ON THE BOUNDEDNESS OF CAUCHY SINGULAR OPERATOR FROM THE SPACE L p TO L q, p > q 1

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1 Geogia Mathematical Joual 1(94), No. 4, ON THE BOUNDEDNESS OF CAUCHY SINGULAR OPERATOR FROM THE SPACE L TO L q, > q 1 G. KHUSKIVADZE AND V. PAATASHVILI Abstact. It is oved that fo a Cauchy tye sigula oeato, give by equality (1), to be bouded fom the Lebesgue sace L () to L q (), as =, = {z : z = }, it is ecessay ad sufficiet that eithe coditio (4) o (5) be fulfilled. 1. Let be a lae ectifiable Joda cuve, L (), 1, a class of fuctios summable to the -th degee o, ad S a Cauchy sigula oeato S (f)(t) = 1 πi f(τ) dτ τ t, f L (), t. (1) Numeous studies have bee devoted to oblems of the existece of S (f)(t) ad boudedess of the oeato S : f S (f) i the sace L () (see, e.g., [1 3]). The fial solutio of these oblems is give i [4,5]. It was oved by G.David that fo the oeato S to be bouded i L (), it is ecessay ad sufficiet that the coditio l(t, ) C (2) be fulfilled, whee l(t, ) is a legth of the at of cotaied i the cicle with cete at t ad adius, ad C is a costat. 1 The eset ae is devoted to the oblem of boudedess of the oeato S fom L () to L q (), > q 1 (see also [6 9]). 2. Thoughout the est of this ae by { } is meat a stictly deceasig sequece of ositive umbes satisfyig the coditio <, ad by, the family of cocetic cicumfeeces o a comlex lae = {z : z = }, = 1, 2, Mathematics Subject Classificatio. 47B38. 1 Followig [7,8], the ecessity of coditio (2) is show also i [6]. I the same wo its sufficiecy is oved fo secial classes of cuves X/94/ $12.50/0 c 1994 Pleum Publishig Cooatio

2 396 G. KHUSKIVADZE AND V. PAATASHVILI It has bee show i [10,11] that fo the oeato S to be bouded i L (), > 1, it is ecessay ad sufficiet that the coditios C, = 1, 2,..., (3) = be fulfilled, whee C is a absolute costat. We shall ove Theoem. Let > q 1 ad σ = q/( q). The the followig statemets ae equivalet: (A) oeato S is bouded fom L () to L q (); ( = (B) ) σ < ; (4) (C) σ <. (5) Rema. A family of cocetic cicumfeeces the sum of whose legths is fiite, as a set of itegatio, icially, simulates ectifiable cuves with isolated sigulaities. Aalogy of coditios (2) ad (3) also idicates this fact. Taig ito accout the above, we assume that the followig statemet (a aalogue of the theoem fom Subsectio 2) is valid: fo the oeato S to be bouded fom L () to L q (), whee is a abitay ectifiable cuve, > q 1, it is ecessay ad sufficiet that the coditio [χ(t)] q/( q) dt < be fulfilled, whee χ(t) = su l(t, τ), t. 3. I ovig this theoem, use will ofte be made of the well-ow Abel equality (see, e.g., [12],.307) ( ( u )v = u v ), (6) whee {u } ad {v } ae sequeces of ositive umbes ad v <, as well as of its aticula case We shall also eed = = v = v. (7)

