����������� ������ 54 (2002), 125–131

UDK 517.927

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AN L_{p} ESTIMATE FOR THE DIFFERENCE OF DERIVATIVES OF SPECTRAL EXPANSIONS

ARISING BY ONE-DIMENSIONAL SCHRÖDINGER OPERATORS

Nebojša L. Lažetić and Olivera R. -Dor - dević

Abstract. We prove the estimate

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{Lp(G)} ≤ C∥f∥_{BV (G)} · μ^{1−1/p},

where 2 ≤ p < +∞, and σ_{μ}(x, f), σ̃_{μ}(x, f) are the partial sums of spectral expansions of a function f(x) ∈ BV (G), corresponding to arbitrary non-negative self-adjoint extensions of the operators Lu = −u′′ + q(x)u, L̃u = −u′′ + q̃(x)u (x ∈ G) respectively; the operators are defined on an arbitrary bounded interval G ⊂ ℝ.

1. Introduction

Let G = (a, b) be an arbitrary bounded interval, and let the operators

Lu = −u′′ + q(x)u,  L̃u = −u′′ + q̃(x)u  (1)

be defined on G, with potentials q(x), q̃(x) ∈ L_{s}(G), 1 < s ≤ 2. Denote by L, L̃ arbitrary non-negative self-adjoint extensions, with discrete spectrum, of the operators (1) respectively (see §17, [1]). Let {u_{n}(x)}_{n=1}^{∞}, {ũ_{n}(x)}_{n=1}^{∞} be complete (in L_{2}(G)) and orthonormal systems of eigenfunctions of those extensions, and {λ_{n}}_{n=1}^{∞}, {λ̃_{n}}_{n=1}^{∞} the corresponding systems of non-negative eigenvalues, enumerated in non-decreasig order. If f(x) ∈ L_{1}(G) and μ ≥ 2, we can form the partial sums of order μ:

σ_{μ}(x, f) = ∑_{\sqrt{λ_{n}}<μ} f_{n}u_{n}(x),  σ̃_{μ}(x, f) = ∑_{\sqrt{λ̃_{n}}<μ} f̃_{n}ũ_{n}(x),

where f_{n} = ∫_{a}^{b} f(x)u_{n}(x) dx, f̃_{n} = ∫_{a}^{b} f(x)ũ_{n}(x) dx. Let AC(G) be the set of absolutely continuous functions on the closed interval \overline{G}. Denote by BV (G) the Banach space of functions having bounded variation on \overline{G}, with the norm ∥f∥_{BV (G)} = sup_{x∈\overline{G}} |f(x)| + V_{a}^{b}(f), where V_{a}^{b}(f) stands for the total variation of f(x) on \overline{G}.

AMS Subject Classification: 34 L 10, 47 E 05

Communicated at the 5th International Symposium on Mathematical Analysis and its Applications, Niška banja, Yugoslavia, October, 2–6, 2002.

The problem of behavior of function σ_{μ}(x, f) (and its derivatives) on subsets of \overline{G}, as μ → +∞, is the classical one. One of the most fruitful approaches to the problem is so-called “equiconvergence approach”: one studies the behavior of the difference σ_{μ}(x, f) − S_{μ}(x, f), as μ → +∞, where S_{μ}(x, f) is the corresponding partial sum of the trigonometrical Fourier series of function f (for a review see [2]). It seems that the first results concerning the equiconvergence rate estimates, in the case of arbirtary self-adjoint Sturm-Liouville operators, were obtained by V.A. Il’in and I. Joo in [3]. They obtained the following estimate:

If q(x), q̃(x) ∈ L_{s}(G) (s > 1), f(x) ∈ AC(G), and K ⊂ G is an arbitrary compact set, then there exists a constant C(K, f) > 0 such that

max_{x∈K} |σ_{μ}(x, f) − σ̃_{μ}(x, f)| ≤ C(K, f) · \frac{1}{μ},  μ ≥ 2;  (2)

C(K, f) does not depend on μ. The estimate is exact in order with respect to μ.

In order to “globalize” the estimate (2), I.S. Lomov has considered the L_{p} metric instead of the uniform one; in paper [4] he proved the following assertion: If q(x), q̃(x) ∈ L_{s}(G) (s > 1), f(x) ∈ BV (G), and 2 ≤ p < +∞, then the estimate

∥σ_{μ}(x, f) − σ̃_{μ}(x, f)∥_{L_{p}(G)} ≤ C∥f∥_{BV (G)} · \frac{1}{μ^{1/p}} ,  μ ≥ 3,  (3)

holds, where C > 0 does not depend on f and μ. (Note that in earlier paper [5] Lomov obtained estimate (3) with μ^{−1/p} ln μ instead of μ^{−1/p}.)

