EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS

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1 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS YÜKSEL SOYKAN Bülent Ecevit University, Department of Mathematics, Art and Science Faculty, 6700, onguldak, Turkey Abstract. In this talk, we exhibit canonical positive de nite integral kernels associated with simply connected domains. We give lower bounds for the eigenvalues of the sums of such kernels Mathematics Subject Classi cation.45c05, 45H05 (primary), 42B30, 4Q05, 30D55 (secondary). Keywords: Eigenvalue Problems, Special Kernels, Integral Operators, Hardy Spaces, Smirnov Classes, Closed Analytic Curves, Ahlfors Regular Curves.. Introduction Let be a simply connected domain (with an in nite complement) in the extended complex plane C = C [ (): Given a Riemann mapping function (RMF) ' for, that is a conformal map of onto the unit disc ; de ne a function K on by (.) K (; z) = '0 () 2 ' 0 (z) 2 ; (; z 2 ) ; '()'(z) for either of the branches of ' 0 2 : This function is independent of the choice of mapping function ', see [9, p.40]. Little s work is on integral operators de ned by restricting the function K to a square. Speci cally, suppose J is a (bounded) closed real interval then we obtain a compact symmetric operator T on L 2 (J) de ned by T f(s) = K (s; t)f(t)dt (f 2 L 2 (J); s 2 J): J This operator is always positive in the sense of operator theory, i.e. ht f; fi 0; for all f 2 L 2 (J), see [9]. Simple interesting examples can be constructed from standard conformal maps; for instance, if = (and J ( ; )) we obtain K (s; t) = =( st); and if is the strip jim zj < =4 and '(z) = tanh z we obtain K (s; t) = sech(s t): For more examples, see [8].

2 2 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS The main object of study in Little s work is the asymptotic behaviour of the eigenvalues n (T ) of T and related operators. These are always positive and we assume they are arranged in decreasing order with repetitions to account for multiplicities. This work is most complete in the case when is symmetric about the real line R, as in the two examples above, for in [8, Proposition 4, p.28] it is shown how to construct a q, 0 < q < such that n (T ) ' q n in the sense that n (T ) = O(q n ) and q n = O( n (T )): If we think of operators of the form T as standard examples then an obvious problem is to consider natural combinations of such operators (they will always be positive). In [9] Little considered distinct simply connected domains ; 2 ; :::; N each of which is bounded by a line or circle. He proves, for instance, that if J \ N i= i is a closed interval then n (T + ) ' n (T ) where T + is just the sum T + T 2 + ::: + T N with kernel K (s; t) + K 2 (s; t) + ::: + K N (s; t); and T is the integral operator on L 2 (J) with whose kernel is the pointwise product K (s; t)k 2 (s; t):::k N (s; t): we will continue in the spirit of [9]. Our main result is Theorem.. Let C = (C ; C 2 ; C 3 ; :::; C N ) be an N-tuple of closed distinct curves on the sphere C and suppose that for each i; i N; C i is a circle, a line [fg, an ellipse, a parabola [fg or a branch of a hyperbola [fg. Let D be a complementary domain of [ N i= C i and suppose that D is simply connected and contains the closed interval J. If D i is the complementary domain of C i which contains D and J then there exists some constant m > 0 such that in the operator sense. In particular, for all n 0, mt D T D + T D2 + ::: + T DN m n (T D ) n (T D + T D2 + ::: + T DN ): We shall prove Theorem. in sections 4 and 6 using Theorems 4. and 4.2 of section 4. Recall that a complementary domain of a closed F C is a maximal connected subset of C F; which must be a domain. Remark.2. From now on, we shall x the notation as in Theorem.. 2. An Outline of Proof of Theorem. An outline of our method of proof Theorem. is as follows. Suppose C is a simply connected domain. Then there is a canonical Hilbert Space E 2 () of analytic functions on. The precise de nition of these spaces will be recalled in Section 3; so these spaces will be taken for granted for the moment. If contains our closed interval J then there is a natural restriction map S : E 2 ()! L 2 (J)

