On the Singular Numbers for Some Integral Operators

Two–sided estimates of Schatten–von Neumann norms for weighted Volterra integral operators are established. Analogous problems for some potential-type operators defined on R are solved. Let H be a separable Hilbert space and let σ∞(H) be the class of all compact operators T : H → H, which forms an ideal in the normed algebra B of all bounded linear operators onH. To construct a Schattenvon Neumann ideal σp(H) (0 < p ≤ ∞) in σ∞(H), the sequence of singular numbers sj(T ) ≡ λj(|T |) is used, where the eigenvalues λj(|T |) ( |T | ≡ (T ∗T )1/2 ) are non-negative and are repeated according to their multiplicity and arranged in decreasing order. A Schatten-von Neumann quasinorm (norm if 1 ≤ p ≤ ∞) is defined as follows: ‖T‖σp(H) ≡ (∑ j spj (T ) )1/p , 0 < p < ∞, with the usual modification if p = ∞. Thus we have ‖T‖σ∞(H) = ‖T‖ and ‖T‖σ2(H) is the HilbertSchmidt norm given by the formula ‖T‖σ2(H) = (∫ ∫ |T1(x, y)|2dxdy )1/2 (1) for an integral operator Tf(x) = ∫ T1(x, y)f(y)dy. We refer, for example, to [2], [6], [7] for more information concerning Schatten-von Neumann ideals. 2000 Mathematics Subject Classification: 26A33, 42B15, 47B10, 47G10. Servicio de Publicaciones. Universidad Complutense. Madrid, 2001 379 http://dx.doi.org/10.5209/rev_REMA.2001.v14.n2.16982 a. meskhi on the singular numbers for some integral operators. . . In this paper necessary and sufficient conditions for the weighted Volterra integral operator Kvf(x) = v(x) ∫ x 0 f(y)k(x, y)dy, x ∈ (0, a), to belong to Schatten-von Neumann ideals are established, where v is a measurable function on (0, a) (0 < a ≤ ∞). Two-sided estimates of Schatten-von Neumann p-norms for the weighted Riemann–Liouville operator Rα,vf(t) = v(x) ∫ x 0 f(t)(x− t)α−1dt, when α > 1/2 and p > 1/α, were established in [13] (for α = 1 and p > 1 see [14]). Analogous results for the weighted Hardy operator Hv,uf(x) = v(x) ∫ x 0 u(y)f(y)dy were obtained in [3]. Similar problems for the Riemann-Liouville operator with two weights Rα,v,uf(x) = v(x) ∫ x 0 u(t)f(t)(x− t)α−1dt, when α ∈ N and p ≥ 1, were solved in [4]. Further, upper and lower bounds for Schatten–von Neumann p-norms (p ≥ 2) of certain Volterra integral operators, involving Rα,v,u only for α ≥ 1, were proved in [4] and [18]. Our main goal is to generalize the results of [13] and [14] for integral transforms with kernels and to give two-sided estimates of the above-mentioned norms for these operators in terms of their kernels. We denote by Lw(Ω), Ω ⊆ Rn, a weighted Lebesgue space with respect to the weight w defined on Ω. Throughout the paper the expression A ≈ B is interpreted as c1A ≤ B ≤ c2A with some positive constants c1 and c2. Let us recall some definitions from [10] (see also [8]). We say that a kernel k : {(x, y) : 0 < y < x < a} → R+ belongs to V (k ∈ V ) if there exists a positive constant d1 such that for all x, y, z with 0 < y < z < x < a the inequality k(x, y) ≤ d1k(x, z) 380 REVISTA MATEMÁTICA COMPLUTENSE (2001) vol. XIV, num. 2, 379-393 a. meskhi on the singular numbers for some integral operators. . . holds. Further, k ∈ Vλ (1 < λ < ∞) if there exists a positive constant d2 such that for all