (ii) T is bounded from L (µ) to L for < <, that is, l q A lb(µ), q } T f q q } B f q q A L (µ) L (µ). } q dµ(x). In what follows T will be a (scalar) alderón-zygmund oerator with ndimensional kernel K (see Definition.). For such a T recall the definition of the maximal oerator associated with it, T f(x) = su r>0 T r f(x) = su r>0 \B(x,r) K(x, y) f(y) dµ(y), which can be viewed as a vector-valued oerator. First observe that it is enough to take the suremum ust over Q +, the set of ositive rationals. 5

6 Next consider the l (Q + )-valued oerator defined by T f(x) = {T r f(x)} r Q +. That is, we choose A = and B = l (Q + ). The kernel will be K(x, y) = {χ \B(x,r) (y) K(x, y)} r Q + taken like an element of L(, l (Q + )) = l (Q + ). However it will not be a alderón-zygmund kernel, since in general condition (ii) is not satisfied. This haens because we are cutting off with characteristic functions which give us a rough kernel. Then we can not obtain, in a straightforward way, vector-valued inequalities like in Theorem 2. for the oerator T. We avoid this roblem by introducing a smooth aroximation to the characteristic function. When we will estimate the difference between the smooth oerator and the revious one, the following version of the Hardy-Littlewood maximal function will be needed: Mf(x) = su r>0 r n B(x,r) f(y) dµ(y). For this oerator we observe the same behaviour as for T, although we can look at this oerator under a vector-valued oint of view, the kernel does not verify, in general, condition (ii). So, it will not be a alderón-zygmund oerator and we have to get a smooth version of it. To be more recise, take ϕ, ψ ([0, )), such that, χ [0,] (t) ϕ(t) χ [0,2] (t) and χ [, ) (t) ψ(t) χ [ 2, )(t). Let us consider the following smooth version of the maximal function M: ( d(x, y) ) M ϕ f(x) = su ϕ f(y) dµ(y). r>0 r n r Then, Mf(x) Mϕ f(x) 2 n Mf(x). On the other hand it is also clear that Mf(x) 3 n Mf(x), where Mf(x) = su f dµ r>0 µ(b(x, 3 r)) B(x,r) is the maximal oerator considered in [NTV2]. The boundedness of M (which is roved there, as an easy consequence of Vitali covering theorem) allows us to obtain that the oerators M and M ϕ are of weak tye (, ) and strong tye (, ), for <. For the maximal oerator associated with T, we erform the regularization given by ( d(x, y) ) T ψ, f(x) = su ψ K(x, y) f(y) dµ(y). r>0 r 6

7 Hence, by using the roerties of K, T ψ, f(x) T f(x) 2 n A Mf(x) 2 n A M ϕ f(x). () This estimate gives the boundedness of T ψ, in L 2 (µ), because M ϕ is bounded and, as it is roved in [NTV2], T is also continuous in this sace. On the other hand, () leads to T f(x) T ψ, f(x) + 2 n A M ϕ f(x), (2) which will be useful to get vector-valued inequalities for T, once we have roved these inequalities for the smooth oerators T ψ, and M ϕ. The next ste is to rove that these two oerators are in fact vectorvalued alderón-zygmund oerators. Then, after using the self-imrovement (Theorem 2.) we can extend them to sequence saces. Theorem 2.2 Under the revious assumtions, T ψ, and M ϕ can be viewed as vector-valued alderón-zygmund oerators. Therefore these oerators, and, consequently, also T will be of weak tye (, ) and strong tye (, ), for < < for T these estimates were roved in [NTV2]. Furthermore, if < q <, the following vector-valued inequalities hold (i) T ψ,, M ϕ and T are bounded from L lq(µ) to L, l (µ). q (ii) T ψ,, M ϕ and T are bounded from L l q (µ) to L l q (µ), for < <. Proof. Note that by (2) we only have to get these inequalities for T ψ, and M ϕ. For the first one, by the monotone convergence theorem, it is enough to consider the l (J)-valued oerator { ( d(x, y) ) } Tψ J f(x) = {T ψ,r f(x)} r J = ψ K(x, y) f(y) dµ(y), r r J with J Q + finite, and to obtain boundedness roerties indeendently of J. We choose A = and B = l (J). The a riori estimate of the oerator is obtained by using that T ψ, is continuous in L 2 (µ), T J ψ f L 2 l (J) (µ) T ψ, f L 2 (µ) f L 2 (µ), where is indeendent of J. This oerator is given by the kernel K J ψ(x, y) = { ψ ( d(x, y) r ) } K(x, y) r J 7 L(, l (J)) = l (J).