3 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 3 S f(s) = f(s) (f 2 E 2 (); s 2 J): It was shown in section 3 of [9, p ] that S is compact and that S S = T. That is, S S is the positive integral operator on L 2 (J) with kernel K. Similarly if ; 2 ; :::; N C, each containing J, are simply connected domains and S + : L E 2 ( i )! L 2 (J) is given by S + (f ; f 2 ; : : : ; f N )(s) = S f (s) + S 2 f 2 (s) + : : : + S N f N (s) = f (s) + f 2 (s) + : : : + f N (s) (f i 2 E 2 ( i ); i N; s 2 J) then (it was shown in [9, p.43] that) S + S+ = T + T 2 + : : : + T N is the compact positive integral operator on L 2 (J) with kernel K D + K D2 + ::: + K DN where T i is the integral operator on L 2 (J) with kernel K i. To prove Theorem. it will su cient to nd a continuous linear operator A : E 2 (D)! N i= E2 (D i ) (see section 4, Theorem 4. and Theorem 4.2). A E 2 N i= E2 (D i ) @R S +? L 2 (J) Figure. Operator A We shall be able to put m = kak 2. Suppose now that D is as in Theorem. and the operator A as in Figure exists and is continuous. Then, for all f 2 L 2 (J), ht D f; fi = hs D S Df; fi = ks Dfk 2 = A S +f 2 ka k 2 S +f 2 = kak 2 hs + S +f; fi = kak 2 h(t D + T D2 + : : : + T DN )f; fi: Hence (2.) T D kak 2 (T D + T D2 + : : : + T DN ):

4 4 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS Hence, by (2.) in the operator sense. In particular, for all n 0; So this proves Theorem.. kak 2 T D T D + T D2 + ::: + T DN kak 2 n(t D ) n (T D + T D2 + ::: + T DN ): continuity, the operator A is quite easy to understand. Although it takes a lot of e ort to verify technical details, especially The operator A comes from Cauchy s Integral Formula. The of D can be decomposed naturally as i C i. Cauchy s Integral Formula f(z) = NX 2 [ ::: j e f() z d (f 2 E2 (D); z 2 D) can be validated (see Theorem 3.2). The function e f is the non-tangential limit function of f. In Theorem. it is insisted that the curves (C ; C 2 ; C 3 ; :::; C N ) are distinct. But once Theorem. has been proved this restriction can be removed. Suppose (C ; C 2 ; C 3 ; :::; C R ) are curves as in Theorem.. Suppose R > N, that (C ; C 2 ; C 3 ; :::; C N ) are distinct and that each C i with i > N already appears in the list (C ; C 2 ; C 3 ; :::; C N ). Clearly N[ R[ C i = C i ; i= so that both sets of curves have the same complementary domains. Suppose D is one of them that D is simply connected and contains J. Let k be a positive integer such that each C i appears at most k times in the list (C ; C 2 ; C 3 ; :::; C R ). Then i= T D + T D2 + ::: + T DN T D + T D2 + ::: + T DR so m(t D ) T D + T D2 + ::: + T DR : 3. Preliminaries 3.. Hardy and Smirnov Classes. We will need the theory of Hardy and Smirnov classes of analytic functions. The reader is referred to [5], [6], [7],[], [3], and references therein for a basic account of the subject. Here is a synopsis. If is a recti able arc (see subsection 3.2 for de nition) in C and f is integrable with respect to arc-length measure the notations f(z)dz; f(z) jdzj

5 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 5 will be used to denote, respectively, the complex line integral of f along and the integral of f with respect to arc-length measure. In the rst case we assume has an orientation. The notation L p () will denote the L p space of normalized arc length measure on with the factor 2. T will be used to denote the boundary of the open unit disc. The space H () is just the set of all bounded analytic functions on with the uniform norm. For p <, H p () is the set of all functions f analytic on such that (3.) sup 0<r< f(re i ) p d < : For p, H p (T) is de ned as the set of all f e 2 L p (T) such that (3.2) ef(z)z m dz = 0; m = 0; ; 2; :::. T We have additional de nitions of H 2 () and H 2 (T). H 2 () is the set of all analytic functions f on of the form X (3.3) f(z) = a n z n with a = (a n ) 2 `2 and H 2 (T) is the set of all f e 2 L 2 (T) of the form X (3.4) f(e e i ) = a n e ni with a = (a n ) 2 `2: n=0 n=0 For a more general simply-connected domain in the extended plane C = C [() and a conformal mapping ' from onto, a function g analytic in is said to belong to the Smirnov class E 2 () if and only if g = (f ')' 0=2 for some f 2 H 2 () where ' 0=2 is an analytic branch of the square root of ' 0 : The following Theorem is proved in Duren [5]. Theorem 3.. Let be the inner domain of a recti able Jordan curve and suppose :! is a conformal equivalence. Then a: 0 2 H (); b: Every f 2 E 2 () has a nontangential limit function e f 2 L 2 (@); c: The map f! e f (E 2 ()! L 2 (@)) is an isometric isomorphism onto a closed subspace E 2 (@) of L 2 (@), so d: For all f 2 E 2 () and z 2 jjfjj 2 E 2 () = jj fjj e 2 L 2 (@) = jf(z)j 2 e 2 f(z) ef() z d: (f 2 E 2 ());

