ISSN 75-987 e-issn 313-1 http://www.journals.pu.if.ua/index.php/cmp Carpathian Math. Publ. 16, 8 (), 4 9 doi:1.1533/cmp.8..4-9 Карпатськi матем. публ. 16, Т.8,, С.4 9 DILNYI V.M., HISHCHAK T.I. ON THE INTERSECTION OF WEIGHTED HARDY SPACES Let H p σ(c + ), 1 p <, σ <, be the space of all functions f analytic in the half plane C + = {z : Rez > } and such that f := ϕ ( π ; π ) f (re iϕ ) p e pσr sin ϕ dr 1/p <. We obtain some properties and description of zeros for functions from the space σ> H p σ(c + ). Key words and phrases: zeros of functions, weighted Hardy space, angular boundary values. Ivan Franko State Pedagogical University, 4 Franka str., 81, Drohobych, Ukraine E-mail: dilnyi@ukr.net (Dilnyi V.M.), taniosered@mail.ru (Hishchak T.I.) INTRODUCTION Let H p (C + ), 1 p <, be the Hardy space of holomorphic in C + = {z : Rez > } functions f such that f p = f (x + iy) p dy x> <. Let H p σ(c + ), 1 p <, σ <, be the space of all functions f analytic in the half plane C + and such that f := ϕ ( π ; π ) f (re iϕ ) p e pσr sin ϕ dr 1/p <. We denote by Hσ (C + ), σ <, the space of all functions analytic in the right halfplane satisfying the condition { f := f (z) e σ Imz } <. z C + The space H p σ(c + ), 1 p, σ <, is a weighted Hardy space, as it follows from results of А. M. Sedletskii [9]. The theory of weighted Hardy space for the case if the weight is an exponential function considered in [, 3, 1 13]. Functions f H p σ(c + ) have angular boundary values almost everywhere on C + (we denote the extension by the same УДК 517.5 1 Mathematics Subject Classification: 3D15, 3H1. c Dilnyi V.M., Hishchak T.I., 16
ON THE INTERSECTION OF WEIGHTED HARDY SPACES 5 symbols f ) and f L p ( C + ). Thus, the space H p σ(c + ), p 1, is a Banach space. For functions f H p σ(c + ) there exists [4, 1] an integral boundary function defined by the equality h(t ) h(t 1 ) = lim x + t t 1 t ln f (x + it) dt t 1 ln f (it) dt, t 1 < t up to an additive constant and to values at continuity points. The integral boundary function h is nonincreasing on R and h (t) = almost everywhere on R. The interest to the space Hσ(C p + ) is generated by studies of completeness [3], by the theory of integral operators and the shift operator [1, 8]. A number of papers have been devoted to the intersection of Hardy and related spaces (see [5, 7]). The aim of our research is to describe some properties of the following space H (C p + ) = Hσ(C p + ). σ> Obviously, H p (C + ) H p (C + ) and H p (C + ) H p ε (C + ) for all ε. Theorem 1. H p (C + ) = H p (C + ). 1 THE MAIN RESULTS Proof. Let f (z) = e zln(z+). We choose the branch of the logarithm that ln 1 = and 1 = 1. Let us prove that the function f belongs to Hσ(C p + ) for all σ >. Indeed, ln f (re iϕ ) = r 4 ln 4r cos ϕ + r + 4 + r cos ϕ + cos ϕ cos r 4 r cos ϕ+ ln 4r cos ϕ+r +4 ln 4r cos ϕ + r + 4 + r ϕ sin ϕ 1 ln r, r. sin ϕ sin r cos ϕ+ ln 4r cos ϕ+r +4 r cos ϕ + sin ϕ sin It follows easily that f H p σ(c + ). Consequently, f H p (C + ). Let us show that f (z) = e z ln(z+) H p (C + ). Indeed, y ln 4+y ln f (iy) = y 4 ln 4 + y + y sin = y 4 ln 4 + y + y ln 4 + y 1 y ln(4 + y ) for y C >. Therefore f (iy) L p (; ). Hence, f H p (C + ). ln 4 + y + y r cos ϕ+ ln 4r cos ϕ+r +4