8 The roerties of K and the assumtions on ψ yield that Kψ J is a alderón- Zygmund kernel with constants which do not deend on J. Thus, Tψ J is a vector-valued alderón-zygmund oerator such that every constant involved is indeendent of the chosen subset J. So, by the observations above and by the vector-valued Theorem obtained in [GM], we conclude that T ψ is continuous from L (µ) to L, l (µ) and from L (µ) to L l (µ), < <. Or equivalently, that T ψ, is of weak tye (, ) and strong tye (, ) for < <. Furthermore, by using the self-imrovement (Theorem 2.), for < q < we have T ψ : L lq(µ) L, (µ), l q l T ψ : L l (µ) L (µ), < <, q l q l which, indeed, are the desired inequalities for T ψ,. For M ϕ, we can also take the suremum over Q +. For some finite J Q +, define the l (J)-valued oerator { Tϕ J f(x) = {T ϕ,r f(x)} r J = r n ( d(x, y) ) } ϕ f(y) dµ(y). r r J Take the Banach saces A = and B = l (J). Observe that T J ϕ f L 2 l (J) (µ) M ϕ f L 2 (µ) f L 2 (µ), where the constant does not deend on J. The kernel of Tϕ J is { ( d(x, y) )} Kϕ(x, J y) = r ϕ L(, l (J)) = l (J). n r r J The roerties of ϕ imly that it is a alderón-zygmund kernel. Thus, Tϕ J is a vector-valued alderón-zygmund oerator and, moreover, every constant involved is indeendent of J. Just as before, if we evaluate this oerator on f and we use the monotone convergence theorem, we obtain the desired inequalities for M ϕ. onsider the l (J)-valued oerator defined by T J f(x) = {T r f(x)} r J, for J Q + finite. Theorem 2.2 guarantees that T J is continuous from L (µ) to L, l (J) (µ), and from L (µ) to L l (J)(µ), < <. Moreover, when these saces are l q -valued ( < q < ), the corresonding vector-valued inequalities hold (see Theorem 2.2). In fact, each estimate is uniform in J. The adoint oerator of T is T, whose kernel is K (x, y) = K(y, x). For f = {f r } r J, we define T J f(x) = Tr f r (x) = K(y, x) f r (y) dµ(y). r J r J \B(x,r) 8

9 Then, it is clear that the adoint of T J is T J and that T J is bounded between L 2 l (J) (µ) and L2 (µ) with constant indeendent of J. It will not be a alderón- Zygmund oerator, because the kernel does not satisfy the smoothness condition (ii). Then, by using the functions ψ, ϕ, define T J ψ f(x) = r J T ψ,rf r (x) = r J Just as we did in (), we can estimate ( d(x, y) ) ψ K(y, x) f r (y) dµ(y). r T J ψ f(x) T J f(x) 2 n A T J ϕ ( f )(x), (3) where, for g = {g r } r J, g = { g r } r J and T ϕ J g(x) = T ϕ,r g r (x) = r n r J r J ( d(x, y) ) ϕ g r (y) dµ(y). r Observe that the adoint oerator of T J ϕ is T J ϕ. Then, we obtain that T J ϕ is bounded from L 2 l (J) (µ) to L2 (µ) indeendently of J. On the other hand, by means of (3) we can see that T J ψ is continuous from L2 l (J) (µ) to L2 (µ) uniformly in J. But, it is also true that T J f(x) T J ψ f(x) + 2 n A T J ϕ ( f )(x), (4) and this inequality allows us to get vector-valued estimates for T J, once we have roved them for the other two oerators. The next result will be a consequence of Theorem 2.2. orollary 2.3 Under the revious hyotheses T ψ J and T ϕ J are vector-valued alderón-zygmund oerators. onsequently for <, q <, we obtain that these two oerators together with T J are continuous between the following saces: (i) L l (J) (µ) L, (µ); L l (J) (µ) L (µ). (ii) L l q (µ) L, l l (µ); L L q l q (J) l (J)(µ) l (µ). q Furthermore, every boundedness does not deend on J. Proof. By the vector-valued results in [GM] and Theorem 2., it is enough to check that T J ψ and T J ϕ are vector-valued alderón-zygmund oerators. 9

10 For the first one, the kernel is K J ψ (x, y) = KJ ψ (y, x) L(l (J), ) = l (J). Since, we roved in Theorem 2.2 that Kψ J, satisfies (i), (ii), then K ψ J also does with the Banach saces A = l (J), B =. For the other oerator the kernel is K ϕ(x, J y) = Kϕ(x, J y) L(l (J), ) = l (J). We showed in Theorem 2.2 that this kernel verifies the required hyotheses; as a consequence, T ϕ J will be a vector-valued alderón-zygmund oerator (here we also have the revious Banach saces). Remark 2.4 In the last result, there is no deendence on J, so by taking finite subsets J Q +, the limit oerator T does not deend on this sequence. This new oerator acts continuously between the revious saces with l (Q + ) instead of l (J). Moreover, the restriction of T to a l (Q + )-valued sequence with a finite number of non-zero coordinates can be written as T J for some J. 3 Two-weight inequalities for T. If < <, consider the two-weight inequality for T : (T f(x)) u(x) dµ(x) f(x) v(x) dµ(x), (5) for f L (v) = L (v dµ) and u, v µ-a.e. ositive functions. Throughout this section we shall be concerned with the following roblem: Find conditions on 0 v < µ-a.e. (res. u > 0 µ-a.e.) such that (5) is satisfied by some u > 0 µ-a.e. (res. 0 v < µ-a.e.). This roblem has been reviously studied in [GM] for T. We would like to follow those ideas to conclude similar results for T. As there, we shall need the following theorem, roved in [FT], which establishes the relationshi between vector-valued inequalities and weights (for closely related results see [GR] ). Theorem 3. Let (Y, dν) be a measure sace; F, G Banach saces, and {A k } k Z a sequence of airwise disoint measurable subsets of Y such that Y = k A k. onsider 0 < s < < and T a sublinear oerator which satisfies the following vector-valued inequality T f G } L s (A k,d ν) k f F 0 }, k Z, (6)

