Alternatively one can assume a Lipschitz form of the above,

Transcription

1 Dedicated to the memory of Cora Sadosky MINIMAL REGULARITY CONDITIONS FOR THE END-POINT ESTIMATE OF BILINEAR CALDERÓN-ZYGMUND OPERATORS CARLOS PÉREZ AND RODOLFO H. TORRES ABSTRACT. Minimal regularity conditions on the kernels of bilinear operators are identified and shown to be sufficient for the existence of end-point estimates within the context of the bilinear Calderón-Zygmund theory.. INTRODUCTION A crucial property addressed in the linear Calderón-Zygmund theory, going back to the founding article [], is the fact that operators bounded on L 2 whose kernels possess certain regularity are in fact bounded on every L p space for < p <. Moreover, such regularity assumption implies, together with the L 2 -boundedness of the operator, a weaktype end-point estimate in L. Form these continuity properties, the whole range of values of p follows by duality and interpolation. The quest for the minimal amount of regularity needed to guarantee the existence of such end-point estimate has a rich history with several important results that promoted as byproduct numerous developments in harmonic analysis. For classical singular integrals operators with homogeneous kernels, the question was finally settled in the work of Seeger [2] who showed that the kernel of the operators could be quite rough. See also previous work of Christ [5] and Christ and Rubio de Francia [7]. We refer to [2] and its featured review by Hofmann [3] for precise technical details, an account of the history, and relevant references. For more general multiplier operators as well as operators of non-convolution type the regularity of the kernel is very closely related to a Lipschitz-type one. It is convenient for our purposes to recall some well-known facts related to this. Assume that T : L 2 (R n L 2 (R n is an operator that, at least for x / p f, is given by T f (x = K(x,y f (ydy. Here, the kernel K is assumed to be an integrable function on any product E E 2 of compact sets in R n, with E E 2 = /0. Then, a sufficient condition for the operator to satisfy T : L L, is the gradient conditions on the kernel given by (. y K(x,y for x y, x y n+ Alternatively one can assume a Lipschitz form of the above, (.2 K(x,y K(x,y y y ε x y n+ε for y y c x y, 2000 Mathematics Subject Classification. 42B20, 42B25. Key words and phrases. Multilinear singular integrals, Calderón-Zygmund theory, weak-type estimates, endpoint estimates. First author s research ported in part by the Spanish Ministry of Science and Innovation grant MTM and by the Junta de Andalucía grant FQM Second author s research ported in part by the National Science Foundation under grant DMS

2 2 CARLOS PÉREZ AND RODOLFO H. TORRES for some 0 < ε and 0 < c <. Moreover, one can also consider a weaker condition which takes the form (.3 K(x,y K(x,y dx <. y y c x y This is sometimes referred to as the integral regularity or Hörmander condition. This condition is very convenient when working on more general geometric or measure theoretic contexts where other types of regularity are absent or hard to formalize. Actually, in R n it can be written in the more geometric form (.4 K(x,y K(x,y Q dx <, Q y Q R n \Q where the remum is taken over all cubes Q in R n with sides parallel to the axes, and where Q is the cube with same center y Q as Q and side length an appropriately chosen large multiple c n of the side length of Q (for example c n = 0 nl(q will do. To some extent, the condition (.3 is barely enough to prove the end-point estimate L L, but, unlike (. or (.2, it is not enough for other aspects of the Calderón- Zygmund theory. In particular it does not suffice to establish good- inequalities between T and the the Hardy-Littlewood maximal function M, and there is no weighted theory for operators satisfying only (.3; see the work of Martell-Pérez-Trujillo [8]. We also recall that in the case of multiplier operators, that is kernel of the form K(x y, (.3 follows from the Hörmander-Mihlin conditions. Let T given by Then, both the Mihlin [9] condition T f (ξ = m(ξ f (ξ. (.5 α m(ξ ξ α for all α [n/2] + ; as well as the weaker Hörmander [4] condtion (.6 m(2 j ϕ L 2 s (R n < for some s > n/2, j Z where ϕ is a smooth bump ported away from the origin and Ls 2 is the usual Sobolev space, imply (.3. We now describe the situation of the bilinear Calderón-Zygmund theory. Under regularity assumptions analogous to (. or (.2 the theory has been by now developed through works of Coifman-Meyer [2], [3], [4], Christ-Journé [6], Kenig-Stein [5] and Grafakos- Torres [0] and []. In particular, it was shown in [] that a bilinear operator bounded from L p L q L r for some /p + /q = /r, and given by T ( f,g(x = K(x,y,z f (yg(zdydz R2n for x / p f pg, also satisfies T : L L L /2, provided the Schwartz kernel of the operator has the Lipschitz regularity properties (.7 K(x,y,z K(x,y,z whenever y y 2 max{ x y, x z }, and (.8 K(x,y,z K(x,y,z y y ε ( x y + x z n+ε, z z ε ( x y + x z n+ε,