6 6 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS Proof. See [5, p.69-70]. We need the following Theorem (see [2]) to show the existence of the operator A. When D and the D i s are as in Theorem., the boundaries of D and the D i s are reasonable and it will make sense for us to discuss to notion of a nontangential limit function and spaces E 2 (@D); E 2 (@D i ) analogous to H 2 (); H 2 (T). Theorem 3.2. Let D be as in Theorem.. Then i: Every f 2 E 2 (D) has a non-tangential limit function e f 2 L 2 (@D); ii: (Parseval s identity) The map f! e f is an isometric isomorphism of E 2 (D) onto a closed subspace E 2 (@D) of L 2 (@D), so jjfjj 2 E 2 (D) = jj e fjj 2 L 2 (@D) = j e f(z)j 2 jdzj iii: (Cauchy s Integral Formula) For all f 2 E 2 (D) and z 2 D f(z) = ef() z (f 2 E 2 (D)); 3.2. Arcs and Curves. An arc or closed curve is called rectiable if and only if it is a countable union of rectiable arcs in C, together with () in the case when 2. For instance, a parabola without is rectiable arc, and a parabola with is rectiable Jordan curve. Recall that a compact arc is called smooth if there exists some parametrization g : [a; b]! such that g 2 C [a; b] and g 0 (t) 6= 0, 8t 2 [a; b]. Remark 3.3. Note that if is smooth then it is recti able, i.e., l() = b a jg 0 (t)j dt < : To de ne the arc length parametrization of put s = s(t) = R t a jg0 (u)j du for a t b so that 0 s `(). Then s 0 (t) = jg 0 (t)j and t! s(t) ([a; b]! [0; `]) is C with strictly positive derivative. Hence also its inverse s! t(s) ([0; `]! [a; b]) is C with strictly positive derivative. The arc length parametrization of the smooth arc is the map h : [0; `]! satisfying h(s) = fthe point on length s from the initial point (g(a)) g, i.e., h(s) = g(t(s)) 0 s `. Since h 0 (s) = g 0 (t(s))t 0 (s), h 2 C [0; `] with non-zero derivative. Necessarily jh 0 (s)j = since We need the following Lemma. (3.5) (3.6) h 0 (s(t)) = g 0 (t)t 0 (s) = g0 (t) s 0 (t) = g0 (t) jg 0 (t)j : Lemma 3.4. Let g 2 C [0; ) with g 0 (t) 6= 0 (t 0). Suppose that c 2 C and lim g(t) = c t! lim t! exist. De ne = g([0; )) [ (c). Then g 0 (t) jg 0 (t)j =! (j!j = )

7 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 7 i: is a compact arc, ii: is recti able, iii: is smooth. Proof. The proof is straightforward and will be omitted. Definition. A simple (hence all) a 2 C Equivalently, if the function f(z) = z rectiable arc C is said to satisfy hypothesis-r if and only if, for some jdzj jz aj 2 < : a belongs to L2 (): Lemma 3.5. Let be a simple, rectiable arc in C. Suppose that a 2 C and that (z) = z a : Then satis es hypothesis-r if and only if () is recti able. Proof. Note that So the result is clear. l(()) = j 0 (z)j jdzj = jdzj jz aj 2 < : Example. An analytic Jordan curve in C obviously satis es hypothesis-r: It is also easy to see that if satis es hypothesis-r so does (); where is a linear equivalence: (z) = z +. Obviously R satis es hypothesis-r and therefore so does any line; and it can be easily seen that every parabola and hyperbola component satis es hypothesis-r: The main result in this section is 4. The Operator A and Cauchy Integrals and easy cases Theorem 4.. There exists a continuous linear operator A : E 2 (D)! N i=e 2 (D i ); say f! (f ; f 2 ; :::f N ); such that f(z) f (z) + f 2 (z) + ::: + f N (z) on D, for all f 2 E 2 (D): Once this is proved our work will be completed, for we will then have as proved in Section 3. T D kak 2 (T D + T D2 + ::: + T DN );

