Estimates on the Bergman Kernels in a Tangential Direction on Pseudoconvex Domains in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3067999pt" id="M1" height="15.6075pt" version="1.1" viewBox="-0.0657574 -15.3007 18.804 15.6075" width="18.804pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-14" d="M632 159C554 79 496 30 378 30C330 30 251 50 213 80V583C258 614 324 632 379 632C497 632 552 583 630 503L661 534C584 627 498 676 375 676C172 676 45 529 45 332C45 140 176 -14 374 -14C497 -14 586 35 663 128L632 159ZM169 118C160 125 155 134 148 145C111 201 89 264 89 332C89 416 122 490 169 558V118Z"/></g><g transform="matrix(.012,0,0,-0.012,12.098,-7.578)"><path id="g50-52" d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"/></g></svg>

Cho, Sanghyun

doi:https://doi.org/10.1155/2018/5360857

Abstract and Applied Analysis

On this page

Abstract Introduction Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 5360857 | https://doi.org/10.1155/2018/5360857

Estimates on the Bergman Kernels in a Tangential Direction on Pseudoconvex Domains in

Sanghyun Cho¹

Academic Editor: Milan Pokorny

Received02 Mar 2018

Accepted05 Jul 2018

Published18 Jul 2018

Abstract

Let be a smoothly bounded pseudoconvex domain in and assume that where , the boundary of . Then we get optimal estimates of the Bergman kernel function along some “almost tangential curve” .

1. Introduction

Let be a bounded domain in . A natural operator on is the orthogonal projection where denotes the holomorphic functions on . There is a corresponding kernel function , called the Bergman kernel function on . The nature of the singularity of tells us much about the holomorphic function theory of the domain in question and has been studied extensively since Bergman’s original inquiries [1].

One of the methods for the estimates of the Bergman kernel is to construct maximal size of polydiscs in where we have a plurisubharmonic function with maximal Hessian. For strongly pseudoconvex domains in , these polydiscs are of size in normal direction and of size in tangential directions. For weakly pseudoconvex domains, the size of the polydisc in tangential directions depends on the boundary geometry of near , and hence we need complete analysis of the boundary geometry near .

However these analyses and hence the optimal estimates on the Bergman kernels are done only for special type of pseudoconvex domains of finite type in . These domains are, for example, pseudoconvex domains of finite type in [2–4], decoupled, convex, or uniformly extendable domains of finite type in [5–7], or pseudoconvex domains in with positive eigenvalues [8, 9]. For the estimates for weighted Bergman projections, one can also refer to [10–12]. Nevertheless, the optimal estimates for general pseudoconvex domains of finite type in , , are not known, even for case.

Assume that is a smoothly bounded domain in with smooth defining function with smooth boundary, . Regular finite 1-type at , denoted by , is the maximum order of vanishing of for all one complex dimensional regular curve , , and . Thus satisfieswhere , , denotes finite -type in the sense of D’Angelo [13]. Note that where is the type in the sense of Bloom-Graham.

Remark 1. Consider the domain [13] in defined by Then and while as the complex analytic curve lies in the boundary. Note that is not regular curve.

In the sequel, we let be a smoothly bounded pseudoconvex domain in , and assume that where . Let be the “almost tangential curve" connecting a point and as defined in (20). Note that for each . Set , where is defined in (51).

Theorem 2. Let be a smoothly bounded pseudoconvex domain in and assume that where . Then , the Bergman kernel function of at , satisfies

Theorem 3. Let and be as in Theorem 2. For each , there is a constant , independent of , such thatfor and .

Remark 4. (1) In Theorems 2 and 3, we do not assume that , but we assume only that (see Remark 1). With this weaker condition, we get optimal estimates for Bergman kernel function along special “almost tangential” direction, , but not normal or arbitrary direction.
In [14], Herbort gives an example of a domain where the Bergman kernel grows logarithmically when approaches to in normal direction. Setand for each small , set . Thus approaches to in normal direction as . In this case, Herbort shows that ; that is, the kernel grows logarithmically. For the same domain in (6), we note that and hence , in (4). Set . Then approaches to in “almost tangential direction”. In the Appendix of this paper, we will show that

In Section 2, we will construct special coordinates which reflect the regular finite type condition, , and then show that vanishes to order in -direction. We then consider the slices of by fixing . Then the domains become domains in , and hence we can handle them. Also, the condition acts like the condition on these slices.