A local uniform estimate for the difference of the first derivatives σ′_{μ}(x, f), σ̃′_{μ}(x, f) was obtained by I. Joo and N. Lažetić in paper [6]. They proved: If q(x) and q̃(x) belong to L_{s}(G) (1 < s ≤ 2), f(x) ∈ AC(G), and K ⊂ G is an arbitrary compact set, then the estimate

max_{x∈K} |σ′_{μ}(x, f) − σ̃′_{μ}(x, f)| ≤ C(K, f),  μ ≥ 2,  (4)

holds, where C(K, f) > 0 is independent of μ. This estimate is exact in order with respect to the spectral parameter μ.

Recently, the estimate (4) has been extendend on the set BV (G). Namely, the authors of this paper have proved ([7]) that for every function f(x) ∈ BV (G) and every compact set K ⊂ G the following estimate is valid:

max_{x∈K} |σ′_{μ}(x, f) − σ̃′_{μ}(x, f)| ≤ C(K)∥f∥_{BV (G)},  μ ≥ 2.  (5)

It is supposed that q(x), q̃(x) ∈ L_{s}(G)(s > 1).

In this paper we propose an L_{p} estimate for the difference mentioned above. That estimate “globalizes” (5), and shows how the estimate (3) is affected by the operation of differentiation (compare with estimates (8)-(9) below). Hence, our result is the following assertion.

Theorem. Suppose q(x), q̃(x) ∈ L_{s}(G)(1 < s ≤ 2), f(x) ∈ BV (G), p ∈ [2, +∞), and μ ≥ 2. There exists a constant C > 0, independent of f and μ, such that the following estimate holds:

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{L_{p}(G)} ≤ C∥f∥_{BV (G)} · μ^{1−1/p}.  (6)

An L_{p} estimate for the difference of derivatives … 127

2. Auxiliary results

Proof of the theorem is based on estimate (5). But we will also use a variety of known results listed below.

Let q(x) ∈ L_{1}(G). Then for systems of eigenfunctions and eigenvalues of an arbitrary non-negative self-adjoint extension L of the operator L the following estimates are valid:

∑_{|\sqrt{λ_{n}}−μ|<1} 1 ≤ A,  μ > 0,  (7)

where A > 0 does not depend on μ (see [8] and [9]);

sup_{x∈G} |u_{n}(x)| ≤ C(G),  (8)

where C(G) > 0 is independent of n ∈ ℕ ([8]);

sup_{x∈G} |u′_{n}(x)| ≤ C_{1}(G)(\sqrt{λ_{n}} + 1),  (9)

with C_{1}(G) > 0 non-depending on n ∈ ℕ ([10]).

If f(x) ∈ BV (G), then for its Fourier coefficients f_{n} (with respect to the system {u_{n}(x)}_{n=1}^{∞}) the estimate

|f_{n}| ≤ \frac{C}{\sqrt{λ_{n}}} · ∥f∥_{BV (G)}  (10)

holds, where C > 0 does not depend on n ∈ ℕ (see [5]).

We will also use so-called “mean value formula” for the derivatives u′_{n}(x) ([10]): If x ∈ G and t > 0 are such that x ± t ∈ G, then

u′_{n}(x + t) − u′_{n}(x − t) = −2\sqrt{λ_{n}}u_{n}(x) sin \sqrt{λ_{n}}t+

+ ∫_{x−t}^{x+t} q(ξ)u_{n}(ξ) cos \sqrt{λ_{n}}(|x − ξ| − t) dξ. (11)

(Note that a function u_{λ}(x) is called an eigenfunction corresponding to an eigenvalue λ of the operator L if u_{λ}(x), u′_{λ}(x) ∈ AC(G) and the equality

−u′′_{λ}(x) + q(x)u_{λ}(x) = λu_{λ}(x)

holds a.e. on G.)