8 8 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS j \C j ( j N) then, by Cauchy s integral formula (Theorem 3.2 iii): So the natural choice for f j is f(z) = NX j= f j (z) j e f() z d (f 2 E2 (D); z 2 D): e f() z d (z 2 D j; j N): Let F j be the extension of fj to C j : 0 F j (z) ef(z); if z j : 0; if z 2 C j Of course, j is nite or empty F j = 0. In general jf j (z)j 2 jdzj = f(z) 2 C j 2 e 2 jdzj f(z) e 2 jdzj = kfk 2 by Parseval s identity (see Theorem 3.2 ii). So F j 2 L 2 (C j ) and the map f! F j is continuous. Hence Theorem 4. will be proved if the following continuity property of the Cauchy integral can be proved. Theorem 4.2. For each j ( j N) the formula (4.) P j g(z) = g() z d (g 2 L2 (C j ); z 2 D j ) de nes a continuous linear operator P j mapping L 2 (C j ) into E 2 (D j ). C j The operator P j is a special case of a more general Cauchy integral operator or Cauchy transform. Definition 2. Let C be a simple oriented the function C g on C is de ned by C g(z) = recti able arc satisfying hypothesis R. For g 2 L 2 () g() z d (g 2 L2 (); z =2 ): Corollary 4.3. If g 2 L 2 () then C g is analytic on C C g is analytic at and C g() = 0:. If also is bounded and recti able then Proof. If z 2 C then, by Schwarz s inequality 2 jg()j j zj jdj kgk 2! =2 jdj j zj 2 < +; by hypothesis R, so C g is de ned on C : To prove that C g is analytic, let z 2 C and choose an r > 0 such that the open disc N 2r (z) centre z, radius 2r is contained in C such that 0 < jh n j < r; for all n and h n! 0. For each n; de ne I n = h n (C g(z + h n ) C g(z)):. Let (h n ) be a sequence in C

9 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 9 Then (4.2) I n = = = g() ( h n z h n z )d g() ( z h n )( z) d g()!() jdj ( z h n )( z) where!() is the unit tangent vector of at (j!()j = a.e.). Now, for all n and almost all 2 ; g()!() ( z h n )( z) jg()j r j zj : Since the last function (of ) is in L () it follows from Lebesgue s theorem, applied to (4.2), that (C g) 0 (z) = lim I n = g() n! ( z) 2 d: If is bounded then 2 C large values of z, ; and if is recti able then L 2 () L (): So if g 2 L 2 () then, for jc g(z)j jg()j 2 jzj jj jdj which tends to 0 as z! ; by Lebesgue s theorem. Therefore C g is analytic and 0 at. Remark 4.4. The orientation here is crucial and we must always be clear what it is before starting any manipulations. Obviously C g = C g. Recall that C j in Theorem 4.2 is oriented so as to wind positively round D and D j. The function P j g is then the restriction of C g to D j ; where = C j : Lemma 4.5. (Invariance lemma) Let ; 2 be simple oriented recti able arcs satisfying hypothesis R, and suppose that ; 2 are simply connected complementary domains of ; 2, respectively. Suppose that is a fractional linear transformation such that ( ) = 2 and ( ) = 2 (with the same orientation). Then the map f! C fj de nes a continuous linear operator mapping L 2 ( ) into E 2 ( ) if and only if the map g! C 2 gj 2 de nes a continuous linear operator mapping L 2 ( 2 ) into E 2 ( 2 ); in which case the two operators are unitarily equivalent. Proof. Let us call the rst map C and the second C 2 2 ; thus C f is the restriction to of the Cauchy integral C f and likewise for C 2 2. By symmetry, we need only prove one of the implications. So, suppose C 2 2 is a continuous linear operator mapping L 2 ( 2 ) into E 2 ( 2 ): Let V = V : L 2 ( )! L 2 ( 2 ) be the unitary operator V f() = f( ()) 0 () =2 (f 2 L 2 ( ); 2 2 ) and let U = U : E 2 ( 2 )! E 2 ( ) be the unitary operator Ug(z) = g((z)) 0 (z) =2 (g 2 E 2 ( 2 ); z 2 ):