3 BOUNDEDNESS OF THE CAUCHY SINGULAR OPERATOR 397 Lemma. Let > 0. If f is a fuctio aalytic i the cicle z < 1, the fo < R < 1, f(z) dz f(z) dz. (8) R z = z =R If f is a fuctio aalytic i the domai z > 1 ad f( ) = 0 the fo 1 < R < ( R ) 1 f(z) dz f(z) dz. (9) z = z =R If, i additio, f belogs to the Hady class H i the domais z < 1 o 2π z > 1, i.e., su ρ f(ρe iϑ ) dϑ < (i aticula, if f is eeseted 0 by a Cauchy tye itegal), the we ca tae R = 1 i iequalities (8) ad (9). Poof. Sice dz = dρe iϑ = ρdϑ, iequality (8) follows fom the fact that the mea value 1 2π 2π f(ρe iϑ ) dϑ of f(ρe iϑ ) is a odeceasig fuctio 0 of ρ (see, e.g., [13],.9). Ude the coditios of the lemma, if z > 1, the the fuctio g(ζ) = 1 ζ f( 1 ζ ) is aalytic i the cicle ζ < 1. Usig iequality (8) fo g, we get ( 1 ) ( f R ) +1 ( dζ 1 ) f dζ. ζ ζ ζ = 1 ζ = 1 R Alyig the tasfomatio of ζ = 1 z, the latte iequality educes to (9). If f H, the by the Riesz theoem 2π 2π lim f(ρe iϑ ) dϑ = f(e iϑ ) dϑ ρ 1 0 (see, e.g., [13],.21), which eables us to suose that R = Let us ove the equivalece of coditios (B) ad (C). This follows fom equality (7) fo σ = 1 ad theefoe we shall assume that σ > 1. (C) follows fom (B). We use Abel Dii s theoem (see, e.g., [12],. 292): if a seies with ositive tems a diveges ad S meas its -th atial sum, the the seies a S (ε > 0) coveges. Assume that the seies 0 also diveges, while the seies a S 1+ε σ diveges. The, settig a = σ ad ω = 1 / σ, we shall see by this theoem that the seies ω σ diveges while the seies ωσ σ coveges, whee σ = σ σ 1 > 1.

4 398 G. KHUSKIVADZE AND V. PAATASHVILI Usig equality (6) ad the Hölde iequality, we obtai ( ω σ 2 ω σ 1) ( 2 ω σ 1) = = 2 ω σ 1 ( = ) [ ( = 2 ) ] σ 1/σ ( ) 1/σ ω σ σ <. The obtaied cotadictio shows that (C) follows fom (B). Let us ow show that (B) follows fom (C). If m, the =m A m = = m + m m m = + ( m) + A. (10) m Let 1 s σ. Usig equality (6) ad iequality (10), we get s 1 A σ s+1 = s 1 A σ s = = ( ) = s 1 A σ s ( s 1[ A + ( ) ] ) σ s 2 σ 2 σ 2 σ ( A σ s A σ s s 1[ A σ s ( s 1) + 2 σ s + 2 σ + ( ) σ s]) ( s 1 ( ) σ s) σ. (11) Let [σ] be the itege at of σ ad α = σ [σ]. Usig iequality (11) successively [σ] times fo s = 1, 2,..., [σ], we aive at the iequality A σ C 1 whee the costats C 1 ad C 2 deed o σ oly. A α [σ] + C 2, (12)

5 BOUNDEDNESS OF THE CAUCHY SINGULAR OPERATOR 399 If σ is a itege, the α = 0, ad cosequetly the oof is comleted. Let α > 0. The maig use of the Hölde iequality ad equality (7), we obtai A α [σ] = A α α(σ 1) σ(1 α) ( ) α ( ) 1 α A σ 1 σ = ( ) 1 α ( = σ σ 1 = = ) α ( ) 1 α ( ) α σ σ 1 = ( ) 1 α ( ) α = σ ( σ 1 ) = σ <, which comletes the oof. 5. Let us show that (A) follows fom (B) o (C). Coside fist the case whe q = 1 ad show that if > 1 ad σ = = /( 1), the S is bouded fom S () to L 1 (). Let φ i ad φ l be the fuctios detemied esectively i It ad Ext by the Cauchy tye itegal 1 2πi ϕ (t) dt t z, ϕ L ( ), 1, z. (13) Usig the Sohotsy Plemelj fomula φ i (t) φ e (t) = ϕ (t), φ i (t) + φ e (t) = S (ϕ )(t) ad the Cauchy fomula 1 2πi ϕ (t) dt t z = 1 2πi φ i (t) φ e (t) t z dt = { φ(z), i z It, φ e (z), z Ext, we obtai by diect calculatios fo t. 1 S (ϕ)(t) = 2 φ(t) i + [φ i (t) + φ e (t)] + 2 =+1 φ e (t) (14)