For the estimates of , Catlin [2, 15] constructed plurisubharmonic functions with maximal Hessian near each thin -strip of (Section of [2]). In this paper, however, we will construct these functions only on nonisotropic polydiscs for each (Proposition 23). This avoids complicated technical parts in Section of [2]. To get estimates of , , , we consider dilated domains for each . Then the polydisc becomes , independent of , where is a polydisc of radius one with center at the origin. Therefore the uniform -subelliptic estimates for -equation hold on , and the estimates for follow.

Remark 5. Let be a smoothly bounded pseudoconvex domain in , and assume that , where . Then the conditions of Theorems 2 and 3 are satisfied. Near future, using the results of Theorems 2 and 3, we hope we can prove some function theories on , for example, the existence of peak function for that peaks at or necessary conditions for the Hölder estimates for -equation.

2. Special Coordinates

In the sequel, we let be a smoothly bounded pseudoconvex domain in and assume that , . Note that and are positive integers. Without loss of generality, we may assume that . In the sequel, we let and be multi-indices and set and , etc. In Theorem 3.1 in [16], You constructed special coordinates which represent the local geometry of near .

Theorem 6. Let be a smoothly bounded pseudoconvex domain in with smooth defining function and assume , . Then there is a holomorphic coordinate system about such thatwhere

(Idea of the proof) by the standard holomorphic coordinate changes, has the Taylor series expansion as in (8). Since , there is a regular curve which we may assume that satisfying for all sufficiently small . Set . Then, in coordinates, has representation satisfying (8). Also (9) follows from the condition that .

Remark 7. (1) The second condition in (8) and property (9) say that vanishes to order along axis and order along axis.
(2) There are much more terms (mixed with , and their conjugates), compared to the -extensible domain cases, in the summation part of (8).

In conjunction with multitype , we need to consider the dominating terms (in size) among the mixed terms in and variables in the summation part of (8). Using the notations of Section in [16], set Then there are for and for , such that

Remark 8. (1) Here, ’s and ’s are the exponents of and , respectively, in the dominating terms in the summation part of (8).
(2) If , then the expression in (8) will be similar to that of case in [2], and hence we need not consider the above complicated pairs.

Set . If , then for some . In this case, setThen , , and are colinear points in the first quadrant of the plane, and (resp., ) is the intercept of -axis (resp., -axis) of this line. Let be the line segment from to for , set , , and set As in Corollary 3.8 and Remark 3.9 in [16], we can rewrite (8) so thatwhere is a nontrivial homogeneous polynomial of degree given by and there are a small constant , and such thatfor all with all . Property (16) means that there is with terms mixed in and variables for . Let be the constant (direction) in (16) and we will fix in the rest of this paper. In the sequel, we set equal to or , .

Remark 9. (a) defined in (12) is strictly decreasing on .
(b) Each of the summation parts of (14) contains the terms of the form where ’s are the pairs, defined in (12), on the polyline .
(c) Each term of the first summation part in (14) is pure in or variables.
(d) Each term of the second summation part in (14) has terms mixed in and , and it corresponds to the pair of integers , the vertices of the polyline .

Lemma 10. Let be the term containing only or variables in the first sum of (14). Then

Proof. From (8) and (14), we see that . On the other hand, since the regular 1-type at is equal to , there is such that .

In the sequel, we let be a small neighborhood of where has expression as in (14). Since , we may assume that for all for a uniform constant by shrinking if necessary. For each fixed and for each satisfying , for a sufficiently small to be chosen, we set , where is the composition of the projection onto plane and then the projection onto along the direction. Using the Taylor series method in variable about , we see that Since and on , it follows from (17) that for near .