11 where, for every k Z, k only deends on F, G, and s. Then, there exists a ositive function u(x) on Y such that { Y T f(x) G u(x) dν(x) } f F, where deends on F, G, and s. Moreover, given a sequence of ositive numbers {a k } k Z with k a k <, and σ = (, s) u(x) can be found in such a way that u χ Ak L σ (A k,dµ) (a k k ). In our context (Y, dν) = (, dµ) which is a σ-finite measure sace. Then, a simle argument shows that the weight u can be also taken so that u < µ-a.e.. Given < < and some x 0, remember the definition of the classes of weights in : D = Z = { 0 w < µ-a.e. : { } w > 0 µ-a.e. : w(x) ( + d(x, x 0 )) n dµ(x) <. } w(x) ( + d(x, x 0 )) n dµ(x) < Note that these classes of weights do not deend on the oint x 0. Remark 3.2 In the case that the diameter of the sace is finite, there exists R such that B(x 0, R) and so µ() R n <. Thus, the revious classes can be given by the equivalent definition: D = Z = { 0 w < µ-a.e. : { } w > 0 µ-a.e. : w(x) dµ(x) <. } w(x) dµ(x) < If the suort of the measure is a bounded set, by restricting the whole sace to this set, we are in the revious case. In the next result we rove the vector-valued inequalities which will be needed to aly theorem 3.. Proosition 3.3 Take 0 < s < < < and v D. diameter of is infinite, we have } (i) (T f ) } s, 2 k n s f L (v dµ), L s (S k,d µ) Then, if the

12 (ii) T } f L s (S k,d µ) s, 2 k n s f L l (Q + ) (v dµ) }, for k = 0,,..., where S 0 = {x : d(x, x 0 ) } and S k = {x : 2 k < d(x, x 0 ) 2 k }, for k =, 2,.... Otherwise, } (i) (T f ) } s, f L (v dµ), (ii) T } f L s (µ) L s (µ) s, f L l (Q + ) (v dµ) }. Proof. The roof is similar to what we did for T in [GM]. First, we work in the case when has infinite diameter. Fix k 0 and set B k+ = B(x 0, 2 k+ ). We decomose every function f = f + f = f χ Bk+ + f χ \Bk+. Then, for x S k, since v D, it follows T f A (x) f(y) dµ(y) \B k+ d(x, y) n 4 n A ( + d(y, x 0 )) n f(y) v(y) v(y) dµ(y) { } 4 n A f(y) v(y) } v(y) dµ(y) dµ(y) ( + d(y, x 0 )) n f L (v dµ). Due to the fact that µ(s k ) µ(b k ) 2 k n, we rove } (T f ) 2 k n s f L (v dµ) L s (S k,dµ) On the other hand, since 0 < s <, by Kolmogorov inequality (see [GR]. 485) and Theorem 2.2, we can obtain } (T f ) } s µ(s k ) s (T f ) µ(s k ) s L s (S k,dµ) B k+ µ(s k ) s f L (v dµ) 2 k n s f L (v dµ) } f (x) } v(x) v(x) dµ(x) }. 2 } { Bk+ } v(x) dµ(x). L, (S k,dµ)

13 The last inequality holds because > 0 and v D s. Thus, collecting these estimates we finish (i). For the other oerator we use the same aroach, but according to Remark 2.4, it is enough to rove that, for any finite J Q +, the inequality holds for T J uniformly in J. If f = {f r } r J, for x S k, we observe that T J f (x) r J T r f r (x) and it can be roved ust as before T } J f L s (S k,dµ) \B k+ 2 k n s A d(y, x) n f(y) l (J) dµ(y), f L l (J) (v dµ) }. On the other hand, since 0 < s < and reeating the revious comutations, T } J f s µ(s k ) s T } J f µ(s k ) s 2 k n s L s (S k,dµ) B k+ L, (S k,dµ) } f (x) l (J) v(x) v(x) dµ(x) f L l (J) (v dµ) } by Kolmogorov inequality (see [GR]. 485) and orollary 2.3. Then collecting both estimates we conclude (ii). In order to deal with the finite diameter case, since the measure of the sace is finite, we do not have to decomose the functions, we only follow the ideas we have used for the functions f (for a similar reasoning and more details see [GM]). With these vector-valued inequalities, we are able to rove the following result. Theorem 3.4 Take, < <. If u Z (res. v D ), then there exists some weight 0 < v < µ-a.e. (res. 0 < u < µ-a.e.) such that (5) holds. Moreover, v (res. u) can be found in such a way that v α Z (res. u α D ), rovided that 0 < α <. Proof. First, we shall rove the case v D for infinite diameter saces. Take 0 < α < and q = + α ( ). Then < q < and we can find s, 0 < s <, such that σ = ( s) > q. We use Theorem 3. with 3