6 400 G. KHUSKIVADZE AND V. PAATASHVILI Let us evaluate the itegals of the sums 1 S 1 (t) = 2 φ i (t) + φ(t), i S 2 (t) = φ e (t) + 2 =+1 φ e (t). Usig the lemma fom Subsectio 3 ad the Hölde iequality, we ca wite S 1 (t) ds 2 φ i (t) ds 2 φ i (t) ds ( ) 2(2π) 1/ 1/, φ(t) i ds 1/ whee φ i is a limitig fuctio of the Cauchy tye itegal (13) o, = 1, 2,...,. Next, chagig the ode of summatio ad usig the Riesz s iequality fo the Cauchy sigula oeato i the case of the cicle as well as the Hölde iequality, we get S 1 (t) ds = S 1 (t) ds 2(2π) 1/ 1/ ( = 2(2π) 1/ = ( 2(2π) 1/ C 1/ = ( 1/ [ ( 2(2π) 1/ = C ) ) 1/ φ i (t) ds = ) 1/ φ(t) i ds ) 1/ ϕ(t) i ds ] 1/ ( ϕ(t) ds) 1/, (15) whee C is the costat fom the Riesz iequality (which deeds o oly). The itegal of S 2 (t) ca be evaluated aalogously. Usig iequality (9), as well as the Hölde ad Riesz iequalities, we obtai S 2 (t) dt 2 φ(t) ds 2(2π) e 1/ ( ) 1/ φ e (t) 1/ ds = =

7 BOUNDEDNESS OF THE CAUCHY SINGULAR OPERATOR 401 2(2π) 1/ = 2(2π) 1/ C ( ) 1 ( = 1/ ( ) 1/ φ e (t) 1/ ds ϕ (t) ds ) 1/. Next, chagig the ode of summatio ad usig the Hölde iequality, we ca wite S 2 (t) dt = S 2 (t) ds 2(2π) 1/ C = ( ) 1/ ( ϕ (t) ds = 2(2π) 1/ C 1/ 1/ ) 1/ ϕ (t) ds ( ) 1/ [ ( ) 1 2(2π) 1/ C ϕ (t) ds ] 1 = 2(2π) 1/ C ( )1/( = ϕ(t) ds) 1/. (16) It follows fom (14),(15) ad (16) that if coditios (B) ad (C) ae fulfilled fo σ =, the the oeato S is bouded fom L () to L 1 (). Let us ow coside the geeal case. Let coditios (B) ad (C) be fulfilled fo > q 1 ad σ = q/( q). The, by vitue of the above agumets, S is cotiuous fom L σ (), σ = σ σ 1, to L 1(). But the S is also cotiuous fom L () to L σ () (L () is a class of fuctios essetially bouded o ). This statemet ca be oved by the well-ow method usig the Riesz equality ϕs ψ dt = ψs ϕ dt, ϕ L σ (), ψ L (), whose validity i ou case ca be immediately veified. Futhe, sice S is bouded fom L σ () ad L () to L 1 () ad L σ (), esectively, accodig to Riesz Toi s theoem o iteolatio of liea oeatos (see, e.g., [14],.144), it follows that S is bouded fom L α (), σ α, to L ασ/(α+σ) (). Lettig α =, we get that S is bouded fom L () to L q (). 6. Let us ow show that (C) ad cosequetly (B) follow fom (A). Let fo a ai ad q, > q 1, σ = q/( q), the seies σ