3 END-POINT ESTIMATE FOR BILINEAR CALDERÓN-ZYGMUND OPERATORS 3 whenever z z 2 max{ x y, x z }. As in the linear theory, if T and its two transposes T and T 2 satisfy the same conditions then interpolation and duality give boundedness for the full range of exponents, L p L q L r for all /p+/q = /r and /2 < r <. Moreover, the regularity assumptions are also good enough to obtain a weighted theory for classical A p weights, as shown by Grafakos-Torres [2] and Pérez-Torres [20], which was later extended to new optimal multilinear classes of weights by Lerner-Ombrosi-Pérez- Torres-Trujillo [6]. There has been after all thsese works some interest in finding the corresponding analog to the minimal regularity assumption (.3 in the multilinear setting. Our purpose with this short note is to contribute in this regard. We note that Maldonado-Naibo [7] have weakened (.7 and (.8 to a Diny-type condition that we shall also consider (see Definition 2.2 below and they simplified the proof of the end-point estimate. However, we are able in this article to identify integral type conditions which are even weaker and further simplify the proof of the L L L /2, result. We only focus on the bilinear setting but the interested reader may find analogous conditions in the m-linear case. The reader may also proceed to the next section were we state the conditions, but we want to provide some formal motivation that lead to the conditions we use. To do so, we recall very recent results for bilinear multiplier operators (corresponding to operators with kernels of the form K(x y,x z. It is of interest that the parallel with the linear theory in terms of minimal regularity for multipliers appears to break down. Suppose that T m is a Coifman-Meyer bilinear multiplier operator. That is, T m ( f,g(x = m(ξ,η f (ξĝ(ηe ix(ξ+η dξdη, where (.9 α m(ξ,η α ( + ξ + η α for all α, then the kernel of T m is of the form K(x y,x z and satisfies α K((x y,x z α ( x y + x z 2n+ α. This follows from considering m as a multiplier in R 2n and restricting the corresponding kernel K(x y,x 2 z to x = x 2. It also follows then that these operators are bounded from L p L q L r for the full range of exponents and also from L L to L /2,. It is very natural to expect that (.9 could be relaxed to a Hörmander-Mihlin condition, limiting the number of derivatives of the symbol needed to be controlled. Certainly the arguments used by Coifman-Meyer only need a sufficiently large number of derivatives (not easy to track in their computations but one should expect to require only about halfthe-dimension number of derivatives. However, the natural dimension in the bilinear setting appears to be 2n and so, some interesting situation occurs. In fact, Tomita [22] recently showed that (.0 m(2 j,2 j ϕ L 2 s (R 2n < for some s > n, j Z where ϕ is an appropriate cut-off function in R 2n, implies the boundedness T m : L p L q L r with /p + /q = /r, but only for r >. That is, essentially half-the-dimension number of derivatives in L 2 to obtain the range r >. However, to obtain other values of /2 <