14 (Y, dν) = (, dµ), F = L (v dµ), G =, {A k } k = {S k } k=0, k = 2 k n s and with the sublinear oerator T. Part (i) of Proosition 3.3 leads to the vector-valued inequality (6). Then, there exists a weight u such that (5) holds. Furthermore, u can be taken so that u L σ (S k,dµ) (a k 2 k n s ), with a k > 0 and a k <. From this oint, we only have to reeat the corresonding roof in [GM] to obtain that u α D. When the sace we are concerned with, has finite diameter, it is easier because we do not have to decomose the sace. For more details see that reference. To get the other case we roceed as follows. all ũ = u and r =. Since u Z, then ũ D = D r. For a fixed 0 < α <, we have q = + α (r ) and thus < q < r. We find 0 < s <, such that σ = ( r s) > q. If the sace has finite diameter, we use Theorem 3. with (Y, dν) = (, dµ), F = L r l (Q + ) (ũ dµ), G =, {A k} k = {S k } k=0, k = 2 k n s and with the oerator T. The vector-valued inequality (6) arises from art (ii) of Proosition 3.3. Therefore, we know that there exists some weight ṽ such that T f(x) r ṽ(x) dµ(x) f(x) r l (Q + ) ũ(x) dµ(x). (7) Moreover, like in [GM], ṽ can be found such that ṽ α D r. Take v so that ṽ = v, then v α Z. For the finite diameter case we roceed analogously and again it is easier because we do not need to decomose the sace. We want to come back to T. We restrict l (Q + ) with the aim of having only a finite number of non-zero coordinates. Take some finite set J Q +. For f = {f r } r J define f = { f r } r Q +, where f r = f r if r J and 0 otherwise. Inequality (7) alied to these sequences and Remark 2.4 allow us to observe that T J f(x) r ṽ(x) dµ(x) = T f(x) r ṽ(x) dµ(x) f(x) r l (Q + ) ũ(x) dµ(x) = f(x) r l (J) ũ(x) dµ(x). By a duality argument and by recalling that r =, this is equivalent to T J f(x) l (J) u(x) dµ(x) f(x) v(x) dµ(x). Besides T J f(x) l (J) leads to (5). = T J f(x) and the monotone convergence theorem 4

15 4 auchy integral oerator. For a non-negative Borel measure µ in the comlex lane, the auchy integral oerator of a comactly suorted function f L (µ),, is defined as f(z) = µ f(z) = f(ξ) dµ(ξ), for µ-a.e. z \ su f. z ξ Note that, in general, this definition makes no sense when z su f. This fact leads to consider the truncated auchy integrals ε and try to find the boundedness of these oerators uniformly in ε. In [To], necessary and sufficient conditions on the measure µ are given to ensure that the truncated auchy integrals are uniformly bounded in L 2 (µ). About the existence of the rincial value, in [To2] it is obtained that the boundedness of the auchy integral oerator in L 2 (µ) is a sufficient condition in order to have that for comactly suorted functions in, ε f(z) converges for µ-a.e. z as ε goes to zero. By using that this sace is densely contained in L (µ), < and the continuity in these saces of which is obtained in [NTV2] and in [To2], it follows that for f L (µ), <, the rincial value lim f(ξ) εf(z) = lim ε 0 ε 0 z ξ >ε z ξ dµ(ξ) exist for µ-a.e. z. Then we can define, at least almost everywhere, the auchy integral oerator. It is clear that this new definition agrees with the definition away from the suort. We have a metric sace with the euclidean metric and µ any - dimensional measure for which the auchy integral oerator is bounded in L 2 (µ). First, we can observe that this oerator falls within the framework of theory develoed in [NTV2]: the auchy integral oerator is defined for comactly suorted functions in L 2 (µ) by means of the -dimensional alderón-zygmund kernel K(z, ξ) =. A bounded extension to the whole z ξ L 2 (µ) arises from the existence of the rincial value and the boundedness of. Then we can aly the results we have obtained to get vector-valued inequalities for. By Theorem 2.2, the following result is established. Theorem 4. Under the above assumtions and for <, q < we have { } } (i) µ z : ( f (z)) q q > λ } f (z) q q dµ(z). λ (ii) } ( f ) q q } f q q L (µ) 5 L (µ).