10 0 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS Then UC 2 2 V : L 2 ( )! E 2 ( ) is a unitarily equivalent copy of C. To elucidate this operator, let f 2 L 2 ( ) and z 2 G 2 ; then C 2 2 V f(z) = = f( 2 ()) 0 () =2 d z f(u) 0 (u) =2 du; (u) z using the substitution = (u): So if w 2 UC 2 2 V f(w) = f(u) 0 (u) =2 0 (w) =2 du: (u) (w) Now because is fractional linear it satis es the functional equation (4.3) 0 (u) =2 0 (w) =2 (u) (w) = u w as the reader will be able to verify. Hence C = UC 2 2 V : L 2 ( )! E 2 ( ) is continuous and unitarily equivalent to C 2 2. Lemma 4.6. (see [9]) Theorem 4.2 is true if C j is a circle or a line. Proof. We can map D j onto and C j onto the positively oriented unit circle using a fractional linear function. Thus in this case P j is unitarily equivalent to the operator C : L 2 (T)! H 2 () : Cf(z) = f() z d; which is continuous, because if with a = (a n ) 2 `2() then Cf(z) = f(e i ) = T X n= X a n z n n=0 a n e ni (z 2 ); it is unitarily equivalent to the orthogonal projection of L 2 (T) onto H 2 (T). 5. David s Theorem The remaining cases of Theorem 4.2 will be proved using David s theorem [0] (and theorem 5.3 below). This has to do with the L 2 space of a recti able Jordan curve. Suppose that is a positively oriented recti able Jordan curve with inner domain and outer domain G. We have two spaces of analytic functions E 2 () on and E 2 (G) on G. Associated with these two we have closed subspaces E 2 (@); E 2 (@G) of L 2 () ( consisting of all nontangential limit functions of elements of E 2 (); E 2 (G); respectively. An obvious question arises is L 2 () = E 2 (@) E 2 (@G); not necessarily orthogonally?. David s theorem gives a su cient condition for the answer to be yes.

11 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS The reason the question is obvious is that, when = T; E 2 () = H 2 () and so E 2 (@) is all functions f on T of the form X f(e i ) = a n e ni n=0 with (a n ) 2 `2: And, because z! z is an RMF for G = C ; E 2 (@G) is all functions f on T of the form X f(e i ) = b n e (n+)i n=0 with (b n ) 2 `2; so David s theorem is true when = T. Suppose now that our recti able Jordan curve does satisfy L 2 () = E 2 (@) E 2 (@G): It follows immediately from Banach s theorem that the (not necessarily orthogonal) projection (f ; f 2 )! f (L 2 ()! E 2 (@)) is continuous. It will be easy to show (Lemma 5.4 ii below) that the Cauchy integral satis es C (f + f 2 )(z) = C f (z) (f 2 E 2 (@); f 2 2 E 2 (@G); z 2 ); so that, by b and c of Theorem 3., C (L 2 ()) E 2 (): And if write f = f + f 2 as above we see that, for all f 2 L 2 (); kc fk E 2 () = kc f k E 2 () = kf k L2 () M kfk L2 () for some M > 0, so that the Cauchy integral C de nes a continuous linear operator mapping L 2 () into E 2 (): There is a complete symmetry here and so C G de nes a continuous linear operator mapping L 2 () into E 2 (G) as well. Now the hypothesis of David s theorem involves the curve only. So in order to prove Theorem 4.2 in the case where C j is an ellipse it is su cient to show that an ellipse satis es the hypothesis of David s theorem, and because of the invariance Lemma 4.5, in order to verify Theorem 4.2 when C j is a parabola or hyperbola component it is su cient to show that some fractional linear image (C j ) of C j satis es the hypothesis of David s theorem. The key hypothesis in David s theorem is that the Jordan curve is Ahlfors regular. Definition 3. An arc or closed curve is called Ahlfors regular if there is a constant k > 0 such that where l is arc length measure. l(n r (z 0 ) \ ) kr; for all r > 0 and z 0 2 ; Corollary 5.. If and k are as above then l(n r (z) \ ) 2kr; for all r > 0 and z 2 C: Proof. We only need to consider the case where N r (z) \ 6= ;: So choose z 0 N r (z) N 2r (z 0 ): Hence the result. 2 N r (z) \ : Then