4 4 CARLOS PÉREZ AND RODOLFO H. TORRES r, which is also natural in the bilinear case, one needs to impose higher regularity. Grafakos-Si [9] showed after Tomita s work that to push p,q to + ε (i.e. r to /2 + ε/2 it appears to be necessary to consider (. m(2 j,2 j ϕ L tε s (R 2n < j Z for appropriate < t ε 2 and s > 2n/t ε. Essentially, one may say that 2n derivatives in L may be required to get the full range of exponents. We refer to [9] for the precise technical details. Similar results on the product of Hardy H p were very recently obtained by Grafakos-Miyachi-Tomita [8] but, as far as we know and unlike the linear case, there are no results of this type that give the end-point estimate L L L /2, (except, of course, for the sufficiently large number of derivatives in the arguments of Coifman-Meyer using (.9 and which give pointwise estimates on the gradient of the kernel. As already mentioned, we want to find some bilinear analog of the integral condition (.3 even for non-convolution operators, but it is instructive to see what such a condition could be in the case of multipliers. If we consider again m as a Fourier multiplier in R 2n, then (.0 is just Hörmander s condition and we have for the kernel of the corresponding linear operator in R 2n for any y,z R n K(x y,x 2 z K(x y,x 2 z dx dx 2 C. y y + z z c x y + x 2 z When z = z and performing a simple change of variables, the above can be written as K(x y,x (x x 2 + z K(x y,x (x x 2 + z dx 2 dx y y c x y + x 2 z = y y c x y + x u K(x y,x u K(x y,x u dudx C. In a more geometric form (and considering a smaller region of integration we can essentially state the above in the form (.2 K(x y,x u K(x y Q,x u dxdu C, Q y Q R n R n \Q and similarly in the other variable, (.3 K(x u,x z K(x u,x y Q dxdu C. Q y Q R n R n \Q In the non-convolution case of interest to us, the conditions (.2 and (.3 would become (.4 K(x,y,u K(x,y Q,u dxdu C Q y Q R n R n \Q and (.5 Q z Q R n R n \Q K(x,u,z K(x,u,z Q dxdu C. Unfortunately these conditions do not seem to be enough to show the end-point estimate we are looking for. This is not surprising, since as observed, in the convolution case they are implied by Tomita s condition (.0 which does not seem to be even enough for boundedness for any r.

5 END-POINT ESTIMATE FOR BILINEAR CALDERÓN-ZYGMUND OPERATORS 5 We find interesting that a small modification of (.4 and (.5 (moving the y Q inside the integral together with a closely related new integral condition do suffice. Moreover we will show that the conditions are implied by a very general Dini-type one, hence showing that they are weaker than all others considered in the literature so far. 2. THE NEW REGULARITY CONDITIONS Definition 2.. We say that the bilinear operator with kernel K satisfy the bilinear geometric Hörmander conditions (BGHC if there exits a fixed constant B such that and for any two families of disjoint dyadic cubes D and D 2, (2. K(x,y,z K(x,y Q,z dxdz B, Q D R n y Q R n \( R D R (2.2 K(x,y,z K(x,y,z P dxdy B, P D 2 R n z P R n \( S D2 S and (2.3 P Q (P,Q D D 2 (y,z P Q R n \( R D R ( S D2 S B( P D P + Q D2 Q. K(x,y,z K(x,y P,z Q dx We want to make some further remarks about these conditions. Write Ω = Q D Q, Ω = Q D Q, Ω 2 = P D2 P, and Ω 2 = P D P. First, note that for any y,y Q, K(x,y,z K(x,y,z dx y Q R n \Ω K(x,y,z K(x,y,z dx + K(x,y,z K(x,y,z dx y Q R n \Ω R n \Ω K(x,y,z K(x,y,z dx, y Q R n \(Ω so the inner remum and integral in (2. can be replaced by either K(x,y,z K(x,y,z dx, y Q y Q R n \Ω or inf K(x,y,z K(x,y,z dx. y Q y Q R n \Ω Similarly with (2.2, while it is also equivalent to replace the remum and integral in (2.3 by K(x,y,z K(x,α,β dx R n \(Ω Ω 2 or (α,β P Q (y,z P Q inf (α,β P Q (y,z P Q R n \(Ω cupω 2 K(x,y,z K(x,α,β dx.