Now for each small , set and set . For a small constant to be chosen, set , and for a fixed small satisfying , set connecting and .

Following the same arguments as in the proof of Proposition 1.2 in [2], for each fixed , we can construct special coordinates about so that, in terms of new coordinates, there is no pure terms in or variables in the first summation part of in (14). We will fix variable and consider the coordinate changes only on variables.

Proposition 11. For each fixed , there is a holomorphic coordinate system such that in the new coordinates defined by where and where , , depends smoothly on , the function given by satisfies

Proof. For , define where . Then we have Assume that (22) and (23) hold for . That is, we have defined so that can be written as If we define , where then satisfies (26) for replaced by . If we proceed up to and set , then by setting , we see that (22) and (23) hold.

In the sequel, we will use the coordinate changes in Proposition 11 only at , (in particular at in Section 3). We want to study the dependence of about . For each , set , and we note that where is defined in the inductive step of the proof of Proposition 11. Set and set . Then and

To study the dependence of and hence dependence of about in (23), we thus need to study the dependence of on variable. For a convenience, set , i.e., remove tilde’s, and assume that satisfies for a sufficiently small to be chosen. In view of (19), we see that satisfies (31). In following we let be the given coordinates, and we let be the coordinates obtained from holomorphic coordinates changes of , as in -th step of coordinate changes in the proof of Proposition 11. Also we let (resp., ), , denote any partial derivative operator of order with respect to and (resp., and ) variables. According to the coordinate changes in Proposition 11, we note that .

Proposition 12. Assume that satisfies (31). Then for each , we haveand for each with , for some in (14), we have

Proof. We will prove by induction on . From (14), (17), and (31) one obtainsand hence (32) follows for . Since , it follows, from (31) and chain rule, that because we are evaluating at . This proves (32) for .
By induction, assume that (32) holds for . For the defined in (29), it follows, from (34) and induction hypothesis, thatfor . Since we are evaluating at , it follows, for , thatBy (30), (36), and (37) and by induction, (32) holds for because if .
Now we prove (33). Assume with , where are the pairs corresponding to the second summation part of (14). Note that the first summation part of (14) will be annihilated by because it contains the pure terms of or mixed with . Thus it follows from (14), (16), and (31) thatSince , it follows from (31) and (38) that Similarly, since , it follows from (36) that because for . This proves (33) for .
By induction assume that (33) holds for . If with and , that is, if has mixed derivatives of and , we note that (37) becomesIf , (33) follows from (41) and induction hypothesis of (33). If , it follows, from (36), (41), and induction hypothesis of (33), that because for . Therefore (33) is proved for .

Recall the expression of and coefficient functions in (23).

Corollary 13. Assume that satisfies (31). Thenand if for some in (14), then

Proof. From (23) we see that where and . Hence it follows from (32) that Assume for some . Thus and it follows from (23), (33), and (41) that because .

Remark 14. Suppose that and . Then , , and are colinear points. From the standard interpolation method, we havefor all sufficiently small . Assume that and for any of . Therefore it follows from (43) and (48) that Therefore the terms of the form , with for some , in the summation part in (23), are the major terms which bounds the other summation terms from above.

In the sequel, we assume that satisfies (31). As in Section in [2], for each , setIn view of Remark 14, we will consider only for , . From (9) and (44) we note thatbecause . For each sufficiently small , setand setFrom (51) and (52), we see that if , then

Lemma 15. For each , .

Proof. Set and . Then Therefore because .

Proposition 16. Assume satisfies (31). Thenwhere .

Proof. By (31), we note that . Assume that . Then by (51). Therefore it follows, from (50) and (52), that and hence it follows from (52) and (53) that Thus follows. Similarly, one can show that .

Let be a small constant to be determined (in Remark 22). By Lemma 15, for each , independent of . Therefore there is a smallest integer , , such thatThen for some by (53). In following, for the fixed integer in (59), set , , and as usual. If we define , where is defined in (22), we may regard that is a biholomorphism. For each , set . For each small , defineand set and when .