Finally, recall the “second part” of the known Riesz theorem ([11]): Let {v_{n}(x)}_{n=1}^{∞} be an orthogonal system of functions defined on a bounded interval G, and such that sup_{x∈G} |v_{n}(x)| ≤ M, where M > 0 is independent on n ∈ ℕ. If 1 < r ≤ 2 and 1/r + 1/p = 1, then for every sequence of (complex) numbers {g_{n}}_{n=1}^{∞}, satisfying (∑_{n=1}^{∞} |g_{n}|^{r})^{1/r} < +∞, there exists a function g(x) ∈ L_{p}(G) such that g_{n} = ∫_{a}^{b} g(y)\overline{v_{n}(y)} dy and

∥g∥_{L_{p}(G)} ≤ M^{2/r−1}( ∑_{n=1}^{∞} |g_{n}|^{r})^{1/r}.  (12)

Note that in proving the estimate (5) we have used all the results (7)–(12).

3. Proof of the theorem

The first step of the proof is the same as the one in the proof of Lemma 2 [5]. Let K = [c, d] ⊂ G be an arbitrary fixed closed interval. Then we have

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{L_{p}(G)}^{p} = ∥(·)∥_{L_{p}((a,c))}^{p} + ∥(·)∥_{L_{p}(K)}^{p} + ∥(·)∥_{L_{p}((d,b))}^{p}. (13)

In estimating the members on the right-hand side, we will assume, with no loss of generality, that λ_{n} ≥ 1 (n ∈ ℕ). (This assumption is based on the equation

−u′′_{n}(x) + [q(x) + 1]u_{n}(x) = (λ_{n} + 1)u_{n}(x).) Set μ_{n} =^{def} \sqrt{λ_{n}}.

Let us consider the first member. Introducing a new variable z = x + h, with h ∈ (0, (d − c)/2) fixed, we obtain

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{L_{p}((a,c))} = ∫_{K_{1}} |σ′_{μ}(z − h, f) − σ̃′_{μ}(z − h, f)|^{p} dz,  (14)

where K_{1} =^{def} [a + h, c + h] ⊂ G. By the mean value formula (11), we can write

σ′_{μ}(z − h, f) = ∑_{μ_{n}<μ} f_{n}u′_{n}(z + h) + ∑_{μ_{n}<μ} 2μ_{n}f_{n}u_{n}(z) sin μ_{n}h−

− ∑_{μ_{n}<μ} f_{n} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − z − h) dξ+

+ ∑_{μ_{n}<μ} f_{n} ∫_{z−h}^{z} q(ξ)u_{n}(ξ) cos μ_{n}(z − ξ − h) dξ.

Analogous equality can be written for σ̃′_{μ}(z−h, f). Therefore, the following equality holds on K_{1}:

σ′_{μ}(z − h, f) − σ̃′_{μ}(z − h, f) = ∑_{μ_{n}<μ} f_{n}u′_{n}(z + h) − ∑_{μ̃_{n}<μ} f̃_{n}ũ′_{n}(z + h)+

+ ∑_{μ_{n}<μ} 2μ_{n}f_{n}u_{n}(z) sin μ_{n}h − ∑_{μ̃_{n}<μ} 2μ̃_{n}f̃_{n}ũ′_{n}(z) sin μ̃_{n}h−

− ∑_{μ_{n}<μ} f_{n} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − x − h) dξ+

+ ∑_{μ̃_{n}<μ} f̃_{n} ∫_{z}^{z+h} q̃(ξ)ũ_{n}(ξ) cos μ̃_{n}(ξ − z − h) dξ+

+ ∑_{μ_{n}<μ} f_{n} ∫_{z−h}^{z} q(ξ)u_{n}(ξ) cos μ_{n}(z − ξ − h) dξ−

− ∑_{μ̃_{n}<μ} f̃_{n} ∫_{z−h}^{z} q̃(ξ)ũ_{n}(ξ) cos μ̃_{n}(z − ξ − h) dξ.

That is why we have the inequality

∥σ′_{μ}(z − h, f) − σ̃′_{μ}(z − h, f)∥_{L_{p}(K_{1})}^{p} ≤ C_{p}∥σ′_{μ}(z + h, f) − σ̃′_{μ}(z + h, f)∥_{L_{p}(K_{1})}^{p}+

+ C_{p}∥ ∑_{μ_{n}<μ} 2μ_{n}f_{n}u_{n}(z) sin μ_{n}h∥_{L_{p}(K_{1})}^{p}+

+ C_{p}∥ ∑_{μ̃_{n}<μ} 2μ̃_{n}f̃_{m}ũ_{n}(z) sin μ̃_{n}h∥_{L_{p}(K_{1})}^{p}+

+ C_{p} ∫_{K_{1}} | ∑_{μ_{n}<μ} f_{n} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − z − h) dξ|^{p} dz+