6 6 CARLOS PÉREZ AND RODOLFO H. TORRES Next, we note that (2.3 is in some sense the strongest of the three conditions, since it gives a substantial part (though not all of (2. and (2.2. In fact, note that D and D 2 could be the same family and so assuming that (2.3 holds, Q D Q Q D Q Q Q (Q,Q D D Ω y Q R n \Ω z Q y Q R n \Ω Q D Q (α,β Q Q (y,z Q Q K(x,y,z K(x,y Q,z dxdz K(x,y,z K(x,y Q,z dx R n \(Ω Ω K(x,y,z K(x,α,β dx B ( Q D Q + Q D Q 2B Q D Q. From this it follows that K(x,y,z K(x,y Q,z dxdz 2B. Ω y Q R n \Ω So, given (2.3, the condition (2. could be replaced with the much weaker one inf K(x,y,z K(x,y,z dxdz B, y Q y Q R n \Ω R n \Ω and similarly with (2.2. We now show that the BGHC are implied by a Dini-type one. The following condition is essentially the one considered by Maldonado and Naibo [7], except that we do not require the function Φ involved to be convex. Definition 2.2. Let Φ increasing and such that Φ(t dt <. 0 t We say that K satisfies a bilinear Dini-type condition if K(x,y,z K(x,y,z whenever y y 2 x y and z z 2 x z. C ( y y ( x y + x z 2n Φ + z z x y + x z By taking Φ(t = t ε it follows that the bilinear Lipschitz regularity conditions imply the Dini-type one. We now show that the BGHC is actually weaker. Proposition 2.3. The Dini-type condition implies the BGHC. Proof. We first show that the Dini-type condition implies K(x,y,z K(x,y Q,z dxdz C R n y Q R n \Ω for some C > 0. Indeed, R n R n y Q R n \Ω y Q R n \Q R n K(x,y,z K(x,y Q,z dxdz ( ( x y + x z 2n Φ y y Q x y + x z ( nl(q R n \Q ( x y Q + x z 2n Φ x y Q dxdz dxdz

7 END-POINT ESTIMATE FOR BILINEAR CALDERÓN-ZYGMUND OPERATORS 7 x >c n l(q By symmetry, the proof of (2.2 is identical. We now prove (2.3. Fix P and Q then Then, (P,Q D D 2 (y,z P Q (y,z P Q inf (y,z P Q (y,z P Q ( ( ( (P,Q D D 2 P Q P Q (P,Q D D 2 = C ( nl(q R n \Q x y Q n Φ x y Q ( x n Φ cn l(q x dx dx <. R n \(Ω Ω 2 K(x,y,z K(x,y P,z Q dx ( y ( x y + x z 2n Φ yp + z z Q x y + x z R n \P Q R n \P Q ( x y + x z 2n Φ ( n(l(p + l(q x y P + x z Q dx dx ( n(l(p + l(q R n \P Q ( x y + x z 2n Φ dx. x y P + x z Q inf (y,z P Q P ( Q K(x,y,z K(x,y P,z Q dx (y,z P Q R n \(Ω Ω 2 ( ( n(l(p + l(q R n \P Q ( x y + x z 2n Φ dx x y P + x z Q ( n(l(p + l(q R n \P Q ( x y + x z 2n Φ dx dydz x y P + x z Q (P,Q D D 2 :l(pl(q + (P,Q D D 2 :l(ql(p = I + II We estimate I, the other term is of course similar. Now, ( I ( x y + x z 2n Φ 2 nl(q x y P + x z Q (P,Q D D 2 Q D 2 Q D 2 P Q R n \Q R n \Q R n \P Q Q P D Q ( 2 nl(q P ( x y + x z 2n Φ x z Q ( 2 nl(q ( x y + x z 2n dydzφ x z Q R n We can estimate the inner most integral by x z n, hence ( 2 nl(q I Q D R n \Q Q 2 x z n Φ x z Q ( 2 nl(q Q D Q R n \Q 2 x z n Φ x z Q ( 2 nl(q x z Q n Φ x z Q Q D 2 Q R n \Q C dxdz dxdz dxdz dy dzdx dx dx dydz