Proposition 17. The function satisfies

Proof. Recall that , and in (32). When , it follows from (12) () and (32) that Assume . Then by (12), for some , and hence it follows that . Therefore one obtains, from (48)–(52), thatFrom (32) and (63), it follows that

Using the coordinates defined in (14), set Then , , are tangential holomorphic vector fields and on for a uniform constant . For any with , define In -coordinates defined by , set , and set , . If we define then by functoriality,

Lemma 18. There is a small constant such thatprovided is sufficiently small.

Proof. Since the level sets of are pseudoconvex, it follows from (61) thatRecall that is the term which contains only or variables in the first summation part of (14). Therefore it follows, from (17), (19), and (23), thatbecause . If , it follows from (61) and (71) and by using the Taylor series method thatprovided is sufficiently small. Thus (69) follows from (68), (70), and (72).

In the sequel, we let and be the constants which may different from time to time but depend only on the derivatives of or up to order . Recall that , . By using (61), and by using Taylor series method, one obtains thatprovided is sufficiently small, where and . Note that we can write where . By applying or successively to , we obtain thatwhere, by (73) and by using induction method, satisfiesCombining the estimate in (61), (75), and (76), one obtains that

Assume that (59) holds. Thus for some , and hence it follows from (53) that . Therefore it follows from (23) and (50) that there exist integers with , such thatFor these , it follows from (61), (75), (76), and (78) and by using the Taylor series method that there are constants such that provided is sufficiently small.

Lemma 19. There is such that

Proof. By functoriality, we have From (23), we see that and it follows from (61) that Therefore (80) follows from (73), (81), and (83) and by using Taylor series method.

Note that , for some , and hence there exist , with . In view of (79), we may assume that is valid (when , we replace by ). Set By using the estimates (73)–(76), one obtains that and similarly, for , where and .

Lemma 20. Assume that (59) holds. Then

Proof. Suppose . In view of (51)–(53), (56), and (59), we see that and hence it follows that This together with (73)–(78) implies the estimate (88).

In the sequel, we write

Lemma 21. There is a positive number , independent of and , such that if and if (59) holds, then there are constants and , independent of , and , such that

Proof. Suppose . From (87) and (88), we note thatfor where and . Using (84)–(88) and (93) and by using small (large) constant method, one obtains that for some and provided is sufficiently small.

Remark 22. From now on, we fix constants and , which depend only on the derivatives of or of order up to on , satisfying (69), (73), (80), and (86)–(92), and set for a convenience. Now we choose and fix and then fix so that

3. Estimates on the Bergman Kernels

Recall that where and where is the projection defined before (19). Also note that where and where is the holomorphic coordinate function defined in Proposition 11 about . Also recall defined in (20). In this section we estimate the Bergman kernel function , for and .

To get optimal estimates of the Bergman kernel, we need to construct a plurisubharmonic function which has maximal Hessian near each thin neighborhood of as in [2, 15]. It contains complicated estimates depending on the type conditions of each boundary points. In this paper, however, we will construct such functions only at . This will make the estimates much simpler than those in [2, 15] but still contain many complicated estimates.

Note that and are fixed in Remark 22 and hence the type and the integer defined in (59) depend only on . Recall that , , , and . From (54) we haveLet us write .

Proposition 23. There exist a smooth plurisubharmonic function on that satisfies the following:
(i) , for , and is supported in .
(ii) There exist a small constant such that if , then (iii) If where is defined in (22), then holds for all where .