+ C_{p} ∫_{K_{1}} | ∑_{μ̃_{n}<μ} f̃_{n} ∫_{z}^{z+h} q̃(ξ)ũ_{n}(ξ) cos μ̃_{n}(ξ − x − h) dξ|^{p} dz+

+ C_{p} ∫_{K_{1}} | ∑_{μ_{n}<μ} f_{n} ∫_{z−h}^{z} q(ξ)u_{n}(ξ) cos μ_{n}(x − ξ − h) dξ|^{p} dz+

+ C_{p} ∫_{K_{1}} | ∑_{μ̃_{n}<μ} f̃_{n} ∫_{z−h}^{z} q̃(ξ)ũ_{n}(ξ) cos μ̃_{n}(x − ξ − h) dξ| dz.  (15)

Here and further, we denote by C_{p} not necessarily equal positive constants.

In order to estimate the first member on the right-hand side of (15), we will use the estimate (5). Having in mind that z + h ∈ K_{2} if z ∈ K_{1}, where K_{2} =^{def} [a + 2h, c + 2h] ⊂ G, we have the inequalities

∥σ′_{μ}(z + h, f) − σ̃′_{μ}(z + h, f)∥_{L_{p}(K_{1})}^{p} ≤ (c − a)C(K_{2})^{p}∥f∥_{BV (G)}^{p}
≤ C_{p}∥f∥_{BV (G)} · μ^{(1−1/p)p}.  (16)

The next two members have the same “structure”, and they will be estimated by the Riesz theorem. First we introduce a new function:

g(z) = ∑_{μ_{n}<μ}(2μ_{n}f_{n} sin μ_{n}h)u_{n}(z), z ∈ G.

It belongs to L_{p}(G) ⊂ L_{2}(G), and its Fourier coefficients (with respect to the system {u_{n}(z)}_{n=1}^{∞}) are given by

g_{n} = \{ \begin{matrix}2μ_{n}f_{n} sin μ_{n}h & if μ_{n} < μ, \\ 0 & if μ_{n} ≥ μ. \end{matrix}

Let r ∈ (1, 2] be a number such that 1/p + 1/r = 1. By estimates (7) and (10), we obtain

( ∑_{n=1}^{∞} |g_{n}|^{r})^{1/r} ≤ C∥f∥_{BV (G)}( ∑_{μ_{n}<μ} 1)^{1/r}

≤ C∥f∥_{BV (G)}( ∑_{k=1}^{[μ]}( ∑_{k≤μ_{n}<k+1} 1))^{1/r} ≤ 2^{1/r}CA^{1/r}∥f∥_{BV (G)} · μ^{1/r}. (17)

Hence, we can use the second part of the Riesz theorem: from estimate (12) it follows that the inequalities

∥g∥_{L_{p}(K_{1})} ≤ ∥g∥_{L_{p}(G)} ≤ (C(G))^{2/r−1}( ∑_{n=1}^{∞} |g_{n}|^{r})^{1/r}

are valid. That is why we can conclude, by (17), that for the second member it holds:

∥ ∑_{μ_{n}<μ} 2μ_{n}f_{n}u_{n}(z) sin μ_{n}h∥_{L_{p}(K_{1})}^{p} ≤ C_{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}.  (18)

The same estimate holds for the third member:

∥ ∑_{μ̃_{n}<μ} 2μ̃_{n}f̃_{n}ũ_{n}(z) sin μ̃_{n}h∥_{L_{p}(K_{1})}^{p} ≤ C̃_{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}.  (19)

In the case of the fourth member, using estimates (7)-(8), (10), and the Hölder inequality, we obtain

∫_{K_{1}} | ∑_{μ_{n}<μ} f_{n} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − x − h) dξ|^{p} dz ≤

∫_{K_{1}}( ∑_{μ_{n}<μ} |μ_{n}f_{n}||\frac{1}{μ_{n}} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − x − h) dξ|)^{p} ≤

( ∑_{μ_{n}<μ} |μ_{n}f_{n}|^{r})^{p/r} ∫_{K_{1}}( ∑_{μ_{n}<μ}|\frac{1}{μ_{n}} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − x − h) dξ|^{p}) dz ≤

C_{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}∥q∥_{L_{1}(G)}^{p}(c − a)( ∑_{μ_{n}<μ} \frac{1}{μ_{n}^{p}} ) ≤ C_{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}.