8 8 CARLOS PÉREZ AND RODOLFO H. TORRES Q Q D x 2 ( x n Φ dx Q D2 Q Φ(t dt x 0 t. Likewise, the term II is bounded by C P D P, which completes the proof of ( THE END-POINT ESTIMATE Theorem 3.. Let T be a bilinear operator satisfying T : L p L p 2 L p 3 for some /p + /p 2 = /p 3, p, p 2 < ; and the BGHC. Then, T : L L L /2,. Proof. It is enough to show that ( (3. {x R n : T ( f, f 2 (x > 2 f (x /2 ( f 2 (x /2 } C dx dx, R n R n for all f, f 2 C c (R n. Moreover, by homogeneity we may assume that f = f 2 = and prove that (3.2 {x R n : T ( f, f 2 (x > 2 } C, with constant C independent of. Fixed > 0, for f we consider the standard Calderón-Zygmund decomposition al level and obtain a collection of dyadic non-overlapping cubes Q,k that satisfy If we set Ω = k Q,k, then < Q,k Q,k f (x dx 2 n. Ω C and f (x a.e.x R n \ Ω. As usual, we write f = g + b, where g is defined by { f (x, x R n \ Ω g (x = f Q,k, x Q,k, and where f Q = Q Q f ; the good function g satisfies for any s (3.3 g s C /s 2s f + s f 2 2s = C /s ; and b is written as b (x = b,k (x = ( f (x f Q,k χ Q,k (x. k k We do the same for f 2. Via the Calderón-Zygmund decomposition at same level, we obtain a collection of of dyadic non-overlapping cubes Q 2,k with union Ω 2 and analogous decomposition f 2 = g 2 + b 2 with the properties described for f. Set Ω = Ω Ω 2. We split the distribution set we are trying to estimate in several parts as follows, {x R n : T ( f, f 2 (x > 2 } {x / Ω : T ( f, f 2 (x > 2 } + Ω {x R n : T (g,g 2 (x > 2 /4} + {x R n \ Ω : T (g,b 2 (x > 2 /4} + {x R n \ Ω : T (b,g 2 (x > 2 /4}

9 END-POINT ESTIMATE FOR BILINEAR CALDERÓN-ZYGMUND OPERATORS 9 + {x R n \ Ω : T (b,b 2 (x > 2 /4} + Ω = E + E 2 + E 3 + E 4 + Ω. Clearly Ω is of the right size, so we only need to estimate the other sets. To estimate E, we use the fact that T : L p L p 2 L p 3 for some /p + /p 2 = /p 3 to obtain E C p T (g,g 2 (x p 3 dx C 3 R n 2p g p 3 3 p g 2 p 3 p 2 C 2p p 3( /p p 3( /p 2 = C 3. By symmetry, the study of E 2 and E 3 are similar, so we only consider E 2. We have E 2 C R 2 T (g,b 2,k (x dx n \Ω k C 2 K(x,y,zg (yb 2,k (zdzdy dx k R n \Ω R n Q 2,k C ( 2 k R n \Ω g (y K(x,y,z K(x,y,z Q2,k b 2,k (zdzdy 2 R Q2,k dx n C 2 g (y b 2,k (z K(x,y,z K(x,y,z k R n Q 2,k R n \Ω Q 2,k dxdzdy C 2 g (y b 2,k (z dz K(x,y,z K(x,y,z k R n Q 2,k z Q 2,k R n \Ω Q 2,k dxdy C g (y Q 2,k K(x,y,z K(x,y,z R n k z Q 2,k R n \Ω Q 2,k dxdy C g Q 2,k R K(x,y,z K(x,y,z n k z Q 2,k R n \Ω Q 2,k dxdy C Ω 2 C, where we have used the condition (2.2. It only remains to estimate Writing we have E 4 = {x R n \ Ω : T (b,b 2 (x > 2 /4}. T (b,b 2 = T (b,l,b 2,k, l,k E 4 C 2 T (b,l,b 2,k (x dx l,k R n \Ω C 2 K(x,y,zb,l (yb 2,k (zdzdy dx. l,k R n \Ω Q,l Q 2,k Fix one of these Q,l and Q 2,k Q,l Q 2,k b,l (yb 2,k (zdzdy = 0