Proof. For each fixed , we note that the integers and , defined in (59), will be fixed. Set and . Note that provided is sufficiently small. Since , it follows from (80) thatfor . From now on, we fix and set .
We may assume that the level sets of are pseudoconvex on and on , where we may assume that . Also by (95). Therefore it follows from (69) and (99) thatfor .
Let be defined by where is a smooth function such that for and for , satisfying . Set . Note that has similar expression as in (22). Thus it follows, from (22), (29), (30), and chain rule, thatHere and . Since , one obtainswhere and .
Suppose that satisfies . Using the fact that , , and the fact that , it follows from (100)–(103) thatWe note that the negative part in (104) contains instead of .
Let be a smooth convex function such that for and for and . Set and set . Suppose . Then , and hence (79) holds for with ; that is,For those with , it follows from (104) (with ) and (105) thatIf , then and hence . Hence is a smooth plurisubharmonic function supported on , and .
Now assume and assume that (59) holds. Then (79) holds for some positive integers with . Let be the function defined in (85). From (88) and (95), we see that, because and . Set where is a smooth function that satisfies , for , for , and for all . Thus and because . By (107) we note that on , and we also note that if , in particular, outside . From (92), we obtain thatfor , because .
Assuming that , we note that the negative coefficient part of of the Hessian of in (104) is controlled by the first term in the third line of (109), and the error terms of the coefficients of and in the third line of (109) are controlled by the corresponding coefficients of the Hessian of in (104). In either or cases, it follows from (104), (106), and (109) thatfor .
Note that parameters, , , , , and , are fixed in Remark 22, independent of . Therefore the upper bound of follows from (84)–(88), (96), (99), (102), and (103). Note that , if , and this property holds on if we take sufficiently small; say, . Also note that on . This fact together with (96) and (110) proves properties (i) and (ii). Property (iii) follows from (22), (30), (32), and (96).

For each , set .

Proposition 24. There is a small constant such that for all sufficiently small .

Proof. From (22)–(29), we obtain thatfor all sufficiently small . Assume and write From (61), and by using Taylor series method, one obtains that for a uniform constant . Similarly, we obtain . Combining these estimates and (111) and if we set , then we obtain that

Remark 25. (1) Set . Then, by functoriality, Proposition 23 holds, where is replaced by , and is replaced by .

For each fixed , and for each fixed , set . Note that on . Thus it follows, from transformation formula, thatIn view of Propositions 23 and 24, there is a smooth plurisubharmonic weight function which has maximal Hessian on . We also note that by (56). If we use these properties and (115), we get the following estimates for the Bergman kernel function at as in Theorem 6.1 in [2]:This proves Theorem 2.

Now we want to get derivative estimates of for and . In view of (115), we will estimate where and . We will follow the methods in [3, 9] which use dilated coordinates. For each fixed , we recall that , and . Define a dilation map given byset and set where and where is defined in Proposition 23.

Set and write . The properties of , which follow from Propositions 23 and 24 and Remark 25, are summarized in the following proposition.

Proposition 26. For each there is , defined on , such that
(1) is smooth plurisubharmonic in , and ;
(2) , for some ;
(3) if ;
(4) .

The weight function with the properties in Proposition 26 is the key ingredient for the derivative estimates of the Bergman kernel function off the diagonal. Set and let be the Neumann operator on . Then we have the following estimates of (Proposition 3.14 in [3]).

Proposition 27. Let be a form and . Then there is , independent of , so that

Note that . Set From (117) and Proposition 24, we note thatindependent of . Let with in a neighborhood of and on . From (123), we see that , independent of . Therefore we have the following elliptic estimates:where is the complex Laplacian on .

Remark 28. The estimates in (124) are on the polydisc , strictly inside of , independent of . Therefore we gain two derivatives in (124) and it is stable; that is, is independent of . Also we note that we do not require that . Since where , we can also apply the estimate (121) on .

Let , , and be polyradial. In terms of -coordinates in (117), we have the following well known representation of Bergman kernel function on . Let with outside and on . Combining (121)–(125), we can prove the following lemma as in the proof of Theorem 4.2 in [3].

Lemma 29. For each there is such that

Now, if we use the estimate (126) with , we can prove Theorem 3 as in the proof of Theorem 4.2 in [3].

