A property of the β-Cauchy-type integral with continuous density

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Datum:	2008
Hauptverfasser:	Abreu Blaya, R., Bory Reyes, J.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2008
Schriftenreihe:	Український математичний журнал
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spelling	irk-123456789-1647762020-02-11T01:27:19Z A property of the β-Cauchy-type integral with continuous density Abreu Blaya, R. Bory Reyes, J. Статті 2008 Article A property of the β-Cauchy-type integral with continuous density / R. Abreu Blaya, J. Bory Reyes // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1443–1448. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164776 517.5 en Український математичний журнал Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Abreu Blaya, R. Bory Reyes, J. A property of the β-Cauchy-type integral with continuous density Український математичний журнал
format	Article
author	Abreu Blaya, R. Bory Reyes, J.
author_facet	Abreu Blaya, R. Bory Reyes, J.
author_sort	Abreu Blaya, R.
title	A property of the β-Cauchy-type integral with continuous density
title_short	A property of the β-Cauchy-type integral with continuous density
title_full	A property of the β-Cauchy-type integral with continuous density
title_fullStr	A property of the β-Cauchy-type integral with continuous density
title_full_unstemmed	A property of the β-Cauchy-type integral with continuous density
title_sort	property of the β-cauchy-type integral with continuous density
publisher	Інститут математики НАН України
publishDate	2008
topic_facet	Статті
url	http://dspace.nbuv.gov.ua/handle/123456789/164776
citation_txt	A property of the β-Cauchy-type integral with continuous density / R. Abreu Blaya, J. Bory Reyes // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1443–1448. — Бібліогр.: 15 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT abreublayar apropertyofthebcauchytypeintegralwithcontinuousdensity AT boryreyesj apropertyofthebcauchytypeintegralwithcontinuousdensity AT abreublayar propertyofthebcauchytypeintegralwithcontinuousdensity AT boryreyesj propertyofthebcauchytypeintegralwithcontinuousdensity
first_indexed	2025-07-14T17:21:43Z
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fulltext	UDC 517.5 R. Abreu Blaya (Univ. de Holguín, Cuba), J. Bory Reyes (Univ. de Oriente, Santiago de Cuba, Cuba) A PROPERTY OF THE ββββ-CAUCHY TYPE INTEGRAL WITH A CONTINUOUS DENSITY* PRO ODNU VLASTYVIST\| INTEHRALA TYPU ββββ-KOÍI Z NEPERERVNOG WIL\|NISTG The aim of this paper is to extend a theorem from classical complex analysis proved by Davydov in 1949 to the theory of solutions of a special case of the Beltrami equation in the z-complex plane ( i.e., null solutions of the differential operator ∂ − ∂z z z z β , 0 ≤ β < 1 ). We prove that if γ is a rectifiable Jordan closed curve and f is a continuous complex-valued function on γ such that the integral f f t t t n n ds t r ( ) − ( ) − / ( ) − ( ) ∈ − ≤{ } ∫     ζ ζ ζ ζ β ζ ζ ζθ γ ζ γ ζ\ : , θ = 2 1 β β− , converges uniformly on γ as r → 0, where n( )ζ is the exterior unit normal vector on γ at a point ζ and ds is the arc length differential, then the β-Cauchy type integral 1 2 1( − )π ( ) − / ( ) − ( )∫    β ζ ζ ζ ζ β ζ ζ ζθ γ f z z n n ds , z ∉ γ, admits a continuous extension to γ and a version of the Sokhotski – Plemelj formulae holds. Metog ci[] statti [ uzahal\nennq teoremy iz klasyçnoho kompleksnoho analizu, wo bula dovede- na Davydovym u 1949 r., dlq teori] rozv’qzkiv okremoho vypadku rivnqnnq Bel\trami u z-komp- leksnij