12 2 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS Corollary 5.2. If C is an Ahlfors regular Jordan curve and is a fractional linear function which is nite on then () is Ahlfors regular. Proof. Choose > 0 such that, for all z 0 2 and r 0 > 0; The pole a of is not on () so l(n r0 (z 0 ) \ ) r 0 : = inf j aj > 0: 2() Let K be the closure of the 2 neighbourhood of () then M = sup 0 (u) < +: u2k If z 2 () and jw zj < 2 then the line segment [z; w] K: Integrating 0 along this arc shows that (w) (z) M jw zj (z 2 (); jw zj < 2 ): Hence also, if r < 2 (N r (z)) N Mr ( (z)): Now N r (z) \ () (N Mr ( (z)) \ ); so l(n r (z) \ ()) j 0 ()j jdj N Mr ( (z))\ k 0 k l(n Mr ( (z)) \ ) (k 0 k M) r (r < 2 ); where k 0 k is the uniform norm of 0 on. Hence for all z 2 () and r > 0 l(n r (z) \ ()) max k 0 k M; 2l(()) r: Theorem 5.3. David s theorem (([0], theorem 8, chapter 2) or [4]) Let be an Ahlfors regular Jordan curve with inner domain and outer domain G. Let R be the set of all rational functions with no poles in and let R G be the set of all rational functions which vanish at and have no poles in G: Then L 2 () = R R G where R ; R G are the closures of R ; R G respectively in L 2 (). curve. To make use of David s theorem one more lemma is needed. It applies to a general recti able Jordan

13 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 3 Lemma 5.4. Let be a recti able Jordan curve with inner domain and outer domain G;and let R ; R G be as in Theorem 5.3. i: R E 2 (@) and R G E 2 (@G): ii: If f 2 E 2 (@) and f 2 2 E 2 (@G) then C f (z) 0 on G and C f 2 (z) 0 on : iii: E 2 (@) \ E 2 (@G) = (0): Proof. We show rst that H () E 2 (): Let g 2 H () and let ' :! be an RMF for with inverse :!. By de nition, g 2 E 2 () i g = (f ')' 0=2 for some f 2 H 2 (); equivalently i (g ) 0=2 2 H 2 (). But g 2 H () and 0=2 2 H 2 () by Theorem 3. a. Taking non-tangential limits gives R H (@) E 2 (@); so R E 2 (@); because E 2 (@) is closed in L 2 (): To prove that R G E 2 (@G) choose a 2 and let X be the set of all function on G of the form g(z) = f(z) with f 2 H (G): z a It is su cient to show that X E 2 (@G) because R G X. Choose a 2 and let (z) = z a. Then () is a recti able Jordan curve with inner domain (G); so that H ((G)) E 2 ((G)); as above. Let U = U : E 2 ((G))! E 2 (G) be the unitary operator Uh(z) = h((z)) 0 (z) =2 (g 2 E 2 ((G)); z 2 G): The map h! h is obviously an isomorphism of H ((G)) onto H (G); and 0 (z) =2 is a multiple of z a. So X E2 (G) and i is proved. To prove ii let f 2 E 2 (@) and z 2 G. Let R() = z ; then R 2 E2 (@); by i, and C f (z) = f ()R()d = f ( (u))r( (u)) 0 (u)du: T The last integrand here is in H (T); because it is the product of two elements of H 2 (T) : (f ) 0=2 and (R ) 0=2. Hence C f (z) = 0 by equation (3.2) with m = 0. The same argument shows that C f 2 0 on : The proof of iii is more subtle. Suppose that m is a ( nite) regular Borel measure with compact support X C: The Cauchy transform bm of m is de ned by bm(z) = X dm() z : It is de ned almost everywhere on C. The Cauchy transform is one of the main tools in the study of function algebras. One of its key properties is (5.) m = 0 if and only if bm(z) = 0; for almost all z 2 C