16 In this framework, for < <, the classes of weights will be { } D = 0 w < µ-a.e. : w(z) ( + z ) dµ(z) < { } Z = w > 0 µ-a.e. : w(z) ( + z ) dµ(z) <. If the measure has bounded suort, these classes admit the equivalent definition given in Remark 3.2. In fact, several results will be easier when this haens. For w 0 µ-a.e. we denote w(a) = w(z) dµ(z), for any measurable set A. A As a consequence of Theorem 3.4 we obtain: orollary 4.2 Let < <. If u Z (res. v D ), then there exists a weight v (res. u) such that f(z) u(z) dµ(z) f(z) v(z) dµ(z), for any f L (v dµ). Moreover, v (res. u) can be taken in such a way that v α Z (res. u α D ), rovided that 0 < α <. For a fixed v D, with v L loc (µ), it follows that there exists another weight u such that the revious inequality holds. The density of continuous functions with comact suort in the sace L (v) and standard arguments yield the existence of the rincial value lim f(ξ) εf(z) = lim ε 0 ε 0 z ξ >ε z ξ dµ(ξ) µ-almost everywhere for any f L (v). Besides it is clear that f is µ- a.e. finite for each f L (v). So, the existence of this rincial value and the finiteness of this maximal oerator follow from the conditions on v. A natural question is whether a converse result can be obtained. The answer is affirmative, that is, the existence of the rincial value or the finiteness almost everywhere of for any function of this weighted sace imly that v is in D. We shall introduce some notation before roving these equivalence. Let us define a new maximal oerator f(z) = su ε f(z) = su 0<ε< 0<ε< ε< z ξ < ε f(ξ) z ξ dµ(ξ). If f is a comactly suorted continuous function, ε f(ξ) f(z) ε f(z) z ξ dµ(ξ) ε f L (µ) 0, as ε 0. z ξ > ε 6

17 Then, lim ε 0 ε f(z) lim ε 0 ε f(z), (8) and, if one of them exists, we get that both limits are equal. On the other hand, if f is such that f(z) <, then for 0 < ε <, and therefore ε f(z) ε f(z) + f(z) 2 f(z), ε f(z) 2 f(z). (9) Lemma 4.3 Take < < and 0 < v < µ-a.e. a measurable function. (i) If v L loc (µ) and f L (v) = L (v dµ), then for 0 < ε <, ε f makes sense because it is defined by means of an absolutely convergent integral. (ii) Assume that for any f L (v) and for any z outside of a set of µ- measure zero, we have some 0 < ε z < in such a way that ε f(z) exists for 0 < ε < ε z, that is, f(ξ) z ξ dµ(ξ) <, for 0 < ε < ε z. ε< z ξ < ε Then, v L loc (µ). Proof. For (i) it is enough to observe that f(ξ) ε< z ξ < z ξ dµ(ξ) ε ε ( ε f L (v) ξ < ε + z f(ξ) v(ξ) v(ξ) dµ(ξ) ξ < ε + z v(ξ) ) dµ(ξ) <. (0) Fix f L (v) and z 0 su µ. For k =, 2,... set B k = B(z 0, 2 k ) and S k = B k \ B k+. Then, there exists k 0 such that µ(s k0 ) > 0. The function f χ Bk0 L (v). By hyothesis we have a zero measure set E, such that, for z \ E, we can find ε z, with f(ξ) χ F ε z (ξ) <, where Fz ε = {ξ B k0 : ε < z ξ < ε }, 7

18 for 0 < ε < ε z. Since µ(s k0 ) > 0, µ(b k0 +3) > 0 and µ(e) = 0, we can find z B k0 +3 \ E and z 2 S k0 \ E. Take 0 < ε 0 < min{ε z, ε z2, 2 k0 3 } and set F = F ε 0 z, F 2 = F ε 0 z 2. Then we know that f χ F, f χ F2 L (µ) and hence f χ F F2 does. For ξ B k0, since z B k0 +3 B k0, we observe that z ξ < ε 0, and therefore F = {ξ B k0 : ε 0 < z ξ < } = {ξ B k0 : z ξ > ε 0 }. ε 0 Since z 2 B k0, we can also obtain that F 2 = {ξ B k0 : z 2 ξ > ε 0 }. Then F F2 = B k0 and f χ F F2 = f χ Bk0 L (µ). In short, for any z 0 su µ there exists r z0 > 0 such that f χ B(z0,r z0 ) L (µ). Then a comactness argument shows that for any comact K, f χ K L (µ), and it follows that L (v) L loc (µ). Thus, the linear oerator P K : L (v) L (µ) f f χ K is well defined. By using closed grah theorem and the fact that 0 < v < µ-a.e., it is easy to see that P K is also bounded. On the other hand χ K L (v ) = χ K v L (µ) = su χ K v f dµ f L (µ) = su P K (v f) L (µ) P K L (v) L (µ) <. f L (µ) = Therefore χ K L (v ) and, since this argument is valid for every comact, we conclude that v L loc (µ). Next, we can rove the result we have announced in Section. Proof of Theorem.2 Imlications (b) = (c), (d) = (e) are trivial. On the other hand, (a) (b) is one of the results we roved in [GM]. In order to rove (a) = (d), it is enough to use orollary 4.2 and (9). The other imlications will be obtained as follows: (e) = (f), (f) = (g), (g) = (a) and (c) = (a). (e) = (f) The inequality of (e) says that the oerator : L (v) L, (u) is bounded. Moreover, by (8), we know that for a continuous function f with comact suort, the rincial value lim ε 0 ε f exists µ-a.e.. Since v L loc (µ), the sace of continuous functions with comact suort is 8