plowyni (tobto nul\ovyx rozv’qzkiv dyferencial\noho operatora ∂ − ∂z z z z β , 0 ≤ ≤ β < 1 ). Dovedeno, wo koly γ [ sprqmlgvanog zamknenog kryvog Ûordana i f [ neperervnog kom- pleksnoznaçnog funkci[g na γ takog, wo intehral f f t t t n n ds t r ( ) − ( ) − / ( ) − ( ) ∈ − ≤{ } ∫     ζ ζ ζ ζ β ζ ζ ζθ γ ζ γ ζ\ : , θ = 2 1 β β− , rivnomirno zbiha[t\sq na γ pry r → 0, de n( )ζ — zovnißnij odynyçnyj normal\nyj vektor na γ u toçci ζ, a ds — dyferencial dovΩyny duhy, todi intehral typu β-Koßi 1 2 1( − )π ( ) − / ( ) − ( )∫    β ζ ζ ζ ζ β ζ ζ ζθ γ f z z n n ds , z ∉ γ, dozvolq[ neperervne rozßyrennq na γ i odyn iz variantiv formuly Soxoc\koho – Plemeq vyko- nu[t\sq. 1. The ββββ -Cauchy type integral and Davydov’s theorem. 1.1. The classical Beltrami equation can be considered as a remarkable generalization of the Cauchy – Riemann equation in the complex plane. Its solutions have many properties analogous to those of analytic functions of one complex variable and numerous problems of * Partially supported by the CAPES-MES project 028/07. © R. ABREU BLAYA, J. BORY REYES, 2008 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1443 1444 R. ABREU BLAYA, J. BORY REYES analysis and geometry can be reduced to solving this equation. For a survey of recent research and historical details on the Beltrami equation we refer the reader to [1 – 3]. It is our purpose to study a Cauchy type integral associated to the theory of solutions of a special case of Beltrami equation, which are called β-analytic functions, namely, solutions of the following linear first order partial differential equation: ∂ = ∂z zf z z fβ , z = x + i y, where 0 ≤ β < 1 and as usual ∂ = (∂ + ∂ )z x yi: 1 2 , ∂ = (∂ − ∂ )z x yi: 1 2 . Suppose that, in C, a domain Ω with boundary γ is given. We refer to [4] for the theory of β -analytic functions in Ω having proved a new integral representation formula. In particular, the Cauchy integral formula ( )( ) = ( ) ∈ ∈    Cγ β f z f z z z , , , \ , Ω Ω0 C where ( )( ) = ( − )π ( ) − +   ∫Cγ β θ γ β ζ ζ ζ ζ β ζ ζ ζf z i f z z d d: 1 2 1 , z ∉ γ, and θ = 2 1 β β− . In this way Cγ β f plays the role of the Cauchy type integral in the theory of β- analytic functions and we shall call it the β-Cauchy type integral. Note that the complex element of integration dζ may be written as dζ = i n ( ζ ) ds, where n ( ζ ) is the exterior unit normal vector on γ at a point ζ, writing it as a complex number and ds is the arc length differential. We can thus write ( )( )Cγ β f z in the form ( )( ) = ( − )π ( ) − ( ) − ( )   ∫Cγ β θ γ β ζ ζ ζ ζ β ζ ζ ζf z f z z n n ds 1 2 1 , z ∉ γ. (1) Here and subsequently γ denotes a rectifiable positively oriented closed Jordan curve in C. Let Ω+ and Ω – be, respectively, the interior and exterior domains bounded by γ. There is no loss of generality in assuming 0 ∈ Ω+. An important point to note here is that if β = 0, i.e., the case of analytic functions, we recovered the standard Cauchy type integral. In the theory of analytic functions of one complex variable, the Cauchy type integral is proved to be a very deep and crucial object of research. Moreover, its study has led to numerous discoveries both in analytic function theory itself and in many other areas, such as the Hardy space theory, singular integral equations as well as potential and elasticity theories. Properties of this integral on a bounded domain with quite enough smooth boundary are sufficiently well understood, see [5, 6]. There are numerous investigations on the evolution of the limit boundary values of the Cauchy type integral in such smooth