6 DILNYI V.M., HISHCHAK T.I. Proposition 1. Suppose that f H p (C + ), 1 p. Then the following conditions are fulfilled : a) angular boundary values exist almost everywhere on ir ; b) f (it) e ε t L p (R) for any ε > ; c) H p (C + ) is a Banach space for uniform convergence on compact sets. Proof. Let f H p (C + ), then f H p ε (C + ) for some ε >. In [11] B. V. Vinnitskii proved that a function f H p σ(c + ), p (1; ), has almost everywhere on ir angular boundary values f (iy) and f (iy)e σ y L p (R). Therefore f (iy)e ε y L p (R) for some positive ε. In [1] B. V. Vinnitskii showed that a function f H σ (C + ) has almost everywhere on ir angular boundary values f (it) and f (it)e ε t L (R) for all ε. In [11] inequality f (z) c exp(c z ) Re(z) 1 p proved for each function f belonging to H p σ(c + ). Furthermore, H p (C + ) is a Banach space with respect to uniform convergence on compact sets. Let B is a class of continuous, increasing functions η : [; ) (; ) such that η(r) = o(r) as r. We denote by H p (C +) the space of functions analytic in C + for which there exists η B where η B. ϕ < π Theorem. If f H p (C +), then f H p (C + ). 1 p f (re iϕ ) p e η(r) sin ϕ dr <, Proof. Let f H p (C +), then f H p σ(c + ) for all σ >. Furthermore, f (re iϕ ) p e prσ sin ϕ dr = f (re iϕ ) p e η(r) sin ϕ e prσ sin ϕ +η(r) sin ϕ dr. Since prσ sin ϕ + η(r) sin ϕ = sin ϕ ( prσ + η(r)) < as r > r, we have f (re iϕ ) p e prσ sin ϕ dr f (re iϕ ) p e η(r) sin ϕ dr <. r r This implies that r f (re iϕ ) p e prσ sin ϕ dr r f (re iϕ ) p e η(r) sin ϕ dr
ON THE INTERSECTION OF WEIGHTED HARDY SPACES 7 and r r f (re iϕ ) p e η(r) sin ϕ dr f (re iϕ ) p exp{ min { η(r)} sin ϕ }dr r [;r ] r r exp{ min { η(r)} sin ϕ } f (re iϕ ) p dr c 1 f (re iϕ ) p dr <. r [;r ] In particular, choosing c = pσr η() we can achieve that f (re iϕ ) p e prσ sin ϕ dr c <. It follows that f H p (C +). B. V. Vinnitskii described [11] zeros for functions f Hσ(C p + ) in terms of the following function S(r) = λ 1< λ n r n λ ) n Reλn r λ n, where λ n C +. We obtain the following statement. Theorem 3. If f H p (C + ), then S(r) = o(ln r), r. Proof. Suppose f H p (C + ), then f H p σ(c + ) for all σ >. Use the following version of the Carleman formula [4, 6, 1] S(r) = 1 πr 1 π π ln f (re iϕ ) cos ϕdϕ + 1 t 1 ) r dh(t) + O(1). t 1 ) r ln f (it) dt (1) In [11] it is shown that for each function f Hσ(C p + ), σ >, the first term on the right side of the last equality is bounded by an independent of r and σ constant. Hence, this term is bounded for each function of the space H (C p + ). Consider the second addend Since 1 1 1 t 1 r ) ln f (it) dt = 1 t 1 r t 1 ) r ( f (it) e σ t + σ t )dt. ) (ln f (it) e σ t + e σ t )dt ( 1 t 1 r ) σ t dt = 1 π σ ln r and f (iy)e σ y L p (R), this yields 1 t 1 ) r ln f (it) dt c 3 + 1 σ ln r. π
8 DILNYI V.M., HISHCHAK T.I. Therefore S(r) = c 4 + 1 π σ ln r 1 Then, the last addend is negative, we deduce t 1 ) r dh(t). S(r) c 4 + σ ln r. π Since the result is true for on of an arbitrary σ, we obtain the statement of the theorem. Theorem 4. If f H (C p + ), then P(r) = o(ln r), r, where P(r) = 1 t 1 ) r dh(t). Proof. Let f H (C p + ), then f Hσ(C p + ) for everyone σ >. Using (1), we get P(r) = K(r) S(r) + O(1), r, where K(r) = 1 t 1 ) r ln f (it) dt. Since K(r) = 1 t 1 ) r ln f (it) e σ t dt + 1 t 1 ) r σ t dt c 3 + 1 σ ln r for all σ >, π we deduce K(r) = o(ln r) as r. From Theorem 3 we get the following S(r) = o(ln r), r. Thus P(r) = o(ln r), r. Theorem 5. Let (λ n ) be an arbitrary sequence in C +. Then S(r) = o(ln r), r, if and only if S (r) = o(ln r), r, where Proof. It is clear that S (r) S(r) = Reλ where s(r) = n λ n. 1< λ n r In [1] B. V. Vinnitskii proved that S (r) = Reλ n λ 1< λ n r n. Reλ n Reλ n r 1< λ n r λ 1< λ n r n r = s(r) r, S(r) 3s(r) 4r. It follows that S (r) S(r) 4rS(r) = 4 3r 3 S(r). Since S(r) = o(ln r), we have S(r) = o(ln r). Hence, S (r) S(r) = o(ln r), r. The converse implication is trivial.