19 densely contained in L (v). These things allow us to conclude that for any f L (v), this rincial value exists µ-almost everywhere. (f) = (g) Fix f L (v). We are under the hyotheses of Lemma 4.3 (ii), then v L loc (µ). This allows us to use (0). Take some z in such a way that the limit exists, then we have some ε 0, 0 < ε 0 <, such that su 0<ε ε0 ε f(z) <. Furthermore, if ε 0 ε <, by (0) we observe ε f(z) ε 0 f L (v) ( ξ < ε 0 + z and therefore f(z) < for µ-a.e. z. ) v(ξ) dµ(ξ) = (f, z, ε 0 ) <, (g) = (a) First, since the assumtions in (ii) of Lemma 4.3 are fulfilled with ε z =, we obtain that v L loc (µ) and we can use (0). We would like to aly Nikishin theorem (see [GR]. 536) and we have to check that is continuous in measure. Decomose as = = B(0, ). Fix 0 < ε <. For f L (v) we know that ε f L 0 (µ), the sace of all µ-measurable functions. We want to show that ε : L (v) L 0 (µ) is continuous in measure, and, in order to do that, it is enough to see that for each, Φ (λ) 0, as λ, where Φ (λ) = For f L (v) =, by using (0) we get ε f(z) ε su µ{z B(0, ) : ε f(z) > λ}. f L (v) = ( ξ < ε + v(ξ) ) dµ(ξ) = (ε, ) < for µ-a.e. z B(0, ). Then Φ (λ) = 0 for λ > (ε, ), and thus ε is continuous in measure. By means of (0), we observe that the maing ε ε f(z) is continuous from the right for each f L (v). onsequently, is the suremum of certain oerators, which are continuous in measure, over a countable set, Q +, and it is µ-almost everywhere finite for f L (v). Then by Banach continuity rincile, (see [GR]. 529), : L (v) L 0 (µ) is continuous in measure. 9

20 Now, Nikishin theorem (see [GR]. 536) guarantees the existence of a measurable µ-a.e. ositive function u, such that, ( ) q f L u(z) dµ(z) (v), f L (v), λ > 0 λ {z : f(z)>λ} where q = min{, 2}. But, if f is continuous with comact suort, ε f f µ-a.e. as ε 0, and, in articular, f(z) = lim ε 0 ε f(z) f(z) for µ-a.e. z. This fact, together with the inequality sulied by Nikishin theorem and a density argument, allows us to extend to the whole L (v) verifying ( ) q f L u(z) dµ(z) (v), () λ {z : f(z) >λ} for f L (v), λ > 0. We want to obtain conditions on u, v from this weak tye inequality. We will do it by using the ideas of [GM]. First, we shall see that for any z 0 su µ, a radius r z0 > 0 can be found in order to ensure u(z) dµ(z) <. (2) B(z 0,r z0 ) We shall use the notation B k and S k introduced in the roof of Lemma 4.3. For z = z + i z 2, we write z = max{ z, z 2 }. Define F = {z : z z 0 = z z 0 }, F 2 = {z : z z 0 = z 2 z 0 2}, F 3 = {z : z z 0 = z 0 z }, F 4 = {z : z z 0 = z 0 2 z 2 }. There is some k 0 0 such that S k0 has ositive measure. Assume for instance that µ(s k0 F ) > 0. We have some measurable set A S k0 F such that µ(a) > 0 and v(a) <. Hence, ust as in [GM], if z B k0 +2, (χ A )(z) Re ((χ A )(z)) A > 0. For 0 < λ 0 < A, we use () to obtain u(z) dµ(z) u(z) dµ(z) {z : (χ A )(z) >λ 0 } λ q v(a) q <, 0 B k0 +2 and it may be sufficient to take r z0 = 2 k0 2. By a comactness argument, (2) yields that u L loc (µ), and in articular, u is a finite µ-a.e. function. (In addition, we can also rove that u Z q+γ for any γ > 0, but this will not be needed in what follows). In order to obtain conditions on v, we shall dualize (). By Kolmogorov inequality (see [GR]. 485), for < r < q, we have ( ) su f r r u dµ E u(e) r r,q f L q, (u dµ) f L (v), (3) q E 20