bounded domains. A well-known first result is that it possesses continuous limit boundary values on γ if its density belongs to the Lipschitz class. By far much more general is the case of Davydov’s theorem (see [7]) which is related to the problem of establishing a sufficient condition for the Cauchy type integral to be continuously extended to the closure of a domain bounded by a rectifiable Jordan ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 A PROPERTY OF THE β-CAUCHY TYPE INTEGRAL WITH A CONTINUOUS DENSITY 1445 closed curve. This result requires the uniform existence of some integral which restricts both the set of curves and classes of densities, it provides a sufficient condition guaranteeing that the Sokhotski – Plemelj formulae hold. As an example of such restriction, Davydov has considered the Lipschitz class and an arbitrary rectifiable closed Jordan curve. Moreover, the result presented in [8] under the much more stronger condition of being γ a regular curve (i.e., the quotient of the measure of γ inside any circle with center at any point of this curve to the radius of the circle is less than some fixed constant) is a very good piece of work on the subject that attracts considerable attention. Generalizations of the Davydov’s theorem were the subjects of research in a number of papers, see [9 – 13], both in the complex and hypercomplex context. Hence, this can be thought of as a good motivation for the analog of the above for the theory of the β-Cauchy type integral and the question arises about the existence of a reasonable extension of Davydov’s theorem. Our paper deals with this situation. For a deep discussion on the existence of continuous limit values of the integral (1) along a regular curve we refer the reader to [14]. In addition, to illustrate how β-Cauchy type integral works, we refer the reader to [15], where some higher order Cauchy – Pompeiu representation formulas for β- analytic functions are given trying to determine particular solutions to some differential equations. 1.2. For the study of the behavior of the β-Cauchy type integral near the integration curve we also need the singular β-Cauchy type integral given by ( )( ) = ( − )π ( ) − ( ) − ( ) − ( )   → ∈ − ≤{ } ∫Sγ β θ γ ζ γ ζ β ζ ζ ζ ζ β ζ ζ ζf t f f t t t n n ds r t r : lim \ : 0 1 2 1 , t ∈ γ. It is a simple exercise to see that the β-Cauchy type integral is an example of a β- analytic function in C \ γ that vanishes at infinity. Another example of a β-analytic function is the function ζ ( z ) : = z \| z \| θ. Let us remark that the transformation ( )z z, → ( )ζ ζ, is continuously differentiable in C \ { 0 } and its Jacobian is J( ) = + − ζ β β ζ θ1 1 2 . Using the above remark and our assumption on γ ( 0 ∉ γ ), it is easy to see that for z1 , z2 sufficiently close to γ we have the following inequalities: c z z z z z z c− ≤ − − ≤1 1 1 2 1 1 2 θ θ , z1 ≠ z2 . In several cases we will make use of these inequalities. We will use the symbol c for constants depending on γ which may vary from one occurence to the next. The generalization of Davydov’s theorem for the β-analytic function theory to be proved is formulated as follows: Theorem 1. L e t γ be a rectifiable Jordan closed curve and let f be a continuous complex-valued function on γ. If the integral f f t t t n n ds t r ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∫ ζ ζ ζ ζ β ζ ζ ζθ γ ζ γ ζ\ : (2) ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1446 R. ABREU BLAYA, J. BORY REYES converges uniformly on γ as r → 0, then the singular β -Cauchy type integral