10 0 CARLOS PÉREZ AND RODOLFO H. TORRES we have K(x,y,zb 2,k (zb,l (ydzdy dx R n \Ω Q,l Q 2,k = (K(x,y,z K(x,y Q,l,z Q2,k b 2,k (zb,l (ydzdy dx R n \Ω Q,l Q 2,k K(x,y,z K(x,yQ,l,z Q2,k dx b 2,k (z b,l (y dzdy R n \Ω Q,l Q 2,k K(x,y,z = K(x,yQ,l,z Q2,k dx b 2,k (z b,l (y dzdy Q,l Q 2,k R n \Ω K(x,y,z b 2,k (z b,l (y dzdy K(x,yQ,l,z Q2,k dx Q,l Q 2,k (y,z Q,l Q 2,k R n \Ω C 2 K(x,y,z Q,l Q 2,k K(x,yQ,l,z Q2,k dx. R n \Ω Finally, (y,z Q,l Q 2,k K(x,y,z E 4 C Q,l Q 2,k K(x,yQ,l,z Q2,k dx l,k (y,z Q,l Q R n \Ω because of the condition (2.3. C ( l Q,l + k Q 2,k = c( Ω + Ω 2 C REFERENCES [] A.P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (952, [2] R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math.Soc. 22 (975, [3] R. R. Coifman and Y. Meyer, Commutateurs d intégrales singuliéres et opérateurs multilinéaires, Ann. Inst. Fourier (Grenoble 28 (978, [4] R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Ast erisque, 57 (978, 85. [5] M. Christ Weak type (, bounds for rough operators, Ann. of Math. (2 28 (988, 942. [6] M. Christ and J-L. Journ, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 59 (987, no. -2, 580. [7] M. Christ and J.L. Rubio de Francia, Weak type (, bounds for rough operators. II, Invent. Math. 93 (988, [8] L. Grafakos, A. Miyachi, and N. Tomita, On multilinear Fourier multipliers of limited smoothness, preprint. [9] L. Grafakos and Z. Si, The Hörmander type multiplier theorem for multilinear operators, J. Reine Angew. Math., to appear. [0] L. Grafakos and R.H. Torres, On multilinear singular integrals of Calderón-Zygmund type, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000, Publ. Mat. 2002, Vol. Extra, [] L. Grafakos and R.H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 65 (2002, no., [2] L. Grafakos and R.H. Torres, Maximal operator and weighted norm inequalities for multilinear singular integrals, Indiana Univ. Math. J. 5 (2002, no. 5, [3] S. Hofmann, Featured Review of Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (996, no., 95 05, by A. Seeger, Math Rev. MR37232 (96f:4202. [4] L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math. 04 (960, [5] C.E. Kenig and E.M. Stein, Multilinear estimates and fractional integration., Math. Res. Lett. 6 (999, 5. [6] A.K. Lerner, S. Ombrosi, C. Pérez, Carlos; R.H. Torres, and R. Trujillo-González, New maximal functions and multiple weights for the multilinear Caldern-Zygmund theory, Adv. Math. 220 (2009, no. 4,

11 END-POINT ESTIMATE FOR BILINEAR CALDERÓN-ZYGMUND OPERATORS [7] D. Maldonado and V. Naibo, Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity, J. Fourier Anal. Appl. 5 (2009, no. 2, [8] J.M. Martell, C. Pérez, and R. Trujillo-González, Lack of natural weighted estimates for some singular integral operators, Trans. Amer. Math. Soc. 357 (2005, no., [9] S. G. Mihlin, On the multipliers of Fourier integrals [in Russian], Dokl. Akad. Nauk. 09(956, [20] C. Pérez and R.H. Torres, Sharp maximal function estimates for multilinear singular integrals, Harmonic analysis at Mount Holyoke (South Hadley, MA, 200, 32333, Contemp. Math., 320, Amer. Math. Soc., Providence, RI, [2] A. Seeger, Singular integral operators with rough convolution kernels, J. Amer. Math. Soc. 9 (996, no., [22] N. Tomita, A Hörmander type multiplier theorem for multilinear operator, J. Funct. Anal. 259 (200, CARLOS PÉREZ, DEPARTAMENTO DE ANÁLISIS MATEMÁTICO, FACULTAD DE MATEMÁTICAS, UNI- VERSIDAD DE SEVILLA, 4080 SEVILLA, SPAIN address: carlosperez@us.es RODOLFO H. TORRES, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KANSAS, 405 SNOW HALL 460 JAYHAWK BLVD, LAWRENCE, KANSAS , USA address: torres@math.ku.edu