21 where the suremum is taken over all measurable sets with 0 < u(e) <. Let A be a measurable set which verifies µ(a) <, 0 < u(a) < and u r dµ <. (4) Then, (χ A ) L (v ) = (χ A) v L (µ) = su g A (χ A ) v g dµ, (5) where the suremum is taken over all continuous functions with comact suort and g L (µ) =. Fix such a g. As µ(a) <, we have χ A L (µ) and thus (χ A ) belongs to this sace. Besides, v g L (µ) since v L loc (µ). We can use the adoint oerator = to obtain (χ A ) v g dµ = χ A (v g) dµ χ A (v g) dµ ( ) (v ( ) g) r r u dµ u r r r dµ A A ( ) u(a) r q u r r dµ, where (3) has been used. Therefore, we have ( (χ A ) L (v ) u(a) r q A A ) u r r dµ. (6) By means of this inequality, we are going to see that v D. For i =,..., 4 set E i by utting z 0 = 0 in the definition of F i. Since we know that v is locally integrable, we only have to find R i > 0 such that v(z) dµ(z) <. (7) ( + z ) E i \B(0,R i ) If su µ E i is a bounded set, it might be enough to enlarge sufficiently R i. Otherwise, we shall do the case i = and in the other regions the roof goes analogously. We can find R > 0 such that µ(b(0, R /2) E ) > 0. Then, since 0 < u < µ-a.e., there exists a measurable set A B(0, R /2) E so that (4) holds and thus the right hand side of (6) is finite. Moreover, if z E \ B(0, R ), like in [GM], it can be roved (χ A )(z) Re ((χ A )(z)) + z > 0. Therefore, we get (χ A )(z) v(z) dµ(z) 2 E \B(0,R ) v(z) ( + z ) dµ(z),

22 and consequently (7) holds for i =. As we have observed before, this fact allows us to obtain that v D. (c) = (a) Now, this imlication is a consequence of the revious one. Observe that the inequality of (c) is () with q =. From this inequality we can obtain as before that u is a finite µ-a.e. function. However, in the revious reasoning, we knew that v L loc (µ) and this allowed us to dualize (). Since, now this condition about v is not assumed, we have to modify this argument. By decomosing the sace in the dyadic level sets where 2 v(z) < 2 + and since 0 < v < µ-a.e., we can rove that boundedly suorted functions in L (µ) L (v ) L (µ) are dense in L (µ). From this, the suremum in (5) can be taken over all these functions. We can use the adoint oerator to get (6) for any measurable set verifying (4) with q = and < r <. Then, (7) holds and we only have to rove that v is locally integrable. For any z 0 su µ, we have some k 0 0 such that S k0 has ositive measure. Suose for examle that µ(s k0 F ) > 0. Since 0 < u < µ-a.e., there is a measurable set A S k0 F such that (4) holds. Then the right hand side of (6) is finite. Moreover like in the reasoning we did for u, we obtain that A B(z 0,r z0 ) v(z) dµ(z) (χ A )(z) v(z) dµ(z) <. where r z0 = 2 k0 2. A comactness argument yields that v L loc (µ) and therefore v D. Remark 4.4 We should mention that (f) and (g) can also be obtained from (a) by the results of [To2]. Namely, if R > 0, f L (v) and v D, by Hölder s inequality one obtains that f(ξ) dµ(ξ) <, for every z B(0, R). (8) z ξ ξ >2 R On the other hand, L (v) L loc (µ) and so f χ B(0,2 R) L (µ). Thus, [To2] yields the existence of the rincial value lim ε(f χ B(0,2 R) )(z), for µ-almost every z B(0, R). (9) ε 0 Let us ut ε f(z) = ε (f χ \B(0,2 R) )(z) + ε (f χ B(0,2 R) )(z). For the first term, by (8), we can rove that there exists the limit as ε goes to 0 and the suremum for 0 < ε < is finite, both facts for every z B(0, R). The 22

23 second term is handled by realizing that, for ε small enough, it is actually ε (f χ B(0,2 R) )(z). By (9) we obtain de existence of the limit and the finiteness of the suremum for µ-almost everywhere z B(0, R). Thus, it is clear that (f) and (g) hold. Next, we want to see that we can relace by. For 0 < ε <, the fact that ε f(z) exists only can mean that the integrand is a function in L (µ). However, for ε f(z), ε > 0, since we have only truncated the integral near z and not at infinity, two interretations are ossibly: either the integrand belongs to L (µ), or the integral at infinity is in fact a rincial value. Since the second one is weaker, we assume that the existence of ε f(z), ε > 0, means lim r ε< z ξ <r f(ξ) z ξ dµ(ξ), and in this case, we write ε f(z) for this limit. Observe that the existence of each integral only admits one interretation: f(ξ) z ξ dµ(ξ) <. ε< z ξ <r orollary 4.5 The statements of Theorem.2 are also equivalent to the following: (d) (e) There exists a µ-a.e. ositive measurable function u, such that, ( f(z)) u(z) dµ(z) f(z) v(z) dµ(z), for any f L (v). There exists a µ-a.e. ositive measurable function u, such that, u(z) dµ(z) f(z) v(z) dµ(z), λ {z : f(z)>λ} for any f L (v), λ > 0. (f) If f L (v), the rincial value exists for µ-a.e. z. lim f(ξ) εf(z) = lim ε 0 ε 0 z ξ >ε z ξ dµ(ξ) (g) If f L (v), then f < µ-a.e.. 23