14 4 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS (see [3, p.55]). Now suppose f 2 E 2 (@) \ E 2 (@G) and let m be the measure on : dm() = f()d = f()!() jdj ; where!() is the unit tangent vector of at. Clearly bm(z) = C f(z); for all z 2 C. Now is recti able, so it has plane measure 0. It follows from ii that bm = 0 a.e. Therefore m = 0; so f = 0. Theorem 5.5. Let be an Ahlfors regular Jordan curve with inner domain and outer domain G. i: L 2 () = E 2 (@) E 2 (@G): ii: The map f! C fj is a continuous linear operator mapping L 2 () into E 2 () and the map f! C fj G is a continuous linear operator mapping L 2 () into E 2 (G). Proof. Let R ; R G be as before. Let f 2 E 2 (@). Then, by David s theorem f = g + h; where g 2 R E 2 (@) and h 2 R G E 2 (@G). Therefore h = f g 2 E 2 (@) \ E 2 (@G) and so h = 0 by Lemma 5.4 iii. So E 2 (@) = R and likewise E 2 (@G) = R G. Hence L 2 () = E 2 (@) E 2 (@G): Now, for f 2 L 2 (); let us write f = f + f 2 with f 2 E 2 (@); f 2 2 E 2 (@G): By Banach s theorem the map f! f (L 2 ()! E 2 (@)) is continuous. By Lemma 5.4 ii C fj = C f j : By b and c of Theorem 3., C f j 2 E 2 () and kc f j k E 2 () M kfk L2 () for some constant M > 0. If a 2 and (z) = then (G) is the inner domain of (), an Ahlfors regular curve, by Corollary z a 5.2. Our previous argument shows that f! C () fj (G) is a continuous linear operator mapping L 2 (()) into E 2 ((G)): Hence, by the invariance Lemma 4.5 f! C fj G is a continuous linear operator mapping L 2 () into E 2 (G): 6. Completion of the proof of Theorem 4.2 We shall nd su cient conditions for a Jordan curve to be Ahlfors regular and show that they apply to C j or to some fractional linear image (C j ) of C j (see Lemma 4.5). Definition 4. A (simple) recti able arc with arc length parametrization h 2 C [0; L] is called a Lavrentiev arc if there is an > 0 such that jh(s) h(t)j js tj (s; t 2 [0; L]):

15 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 5 Theorem 6.. i: A simple smooth arc is a Lavrentiev arc. ii: A Lavrentiev arc is Ahlfors regular. iii: If the arc is the union [ 2 [ ::: [ n of Ahlfors regular arcs then itself is Ahlfors regular. iv: A piecewise smooth Jordan curve is Ahlfors regular. Proof. Recall, from Remark 3.3, that is recti able with total length L, say, and that its arc length parametrization h 2 C [0; L]: Now suppose is not a Lavrentiev arc. Then there are sequences (s n ); (t n ) [0; L] such that (6.) jh(s n ) h(t n )j n js n t n j (n ): Assume that these two sequences are convergent, say s n! s 0 ; t n! t 0 : It follows that h(s 0 ) h(t 0 ) = 0; so that s 0 = t 0 : Now, for all n h(t n ) h(s n ) = tn s n h 0 (u)du = (t n s n ) (t n s n )h 0 (t 0 ); 0 h 0 (s n + v(t n s n ))dv by Lebesgue s bounded convergence theorem. But jh 0 (t 0 )j = ; contradicting (6.). To prove ii let z 0 = h(t) 2 ; with t 2 [0; L]. If z = h(s) is a general point of then, for r > 0; z 2 N r (z 0 ), jh(s) h(t)j < r Remembering that s is arc length we see that ) js tj < r r, s 2 (t ; t + r ): l(n r (z 0 ) \ ) arc length from h(t = 2 r: r ) to h(t + r ) To prove iii suppose that = [ 2 [ ::: [ n where each i is Ahlfors regular, say (6.2) l(n r (z i ) \ i ) k i r (z i 2 i ; r > 0): Let z 0 2 ; say z 0 2 i : Then we certainly have l(n r (z 0 ) \ i ) k i r (r > 0) and by Corollary 5. l(n r (z 0 ) \ j ) 2k j r (j 6= i) : Varying i gives l(n r (z 0 ) \ ) 2(k + k 2 + ::: + k n )r (z 0 2 ; r > 0):

16 6 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS Item iv is now clear and therefore so is. Corollary 6.2. If C j is an ellipse then Theorem 4.2 is true. Now suppose C j is a parabola. By the invariance Lemma 4.5, we can assume that C j is the parabola y 2 = 4( x): Now let (z) = : Another application of the same lemma and Theorem 6. iv shows that it is su cient z to show that (C j ) is piecewise smooth. Now (C j ) has a cusp at 0 = (): But we can show that the parts of (C j ) in the upper and lower half planes are smooth. The upper part of (C i ) has parametric function g 2 C [0; ) so that g 0 (t) = g(t) = : Now g(t)! 0 as t! + and ( + it) 3 g 0 (t) jg 0 (t)j = ( + it) 2 (t 0); j + itj3 i ( + it) 3 = i i + 3 t (i +! t )3 as t! +: So by Lemma 3.4, g[0; ) [ g() is smooth and, similarly g( ; 0] [ g( ) is smooth. Finally, let C j be a hyperbola component, say C j = sin( + ir); where 0 < < 2 : Again put (z) = z : The upper part of (C j) is parametrized by g(t) = Again g(t)! 0 as t! + and sin cosh t + i cos sinh t (t 0): g 0 (t) jg 0 (t)j = =! (sin sinh t + i cos cosh t) jsin sinh t + i cos cosh tj (sin tanh t + i cos ) jsin tanh t + i cos j ie i (ie i ) 2 = iei : Arguing as before we see that (C j ) is piecewise smooth. jsin cosh t + i cos sinh tj 2 (sin cosh t + i cos sinh t) 2 jsin coth t + i cos j 2 (sin coth t + i cos ) 2 Remark 6.3. Theorem. creates the impression that an important role is to be played by the very speci c curves mentioned; but on closer inspection of proof it can be seen that the argument runs on the lines conic section ) piecewise smooth ) Lavrentief arc ) Ahlfors regular and then Ahlfors regularity is used to prove continuity of the mapping A.