Here 1/p + 1/r = 1. Also we have in mind that

∑_{μ_{n}<μ} \frac{1}{μ_{n}^{p}} ≤ ∑_{k=1}^{∞}( ∑_{k≤μ_{n}<k+1} \frac{1}{μ_{n}^{p}} ) ≤ A ∑_{k=1}^{∞} \frac{1}{k^{p}} .

Therefore, the following estimate holds:

∫_{K_{1}} | ∑_{μ_{n}<μ} f_{n} ∫_{z}^{z+h} q(ξ)u_{n}(ξ) cos μ_{n}(ξ − x − h) dξ|^{p} dz ≤

≤ C_{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}. (20)

The estimates of the same form, with possibly different constants C_{p}, are valid for the last three members on the right-hand side of (15). So we get, by (14)–(16) and (18)–(20), the final estimate

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{L_{p}((a,c))}^{p} ≤ C_{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}.  (21)

Using the analogous argument, one can prove the estimate

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{L_{p}((d,b))}^{p} ≤ C_{p}∥f∥_{BV (G)} · μ^{(1−1/p)p}.  (22)

Finally, by estimate (5), we obtain

∥σ′_{μ}(x, f) − σ̃′_{μ}(x, f)∥_{L_{p}(K)}^{p} ≤ (b − a)C(K)^{p}∥f∥_{BV (G)}^{p} · μ^{(1−1/p)p}.  (23)

Now, the estimate (6) follows from (13) and (21)–(23). The theorem is proved.

REFERENCES

[1] M. A. Naímark, Lineínye differentsial’nye operatory, Nauka, Moskva, 1969.

[2] A. M. Minkin, Equiconvergence theorems for differential operators, Journal of Math. Sci., 96, 6 (1999), 3631–3715.

[3] V. A. Il’in, I. Ío, Otsenka raznosti chastichnykh summ razlozhenií, otvechayushchikh dvum proizvol’nym neotritsatel’nym samosopyazhennym rasshireniyam dvukh operatorov tipa Shturma-Liuvillya, dlya absolyutno nepreryvnoí funktsii, Diff. uravneniya 15, 7 (1979), 1175– 1193.

[4] I. S. Lomov, Ob approksimatsii funktsií na otrezke spektral’nymi razlozheniyami operatora Shredingera, Vestn. Mosk. un-ta., Ser.1, mat., meh., 4 (1995), 43–54.

[5] I. S. Lomov, O skorosti ravnoskhodimosti ryadov Fur’e po sobstvennym funktsiyam operatorov Shturma-Liuvillya v integral’noí metrike, Diff. uravneniya 18, 9 (1982), 1480–1493.

[6] I. Ío, N. Lazhetich, Otsenka raznosti proizvodnykh chastichnykh summ razlozhenií, otvechayushchikh dvum proizvol’nym neotritsatel’nym samosopyazhennym rasshireniyam dvukh operatorov tipa Shturma-Liuvillya, dlya absolyutno nepreryvnoí funktsii, Diff. uravneniya 16, 4 (1980), 598–619.

[7] N.L. Lažetić, O. Djordjević, A local uniform estimate for the difference of derivatives of spectral expansions corresponding to self-adjoint one-dimensional Schrödinger operators, submitted.

[8] V. A. Il’in, I. Ío, Ravnomernaya otsenka sobstvennykh funktsií i otsenka sverkhu chisla sobstvennykh znachenií operatora Shturma-Liuvillya s potentsialom iz klassa L^{p}, Diff. uravneniya 15, 7 (1979), 1165–1174.

[9] L. V. Kritskov, Ravnomernayua otsenka poryadka prisoedinennykh funktsií i raspredelenie sobbstvennykh znachenií odnomernogo operatora Shredingera, Diff. uravneniya 25, 7 (1989), 1121–1129.

[10] N. Lazhetich, Ravnomernye otsenki dlya proizvodnyh sobstvennyh funktsií samosopryazhennogo operatora Shturma-Liuvillya, Diff. uravneniya 17, 11 (1981), 1978–1983.

[11] A. Zygmund, Trigonometric Series, Volume I and II, Cambridge, 1968.

(received 27.12.2002)

Faculty of Mathematics, Studentski trg 16, P.O. Box 550, 11000 Beograd, Yugoslavia E-mail: lazetic@matf.bg.ac.yu Faculty of Ogranizational Sciences, Jove Ilića 154, 11000 Beograd, Yugoslavia E-mail: oliveradj@fon.bg.ac.yu