24 Proof. By orollary 4.2, we know that (a) = (d). It is clear that (d) = (e). We can rove that (e) = (f) ust as the corresonding imlication of Theorem.2. On the other hand, (d) = (g) since u > 0 µ-a.e.. Moreover, (g) = (g) follows from (9). Therefore, we shall finish if we see that (f) = (f). Fix f L (v), and take z 0 such that ε f(z 0 ) converges as ε goes to 0. This imlies in articular the existence of ε 0, 0 < ε 0 <, for which ε f(z 0 ) makes sense if 0 < ε ε 0. Given δ > 0, there exists η 0, 0 < η 0 < ε 0, such that ε f(z 0 ) ε2 f(z 0 ) = ε < z 0 ξ ε 2 f(ξ) z 0 ξ dµ(ξ) δ < 2, (20) for 0 < ε < ε 2 η 0. On the other hand, since ε0,rf(z 0 ) converges as r goes to infinity, we can find R 0 such that, if 0 < R 0 < r 2 < r, we have f(ξ) z 0 ξ dµ(ξ) = ε0,r f(z 0 ) ε0,r 2 f(z 0 ) < δ 2. (2) r 2 z 0 ξ <r Set η = min{η 0, R0 }. If 0 < ε < ε 2 < η we can use (20), and (2) holds with r = ε, r 2 = ε 2. Thus, ε f(z 0 ) ε 2 f(z 0 ) f(ξ) z 0 ξ dµ(ξ) + ε < z 0 ξ ε 2 < δ 2 + δ 2 = δ. z ε 0 ξ < 2 ε f(ξ) z 0 ξ dµ(ξ) Since δ > 0 is arbitrary, it follows that ε f(z 0 ) converges as ε 0, and we have (f). References [BP] [hr] [Da] [Da2] A. Benedek, A.P. alderón and R. Panzone, onvolution oerators on Banach sace valued functions, Proc. Nat. Acad. Sci. USA 48 (962), M. hrist, Lectures on singular integral oerators, BMS Regional onference Series in Mathematics 77, Amer. Math. Soc., 990. G. David, Wavelets and singular integrals on curves and surfaces, Lecture Notes in Mathematics 465, Sringer- Verlag, 99. G. David, Analytic caacity, alderón-zygmund oerators, and rectifiability, Publ. Mat. 43 (999), no.,

25 [FT] [GM] [GR] L.M. Férnandez-abrera and J.L. Torrea, Vector-valued inequalities with weights, Publ. Mat. 37 (993), no., J. García-uerva and J.M. Martell, Weighted inequalities and vector-valued alderón-zygmund oerators on non-homogeneous saces, Publ. Mat. 44 (2000), no. 2, J. García-uerva and J.L. Rubio de Francia, Weighted norm inequalities and related toics, North-Holland Math. Stud. 6, 985. [MMV] P. Mattila, M.S. Melnikov and J. Verdera, The auchy integral, analytic caacity, and uniform rectifiability, Ann. of Math. (2) 44 (996), no., [Mur] T. Murai, A real variable method for the auchy transform and analytic caacity, Lecture Notes in Mathematics 307, Sringer- Verlag, 988. [NTV] F. Nazarov, S. Treil and A. Volberg, auchy integral and alderón- Zygmund oerators on nonhomogeneous saces, Internat. Math. Res. Notices 997, no. 5, [NTV2] F. Nazarov, S. Treil and A. Volberg, Weak tye estimates and otlar inequalities for alderón-zygmund oerators on nonhomogeneous saces, Internat. Math. Res. Notices 998, no. 9, [RRT] [To] [To2] [Ver] J.L. Rubio de Francia, F.J. Ruiz and J.L. Torrea, alderón-zygmund theory for oerator-valued kernels, Adv. Math. 62 (986), Tolsa, L 2 -boundedness of the auchy integral oerator for continuous measures, Duke Math. J. 98 (999), no. 2, Tolsa, otlar s inequality without the doubling condition and existence of rincial values for the auchy integral of measures, J. Reine Angew. Math. 502 (998), J. Verdera, On the T () theorem for the auchy integral, Preublicacions Universitat Autònoma de Barcelona 6/998, Ark. Mat., (To aear). José García-uerva and José María Martell Deartamento de Matemáticas, -V Universidad Autónoma de Madrid Madrid, Sain ose.garcia-cuerva@uam.es chema.martell@uam.es 25