ON THE INTERSECTION OF WEIGHTED HARDY SPACES 9 REFERENCES [1] Dilnyi V. M. Splitting of some spaces of analytic functions. Ufa Math. J. 14, 6 (), 5 34. dio:1.1318/14-6-- 5 [] Dil nyi V. M. On the equivalence of some conditions for weighted Hardy spaces. Ukrainian Math. J. 6, 58 (9), 145 143. doi:1.17/s1153-6-141- (translation of Ukrain. Mat. Zh. 6, 58 (9), 157 163. (in Ukrainian)) [3] Dilnyi V. M. On Cyclic Functions in Weighted Hardy Spaces. J. Math. Phys., Anal., Geom. 11, 7 (1), 19 33. (in Ukrainian) [4] Fedorov M.A., Grishin A.F. Some questions of the Nevanlinna theory for the Complex Half-Plane. Interface Science 1998, 6 (3), 3 71. doi:1.13/a:186744744 [5] Fedorov S.I. On a recent result on the intersection of weighted Hardy spaces. NZ J. Mathematics. 1, 3 (), 119 13. [6] Govorov N.V. Riemann boundary value problem with infinite index. Nauka, Moscow, 1986. (in Russian) [7] Jones P.W. Recent advances in the theory of Hardy Spaces. In: Proc. of the International Congress of Mathematicians, Warszawa, Poland, August 16 4, 1983. 1, 89 839. http://www.mathunion.org/icm/icm1983.1/ [8] Nikolskii N.K. Treatise on the shift operator. Spectral function theory. In: Grundlehren der mathematischen Wissenschaften, 73. Springer Verlag, Berlin, 1986. doi:1.17/978-3-64-7151-1 [9] Sedletskii A.M. An equivalent definition of H p spaces in the half-plane and some applications. Mathematics of the USSR-Sbornik 1975, 5 (1), 69 76. doi:1.17/sm1975v5n1abeh198 (translation of Math. Sb. (N.S.) 1975, 96(138) (1), 75 8. (in Russian)) [1] Vinnitskii B.V. On zeros of functions analytic in a half plane and completeness of systems of exponents. Ukrainian Math. J. 1994, 46 (5), 514 53. doi:1.17/bf158515 (translation of Ukrain. Mat. Zh. 1994, 46 (5), 484 5. (in Ukrainian)) [11] Vinnitskii B.V. On zeros of functions analytic in a half plane. Mat. Stud. 1996, 6 (1), 67 7. (in Ukrainian) [1] Vynnytskyi B.V., Dil nyi V.M. On necessary conditions for existence of solutions of convolution type equation. Mat. Stud. 1, 16 (1), 61 7. (in Ukrainian) [13] Vynnytsyi B.V., Sharan V.L. On the factorization of one class of functions analytic in the half-plane. Mat. Stud., 14 (1), 41 48. (in Ukrainian) Received 3.9.16 Revised 8.11.16 Дiльний В.М., Гiщак Т.I. Про перетин вагових просторiв Гардi // Карпатськi матем. публ. 16. Т.8,. C. 4 9. Нехай Hσ(C p + ), 1 p <, σ <, простiр функцiй, аналiтичних у пiвплощинi C + = {z : Rez > }, для яких 1/p f := f (re iϕ ) p e pσr sin ϕ dr <. ϕ ( π ; π ) Отримано деякi властивостi i опис нулiв для функцiй з простору σ> H p σ(c + ). Ключовi слова i фрази: нулi функцiй, ваговий простiр Гардi, кутовi граничнi значення.