8 402 G. KHUSKIVADZE AND V. PAATASHVILI divege. The, accodig to the above-metioed Abel Dii s theoem, if ω = ( σ ) 1/q, the ω σ = σ S /q ω q σ = <, S = σ S =. σ, Coside, o, the fuctio ϕ(t) = ω σ/ fo t, = 1, 2,.... The ϕ(t) dt = ϕ(t) ds = 2π ω σ <. (17) Next, by equality (14) we have 1 S (ϕ)(t) = 2 ω σ/ + ω σ/ > ω σ/ fo t. Cosequetly, ( S (ϕ)(t) q dt = S (ϕ)(t) q dt > 2π ω σ/) q > > 2π ω q ( σ/) q 2π ω q ( σ +1)q = = 2π ω q σ =. (18) It follows fom (17) ad (18) that if coditio (C) is ot fulfilled fo > q 1, the thee exists a fuctio ϕ L () fo which S (ϕ) L q (). Cosequetly, fo coditio (A) to be fulfilled, it is ecessay that coditio (C) (ad hece (B)) be fulfilled. Refeeces 1. B.V. Khvedelidze, Method of Cauchy tye itegals i discotiuous bouday value oblems of the theoy of holomohic fuctios of oe comlex vaiable. (Russia) Cuet oblems of mathematics, v. 7 (Russia), Itogi aui i tehii. Aad. Nau SSSR, Vsesoyuz. Ist. Nauch. i Teh. Ifom. Moscow, E.M. Dyi, Methods of the theoy of sigula itegals (Hilbet tasfom ad Caldeo Zygmud theoy). (Russia) Cuet oblems of

9 BOUNDEDNESS OF THE CAUCHY SINGULAR OPERATOR 403 mathematics. Fudametal diectios, v.15 (Russia) , Itogi aui i tehii. Aad. Nau SSSR, Vsesoyuz. Ist. Nauch. i Teh. Ifom. Moscow, , Methods of the theoy of sigula itegals, II. Littlewood Paley theoy ad its alicatios. (Russia) Cuet oblems of mathematics. Fudametal diectios, v.42 (Russia), , Itogi aui i tehii. Aad. Nau SSSR, Vsesoyuz. Ist. Nauch. i Teh. Ifom. Moscow, A.P. Caldeo, Cauchy itegals o Lischitz cuves ad elated oeatos. Poc. Nat. Acad. Sci. USA, 1977, 4, G. David, L itegale de Cauchy su le coubes ectifiables. Peublicatios Uiv. Pais Sud; Det. Math. 82 T 05(1982). 6. V.A. Paatashvili ad G.A. Khusivadze, O boudedess of Cauchy oeato i Lebesgue saces i the case of o-smooth cotous. Tudy Tbiliss. Mat. Ist. Razmadze, 69(1982), V.P. Havi, Bouday oeties of Cauchy tye itegals ad hamoically cojugate fuctios i domais with ectifiable bouday. (Russia) Mat. Sboi 68(110)(1965), No. 4, G.A. Khusivadze, O sigula Cauchy itegal ad o Cauchy tye itegal. (Russia) Tudy Tbiliss. Mat. Ist. Razmadze, 53(1976), T.S. Salimov, Sigula Cauchy itegal i saces L, 1. (Russia) Dol. Aad. Nau Azebaija SSR, XI(1985), No. 3, A.V. Aize stat, O sigula itegal equatios o a coutable set of closed cuves imbedded i oe aothe. (Russia) Dol. Aad. Nau SSSR, 232(1977), No. 5, V.A. Paatashvili ad G.A. Khusivadze, O sigula Cauchy oeato o a coutable set of cocetic cicles. (Russia) Tudy Tbiliss. Mat. Ist. Razmadze, 69(1982), G.M. Fichteholz, A couse of diffeetial ad itegal calculus, v.ii. (Russia) Fizmatgiz, Moscow, P.L. Due, Theoy of H saces. Academic Pess, New Yo ad Lodo, A. Zygmud, Tigoometic seies, v.2. (Taslated ito Russia) Mi, Moscow, 1965; Eglish oigial: Cambidge, (Received ) Authos addess: A.Razmadze Mathematical Istitute Geogia Academy of Scieces 1, Z.Ruhadze St., Tbilisi Reublic of Geogia