x, x ∈ (0, a), the inequality ∫ x x/2 k ′ (x, y)dy ≤ d2xkλ ( x, x/2 ) , λ′ = λ λ− 1 . is fulfilled. For example, if k1(x) = xα−1, where 1 λ < α ≤ 1, then k(x, y) = k1(x− y) belongs to V ∩Vλ (for other examples of kernel k see [10], [8]). First we investigate the mapping properties ofKv in Lebesgue spaces. The following statements in equivalent form were proved in [10] (see also [8], [11]). Theorem A. Let 1 < p ≤ q < ∞, a = ∞ and let k ∈ V ∩ Vp. Then (a) Kv is bounded from Lp(0,∞) into Lq(0,∞) if and only if D∞ ≡ sup j∈Z D∞(j) ≡ sup j∈Z (∫ 2j+1 2j k(x, x/2)x ′ |v(x)|qdx ) 1 q < ∞. Moreover, ‖Kv‖ ≈ D∞. (b) Kv acts compactly from Lp(0, a) into Lq(0, a) if and only if D∞ < ∞ and lim j→+∞ D∞(j) = lim j→−∞ D∞(j) = 0. Theorem B. Let 1 < p ≤ q < ∞, a < ∞ and let k ∈ V ∩ Vp. Then (a) Kv is bounded from Lp(0, a) to Lq(0, a) if and only if Da ≡ sup j≥0 Da(j) ≡ sup j≥0 (∫ 2−ja 2−(j+1)a |v(x)|qkq(x, x/2)xq/pdx ) 1 q < ∞. Moreover, ‖Kv‖ ≈ Da. (b) Kv acts compactly from Lp(0, a) into Lq(0, a) if and only if Da < ∞ and lim j→+∞ Da(j) = 0; Analogous problems for the Riemann-Liouville operator for α > 1/p were solved in [9] (For boundedness two-weight criteria of general integral operators with positive kernels see [5], Chapter 3). Let 0 < a ≤ ∞, k : {(x, y) : 0 < y < x < a} → R+ be a kernel and let k0(x) ≡ xk2(x, x/2). 381 REVISTA MATEMÁTICA COMPLUTENSE (2001) vol. XIV, num. 2, 379-393 a. meskhi on the singular numbers for some integral operators. . . We denote by lp(L2k0(0, a)) the set of all measurable functions g : (0, a) → R1 for which ‖g‖lp(L2k0 (0,∞)) = (∑ n∈Z (∫ 2n+1 2n |g(x)|k0(x)dx )p/2)1/p < ∞ if a = ∞ and ‖g‖lp(L2k0 (0,a)) = ( +∞ ∑ n=0 (∫ 2−na 2−(n+1)a |g(x)|k0(x)dx )p/2)1/p < ∞ if a < ∞, with the usual modification for p = ∞. We shall need the following interpolation result (see, e.g., [19], p. 147 for the interpolation properties of the Schatten classes, and p. 127 for the corresponding properties of the sequence spaces. See also [1], Theorem 5.1.2): Proposition A. Let 0 < a ≤ ∞, 1 ≤ p0, p1 ≤ ∞, 0 ≤ θ ≤ 1, 1 p = 1−θ p0 + θ p1 . If T is a bounded operator from lpi(L 2 k0 (0, a)) into σpi(L 2(0, a)), where i = 0, 1, then it is also bounded from lp(L2k0(0, a)) into σp(L(0, a)). Moreover, ‖T‖lp(L2k0 )→σp(L2) ≤ ‖T‖ 1−θ lp0 (L2k0 )→σp0 (L2) ‖T‖lp1 (L2k0 )→σp1 (L2). The next statement is obvious when p = ∞; and when 1 ≤ p < ∞ it follows from Lemma 2.11.12 of [15]. Proposition B. Let 1 ≤ p ≤ ∞ and let {fk}, {gk} be orthonormal systems in a Hilbert space H. If T ∈ σp(H), then ‖T‖σp(H) ≥ (∑ n |〈Tfn, gn〉| )1/p . Now we prove the main results. In the sequel we shall assume that v ∈ L2k0(2n, 2n+1) for all n ∈ Z. Theorem 1. Let a = ∞, 2 ≤ p < ∞ and let k ∈ V ∩ V2. Then Kv belongs to σp(L(0,∞)) if and only if v ∈ l(Lk0(0,∞)). Moreover, there exist positive constants b1 and b2 such that b1‖v‖lp(L2k0 (0,∞)) ≤ ‖Kv‖σp(L2(0,∞)) ≤ b2‖v‖lp(L2k0 (0,∞)). 382 REVISTA MATEMÁTICA COMPLUTENSE (2001) vol. XIV, num. 2, 379-393 a. meskhi on the singular numbers for some integral operators. . . Proof. Sufficiency. Note that the fact k ∈ V ∩ V2 implies I(x) ≡ ∫ x 0 k(x, y)dy ≤ ck0(x) (2) for some positive constant c independent of x. Indeed, by the condition k ∈ V ∩ V2 we have I(x) = ∫ x/2