Appendix

We recall Herbort’s example in (6). Therefore and hence , , and in our notations. For each fixed , set . Then and approaches to in “almost tangential direction” as the points do along . In this case, we will show thatwhich is exactly same result as Theorem 2.

Proof of (A.1). SetThen the polydisc about is contained in . Therefore the upper bound follows. Let us show lower bounds.
Let be the unit polydisc in . Since the localization lemma is valid for , we will estimate . Set and setThen and . Set . Then . Note that where we have used . Set and . Then where we have used the polar coordinates: , in the second line.
Set and . ThenWrite . Set . Then Also, Combining (A.5)–(A.9), we obtain that . Therefore .

Remark 30. Set . Then is a peak function that peaks at for the domain .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author was partially supported by the Sogang University Research Fund.

References

S. Bergman, The kernel function and conformal mapping, American Mathematical Society, Providence, R.I., 2nd edition, 1970.
View at: MathSciNet
D. W. Catlin, “Estimates of invariant metrics on pseudoconvex domains of dimension two,” Mathematische Zeitschrift, vol. 200, no. 3, pp. 429–466, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
J. D. McNeal, “Boundary behavior of the Bergman kernel function in C2,” Duke Mathematical Journal, vol. 58, no. 2, pp. 499–512, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, “Estimates for the Bergman and Szegö kernels in C2,” Annals of Mathematics: Second Series, vol. 129, no. 1, pp. 113–149, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
J. D. McNeal, “Local geometry of decoupled pseudoconvex domains,” in Complex analysis (Wuppertal, 1991), Aspects Math., E17, pp. 223–230, Friedr. Vieweg, Braunschweig, 1991.
View at: Google Scholar | MathSciNet
J. D. McNeal, “Estimates on the Bergman kernels of convex domains,” Advances in Mathematics, vol. 109, no. 1, pp. 108–139, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
K. Diederich, G. Herbort, and T. Ohsawa, “The Bergman kernel on uniformly extendable pseudoconvex domains,” Mathematische Annalen, vol. 273, no. 3, pp. 471–478, 1986.
View at: Publisher Site | Google Scholar | MathSciNet
S. Cho, “Boundary behavior of the Bergman kernel function on some pseudoconvex domains in C,” Transactions of the American Mathematical Society, vol. 345, no. 2, pp. 803–817, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
S. Cho, “Estimates of the Bergman kernel function on certain pseudoconvex domains in Cn,” Mathematische Zeitschrift, vol. 222, no. 2, pp. 329–339, 1996.
View at: Publisher Site | Google Scholar | MathSciNet
P. Charpentier and Y. Dupain, “Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form,” Publicacions Matemàtiques, vol. 50, no. 2, pp. 413–446, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
P. Charpentier, Y. Dupain, and M. Mounkaila, “Estimates weighted Bergman projections on pseudo-convex domains of finite type in Cn,” Complex Variables and Elliptic Equations, vol. 59, no. 8, pp. 1070–1095, 2014.
View at: Google Scholar
P. Charpentier, Y. Dupain, and M. Mounkaila, “On estimates for weighted Bergman projections,” Proceedings of the American Mathematical Society, vol. 143, no. 12, pp. 5337–5352, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
J. P. D'Angelo, “Real hypersurfaces, orders of contact, and applications,” Annals of Mathematics, vol. 115, no. 3, pp. 615–637, 1982.
View at: Publisher Site | Google Scholar | MathSciNet
G. Herbort, “Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains in C3 of finite type,” Manuscripta Mathematica, vol. 45, no. 1, pp. 69–76, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
D. Catlin, “Subelliptic estimates for the ∂-Neumann problem on pseudoconvex domains,” Annals of Mathematics: Second Series, vol. 126, no. 1, pp. 131–191, 1987.
View at: Publisher Site | Google Scholar | MathSciNet
Y. You, “Necessary conditions for Hölder regularity gain of ∂ equation in C3,” Purdue University, West Lafayette, IN, 2011.
View at: Google Scholar

Copyright

Copyright © 2018 Sanghyun Cho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

1027

Downloads

1216

Citations