exists and then the β -Cauchy type integral of f has continuous limit values on γ . Moreover, the Sokhotski – Plemelj formulae hold : lim Ω+ → ( )( ) = ( )( ) + ( ) �z t f z f t f tC Sγ β γ β , t ∈ γ, (3) lim Ω− → ( )( ) = ( )( ) �z t f z f tC Sγ β γ β , t ∈ γ. (4) 2. Proof of the theorem. We first show that if the integral (2) converges uniformly on γ as r → 0, then the singular β-Cauchy type integral exists and is continuous on γ. Denote Wf t r r t f f t t t n n ds( ) = ( ) − ( ) − ( ) − ( ) ∈ − ≤{ } ∫, \ : ζ ζ ζ ζ β ζ ζ ζθ γ ζ γ ζ , Vf t r r t f f t t t n n ds( ) = ( − )π ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∫, \ : 1 2 1 β ζ ζ ζ ζ β ζ ζ ζθ γ ζ γ ζ . By assumption, for every ε > 0 there exists δ ( ε ) > 0 such that for all t in γ and all 0 < r1 < r2 < δ ( ε ), we have W Wf fr t r t( ) − ( )1 2, , = = f f t t t n n ds t r t r ( ) − ( ) − ( ) − ( ) < ( − )π ∈ − ≤{ } ∈ − ≤{ } ∫ ζ ζ ζ ζ β ζ ζ ζ β εθ ζ γ ζ ζ γ ζ: \ :2 1 2 1 . From the above it follows that for all t in γ and all 0 < r1 < r2 < δ ( ε ) we obtain V Vf fr t r t( ) − ( )1 2, , = = 1 2 1 2 1 ( − )π ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∈ − ≤{ } ∫β ζ ζ ζ ζ β ζ ζ ζθ ζ γ ζ ζ γ ζ f f t t t n n ds t r t r: \ : ≤ ≤ 1 2 1 2 1 ( − )π ( ) − ( ) − ( ) − ( ) ∈ − ≤{ } ∈ − ≤{ } ∫β ζ ζ ζ ζ β ζ ζ ζθ ζ γ ζ ζ γ ζ f f t t t n n ds t r t r: \ : < ε. Hence, Vf r t( ), converges uniformly on γ to ( )( )Sγ β f t as r → 0. Therefore, the singular β-Cauchy type integral ( )Sγ β f is continuous on γ. Now, let us prove (3). The relation (4) can be proved similarly. Let t be a fixed point of γ and let z ∈ Ω+. If tz ∈ { ζ ∈ γ : \| z – ζ \| = dist ( z, γ ) } we have that ( )( ) − ( )( ) − ( )C Sγ β γ βf z f t f t ≤ ≤ ( )( ) − ( ) − ( )( ) + ( )( ) − ( )( ) + ( ) − ( )C S S Sγ β γ β γ β γ βf z f t f t f t f t f t f tz z z z . By continuity, the last two summands on the right-hand side of the previous inequality tend to zero as z → t. We now turn to the first summand. Note that for any r > 0 we have ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 A PROPERTY OF THE β-CAUCHY TYPE INTEGRAL WITH A CONTINUOUS DENSITY 1447 2 1( − )π ( )( ) − ( ) − ( )( )β γ β γ βC Sf z f t f tz z = =     − − −     ( ) − ( ) ( ) − ( )   ∫ ( )1 1 ζ ζ ζ ζ ζ ζ β ζ ζ ζθ θ γ z z t t f f t n n ds z z z ≤ ≤ f f t t t n n dsz z zt rz ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∫ ζ ζ ζ ζ β ζ ζ ζθ ζ γ ζ: + + f f t z z n n dsz t rz ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∫ ζ ζ ζ ζ β ζ ζ ζθ ζ γ ζ: + +     − − −     ( ) − ( ) ( ) − ( )    ∈ − ≤{ } ∫ ( )1 1 ζ ζ ζ ζ ζ ζ β ζ ζ ζθ θ γ ζ γ ζ z z t t f f t n n ds z zt r z \ : = = I I I1 2 3+ + . Fix ε > 0. By the uniform convergence of Vf r t( ), , there exists δ1 ( ε ) > 0 such that for all r < δ1 ( ε ) we have I1 2 1 3 < ( − )πβ ε for all z ∈ Ω+. Since \| ζ – tz \| ≤ 2 \| ζ – z \| for all ζ ∈ γ, we get that I f f t z z n n dsz t rz 2 ≤ ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∫ ζ ζ ζ ζ β ζ ζ ζθ ζ γ ζ: ≤ ≤ c f f t t t n n dsz z zt rz ( ) − ( ) − ( ) − ( )    ∈ − ≤{ } ∫ ζ ζ ζ ζ β ζ ζ ζθ ζ γ ζ: . The uniform convergence of the integral (2) now assures the existence of a positive number δ2 ( ε ) such that for all r < δ2 ( ε ) and all z ∈ Ω+ we have I2 2 1 3 < ( − )πβ ε . Now fix any r > 0 strictly less than min { δ1 ( ε ), δ2 ( ε ) }. Our next concern is to estimate I3 . First, note that for ζ ∈ γ \ { ζ ∈ γ : \| ζ – tz \| ≤ r } we have r < \| ζ – tz \| ≤ ≤ 2 \| ζ – z \|. In this way 1 1 2 ζ ζ ζ ζ ζ ζ ζ ζ ζθ θ θ θ θ θ θ θ θ − − − ≤ − − − ≤ − z z t t z z t t z z t t c z t r z z z z z z z for ζ ∈ γ \ { ζ ∈ γ : \| ζ – tz \| ≤ r }. Let us take ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1448 R. ABREU BLAYA, J. BORY REYES δ ε β ε ζ γ ζ γ ( ) = ( − )π ( ) ( ) ∈ 2 1 2r c f smax . Then for all z ∈ Ω+ with \| z – t \| < δ ( ε ), we have I z z t t f f t ds z zt r z z 3 1 1≤ − − − ( ) − ( ) ∈ − ≤{ } ∫ ζ ζ ζ ζ ζθ θ γ ζ γ ζ\ : ≤ ≤ c z t r f sz− ( ) ( ) < ( − )π ∈2 2 1 3 max ζ γ ζ γ β ε , which completes the proof. If β = 0, then an immediate consequence of Theorem 1 is the Davydov theorem [7] mentioned in Section 1. Acknowledgments. The authors wish to thank the IMPA, Rio de Janeiro, were the paper was written, for the invitation and hospitality. We are greatly indebted to Prof. Sergiy Plaksa for his active interest in the publication of this paper. 1. Begehr H. Complex analytic methods for partial differential equations. An introductory text. – River Edge, NJ: World Scientific Publishing Co., Inc., 1994. 2. Bojarski B. Old and new on Beltrami equations // Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations: Proc. of the ICTP (8 – 19 February, 1988, Trieste, Italy). – 1988. – P. 173 – 187. 3. Iwaniec T., Martin G. What’s new for the Beltrami equation? // Geometric analysis and applications: Proc. Centre Math. Appl. Austral. Nat. Univ. (Canberra, 2000). – 2001. – 39. – P. 132 – 148. 4. Tungatarov A. Properties of an integral operator in classes of summable functions // Izv. Akad. Nauk Kazakh. SSR. Ser. Fiz.-Mat. – 1985. – # 5. – P. 58 – 62. 5. Gakhov F. D. Boundary value problems. – Third edition. – Moscow: Nauka, 1977 (in Russian). 6. Muskhelishvili N. I. Singular integral equations. Boundary value problems in the theory of function and some applications of them to mathematical physics. – Third edition. – Moscow: Nauka, 1968 (in Russian). 7. Davydov N. A. The continuity of an integral of Cauchy type in a closed region // Doklady Akad. Nauk SSSR. – 1949. – 64. – P. 759 – 762. 8. Salaev V. V., Tokov A. O. Necessary and sufficient conditions for continuity of the Cauchy integral in a closed domain // Akad. Nauk Azerb. SSR. Dokl. – 1983. – 39, # 12. – P. 7 – 11. 9. Abreu Blaya R., Bory Reyes J., Gerus O., Shapiro M. The Clifford – Cauchy transform with a continuous density: N. Davydov theorem // Math. Methods Appl. Sci. – 2005. – 28, # 7. – P. 811 – 825. 10. Abreu Blaya R., Bory Reyes J., Shapiro M. On the Laplacian vector fields theory in domains with rectifiable boundary // Math. Methods Appl. Sci. – 2006. – 29, # 15. – P. 1861 – 1881. 11. Gerus O., Shapiro M. On a Cauchy-type integral related to the Helmholtz operator in the plane // Bol. Soc. Mat. Mexicana. – 2004. – 3, # 10. – P. 63 – 82. 12. Kats B. A. A generalization of a theorem of N. A. Davydov // Doklady Ros. Akad. Nauk. – 2000. – 374, # 4. – P. 443 – 444; English translation: Dokl. Math. – 62. – 2000. – P. 220 – 221. 13. Kats B. A. On a generalization of a theorem of N. A. Davydov // Izv. Vyssh. Uchebn. Zaved. Ser. Mat. – 2002. – # 1. – P. 39 – 44; English translation: Russian Math. – 2002. – 46, # 1. – P. 37 – 42. 14. Abreu Blaya R., Peña Peña D., Bory Reyes J. On the jump problem for β-analytic functions // Complex Variables Theory and Elliptic Equations. – 2006. – 51, # 8 – 11. – P. 763 – 775. 15. Bory Reyes J., Peña Peña D. Some higher order Cauchy-Pompeiu integral representations // Integral Transforms Spec. Funct. – 2005. – 16, # 8. – P. 615 – 624. Received 29.05.07, after revision — 09.01.08 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11

A property of the β-Cauchy-type integral with continuous density

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