17 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS 7 7. Conclusions and further work In this paper we proved Theorem. in detail by showing the continuity of the operator A which is followed by the continuity of the Cauchy integral operators (Theorem 4.2). Also we need Theorem 3.2 which shows that the operator A exists. There are some di culties in generalizing Theorem. for more general case than conic sections. We raise some of them in the following questions. Under what conditions on an N-tuple C = (C ; C 2 ; C 3 ; :::; C N ) of closed distinct curves in the extended plane (a) are the results of Theorem 3.2 true? (so that the operator A exists) and (b) is Theorem 4.2 true?, i.e., is each Cauchy integral operators continuous? i, ii and iii of Theorem 3.2 are consequences of being 0 2 H () for the bounded case of D and 0 ( a) 2 2 H () for the unbounded case of D where a 2 CnD; see [2, Theorems 25 and 30]. Arsove [, 2] gives a complete description of bounded simply connected domains for which the derivative of a conformal mapping of the unit disk onto belongs to H. In fact, according to Theorem of [], if is a bounded simply connected domain and is a Riemann mapping function for then 0 is in the Hardy class H if and only can be parametrized as a recti able closed curve. This description shows that there is no real need to restrict oneself to domains bounded by pieces of quadrics so that Theorem 3.2 can be improved. Therefore, it looks that the more di cult part of future work is to prove the continuity of the Cauchy integral operators (Theorem 4.2). choosing all closed curves as analytic -recti able Jordan curve. Further work includes investigating more complex scenarios such as References [] Arsove M.G., Some boundary properties of the Riemann mapping function, Proc. AMS, 9, No. 3, (968), [2] Arsove M.G., A correction to Some boundary properties of the Riemann mapping function, Proc. AMS, 22, No. 3, (969), [3] Browder A., Introduction to Function Algebras (W.A.Benjamin, Inc., New York, Amsterdam, 969). [4] David J., Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. E.N.S. 7 (984), [5] Duren P. L., Theory of H p Spaces (Academic Press, New York, 970). [6] Goluzin G.M., Functions of a Complex Variable (Amer. Math. Soc., Providence, RI, 969 (translated from Russian). [7] Khavinson D., Factorization theorems for certain classes of analytic functions in multiply connected domains. Paci c. J. Math. 08 (983), [8] Little G., Asymptotic estimates of the eigenvalues of certain positive Fredholm operators. Math. Proc. Camb.Phil. Soc. 9 (982), [9] Little G., Equivalences of positive integral operators with rational kernels. Proc. London. Math. Soc. (3) 62 (99), [0] Meyer Y., and Coifman R., Wavelets, Calderon-ygmund and multilinear operators (Cambridge studies in advanced mathematics, 48, Translation of vols. 2-3 of : Ondelettes et operateurs by D.H. Salinger, 997). [] Privalov I.I., Boundary Properties of Analytic Functions (Moscow, (950)) (Russian; German translation: Randeigenschaften analytischer Funktionen, Deutscher Verlag der Wiss., Berlin, 956).

18 8 EIGENVALUE ESTIMATES OF POSITIVE INTEGRAL OPERATORS WITH ANALYTIC KERNELS [2] Soykan Y., Boundary Behaviour of Analytic Curves. Int. Journal of Math. Analysis. Vol. no. 3 (2007), [Online: [3] Soykan Y., On Equivalent Characterisation of Elements of Hardy and Smirnov Spaces. Int. Math. Forum, Vol. 2 no. 24 (2007), [Online: