Wavelet integral operator on weighted Besov spaces

WAVELET INTEGRAL OPERATOR ON WEIGHTED BESOV SPACES Nguyen Minh Chuong and Dao Van Duong Institute of Mathematics, Vietnam Academy of Sciences and Technology Abstract. In this paper wavelet integral operator is studied in Besov spaces and weighted Besov spaces. MSC: 42C10, 42C15 Keywords: wavelet integral operator, wavelets, basic wavelets, Besov spaces, weighted Besov spaces, properties. 1 Introduction Wavelet integral operator is a new exciting and powerful tool for solving difficult problems in mathematics, physics, engineering. This class of operators is a new one of the well-know pseudodifferential, paradifferential operators. In terms of wavelets, wavelet integral operators, various very important functional spaces, such as H .. older, Zygmund, Sobolev, Besov, Hardy, BMO, VMO, have new characterization. Wavelet theory plays a great role not only in deterministic but also in stochastic analysis as well. Due to wavelet theory, one gets great successes in various very important practice areas such as image processing, pattern recognition, computer vision, etc... The most widespread application of the wavelet integral operator so far has been data compression (see[6,7,9,10] and references therein). Wavelet integral operator is now developed on ultrametric spaces (see [1, 9]). Wavelet theory on Archimedean and Non-Archimedean fields will bring to us greater and greater successes in investigating sciences and technologies. It is well known that although Besov spaces appeared much later than Sobolev spaces, but they include Sobolev, Zygmund and H .. older spaces as particular cases, and in nowadays they play a more important role due to their approximation properties. Wavelet integral operators have been studied in Sobolev spaces Hs,p in [4, 5]. Here we will discuss these operators on Besov spaces and weighted Besov spaces. 2 Some defenitions and properties For an arbitrary f ∈ Lp(Rn), 1 6 p 6 ∞, we define the Lp(Rn)-modulus of continuity as follows ωp(f, h) = ‖f(·+ h)− f(·)‖p. Let us denote the Besov space by B α,q p (Rn), 1 6 p, q 6 ∞ with 0 < α < 1, 1 6 p, q 6 ∞ defined as Bα,qp (R n) =  f ∈ Lp(Rn) : ∫ Rn [ωp(f, h)] q dh |h|n+αq <∞   1 for q <∞, and Bα,∞p (R n) = {f ∈ Lp(Rn) : |h|−αωp(f, h) ∈ L ∞(Rn\{0})} for q = ∞, with |h|, an Euclidean norm of h ∈ Rn. Then Bα,qp (Rn) is a Banach space with norm ‖f‖Bα,qp = ‖f‖p +  ∫ Rn [ωp(f, h)] q dh |h|n+αq   1 q for q <∞, and ‖f‖Bα,∞p = ‖f‖p + ‖|h| −αωp(f, h)‖∞ for q = ∞. We will use the class of basic wavelets by Yves Meyer [10] defined as nontrivial functions ψ ∈ L1(Rn) ∩ L2(Rn) such that ∫ Rn ψ(x)dx = 0, Cψ = (2pi) n ∞∫ 0 |ψ̂(aξ)|2 da a , where Cψ 6= 0 is a constant for every ξ 6= 0. With this basic we consider the wavelet integral operator (Wψf)(a, b) = 1 2n √ Cψ 1√ |a|n ∫ Rn f(t)ψ( t− b a )dt, where a ∈ R\{0}, b ∈ Rn. Theorem 2.1.For each fixed a 6= 0, the operator Wψ : B α,q p (R n) −→ Bα,qp (R n) f 7−→ (Wψf)(a, ·) is linear and bounded. Moreover the following estimate holds true ‖(Wψf)(a, ·)‖Bα,qp 6 |a| n 2 2n √ Cψ ‖ψ‖1‖f‖Bα,qp . (1) Proof. The linearity of Wψ is obvious. For each fixed a 6= 0 it will be proved that (Wψf)(a, ·) ∈ B α,q p (Rn). Setting ψa(x) = 1√ |a|n ψ( x a ), we have (Wψf)(a, ·) = 1 2n √ Cψ (ψ−a∗ f)(·). By Young inequality ‖(Wψf)(a, ·)‖p 6 |a| n 2 2n √ Cψ ‖ψ‖1‖f‖p. (2) With the change of variable x = t− b a , we get (Wψf)(a, b) = |a| n 2 2n √ Cψ ∫ Rn f(ax+ b)ψ(x)dx. 2 For 0 < α < 1, 1 6 p <∞. By Minkowski inequality, we obtain ωp((Wψf)(a, ·), h) = ‖(Wψf)(·+ h)− (Wψf)(·)‖p = |a| n 2 2n √ Cψ  ∫ Rn ∣∣∣∣∣∣ ∫ Rn [f(ax+ b+ h)− f(ax+ b)]ψ(x)dx ∣∣∣∣∣∣ p db   1 p 6 |a| n 2 2n √ Cψ ∫ Rn |ψ(x)|  ∫ Rn |f(ax+ b+ h)− f(ax+ b)|pdb   1 p dx 6 |a| n 2 2n √ Cψ ‖ψ‖1ωp(f, h). Therefore for q <∞, we get  ∫ Rn [ωp((Wψf)(a, ·), h)] q dh |h|n+αq   1 q 6 |a| n 2 2n √ Cψ ‖ψ‖1  ∫ Rn [ωp(f, h)] q dh |h|n+αq   1 q . (3) From (2.2) and (2.3) , it follows that |(Wψf)(a, ·)‖Bα,qp 6 |a| n 2 2n √ Cψ ‖ψ‖1‖f‖Bα,qp . For p = ∞, q = ∞ we also obtain analogous result. 3 The weighted Besov space In this paper let us use the H .. ormander's weight. This is the function k(x) > 0,∀x ∈ Rn, and there exist two constants M > 0 and N ∈ R such that k(x+ y) 6 (1 +M |x|)Nk(y), ∀ x, y ∈ Rn. We use the following notations Lp,k(Rn) =  f ∈ Lp(Rn) : ‖f‖p,k =  ∫ Rn k(x)|f(x)|pdx   1 p <∞   for 1 6 p <∞, and L∞,k(Rn) = {f ∈ Lp(Rn) : ‖f‖∞,k = ‖kf‖∞ <∞}, for p = ∞. ωp,k(f, h) = ‖f(·+ h)− f(·)‖p,k. The weighted Besov space B α,q p,k (Rn) is defined then as follows B α,q p,k (Rn) =  f ∈ Lp,k(Rn) : ∫ Rn [ωp,k(f, h)] q dh |h|n+αq <∞   for q <∞, and B α,∞ p,k (Rn) = { f ∈ Lp,k(Rn) : |h|−αωp,k(f, h) ∈ L ∞(Rn\{0}) } 3 for q = ∞. Then the space Bα,qp,k (R n) is a Banach space with the norm ‖f‖Bα,q p,k = ‖f‖p,k +  ∫ Rn [ωp,k(f, h)] q dh |h|n+αq   1 q , for q <∞, ‖f‖Bα,∞ p,k = ‖f‖p,k + ‖|h| −αωp,k(f, h)‖∞, for q = ∞. We shall consider the above mentioned basic wavelet ψ but with a compact support lying in a ball B(0, r) with radius r, centered at 0. Theorem 3.1. For each fixed a 6= 0, the operator Wψ : B α,q p,k (R n) −→ Bα,qp,k (R n) f 7−→ (Wψf)(a, ·) is linear and bounded. Moreover the following estimate holds true |(Wψf)(a, ·)‖Bα,q p,k 6 (1 +Mr|a|) N p 2n √ Cψ |a| n 2 ‖ψ‖1‖f‖Bα,q p,k . (4) Proof. We have known (Wψf)(a, ·) = 1 2n √ Cψ (ψ−a ∗ f)(·). The theorem will be proved for 1 6 p, q < ∞. The case p = ∞, q = ∞ are discussed quite analogously. The Minkowski inequality yields ‖(Wψf)(a, ·)‖p,k = 1 2n √ Cψ  ∫ Rn k(b)|(ψ−a ∗ f)(b)| pdb   1 p = 1 2n √ Cψ  ∫ Rn k(b) ∣∣∣∣∣∣ ∫ Rn f(b− x)ψ−a(x)dx ∣∣∣∣∣∣ p db   1 p 6 1 2n √ Cψ ∫ Rn |ψ−a(x)|dx  ∫ Rn k(b)|f(b− x)|pdb   1 p . By the weighted function k(x), setting y = b− x, we get k(b) 6 (1 +M |x|)Nk(b− x), and  ∫ Rn k(b)|f(b− x)|pdb   1 p 6 (1 +M |x|) N p  ∫ Rn k(y)|f(y)|pdy   1 p , 4 therefore ‖(Wψf)(a, ·)‖p,k 6 1 2n √ Cψ  ∫ Rn (1 +M |x|) N p |ψ−a(x)|dx    ∫ Rn k(y)|f(y)|pdy   1 p 6 1 2n √ Cψ   ∫ |x|6r|a| (1 +M |x|) N p |ψ−a(x)|dx    ∫ Rn k(y)|f(y)|pdy   1 p 6 (1 +Mr|a|) N p 2n √ Cψ |a| n 2 ‖ψ‖1‖f‖p,k. (5) Moreover it is obvious that ωp,k((Wψf)(a, ·), h) = ‖(Wψf)(·+ h)− (Wψf)(·)‖p,k = |a| n 2 2n √ Cψ  ∫ Rn k(b) ∣∣∣∣∣∣ ∫ Rn [f(ax+ b+ h)− f(ax+ b)]ψ(x)dx ∣∣∣∣∣∣ p db   1 p 6 |a| n 2 2n √ Cψ ∫ Rn |ψ(x)|dx  ∫ Rn k(b)|f(ax+ b+ h)− f(ax+ b)|pdb   1 p 6 |a| n 2 2n √ Cψ ∫ Rn (1 +M |ax|) N p |ψ(x)|dx× ×  ∫ Rn k(ax+ b)|f(ax+ b+ h)− f(ax+ b)|pdb   1 p 6 |a| n 2 2n √ Cψ ωp,k(f, h) ∫ |x|6r (1 +M |ax|) N p |ψ(x)|dx 6 (1 +Mr|a|) N p 2n √ Cψ |a| n 2 ‖ψ‖1ωp,k(f, h). So we obtain ∫ Rn [ωp,k((Wψf)(a, ·), h)] q |h|n+αq dh   1 q 6 (1 +Mr|a|) N p 2n √ Cψ |a| n 2 ‖ψ‖1  ∫ Rn [ωp,k(f, h)] q |h|n+αq dh   1 q (6) By (3.2), (3.3), we get |(Wψf)(a, ·)‖Bα,q p,k 6 (1 +Mr|a|) N p 2n √ Cψ |a| n 2 ‖ψ‖1‖f‖Bα,q p,k . Thus theorem 2 isproved. 5 REFERENCES [1] M. V. Altaiski, 1997. p-adic wavelet decomposition vs Fourier analysis on spheres. Indian J. pure appl. Math, 28(2), pp. 197-205. [2] Nguyen Minh Chuong, Nguyen Minh Tri and Le Quang Trung, 2000. Theory of partial differential equations. Sci. and Techn. Publishng House. [3] Nguyen Minh chuong, 1981. Prabolic pseudodifferential operators of variable order. Dokl. Akad. Nauk SSSR 256, No 6, pp. 1308-1312. [4] Nguyen Minh Chuong, Ta Ngoc Tri, 2000. The intergal wavelet tranform in Lp(R n), 1 6 p 6 ∞, Fractional Calculus and Applied Analysis (An Internat. J. of Math. and Appl.) 3, No 2, pp. 133-140. [5] Nguyen Minh Chuong and Bui Kien Cuong, 2004. Convergence estimates of Galerkin- Wavelet solutions to a Cauchy problem for a class of periodic pseudodifferential equation. Proc. AMS 132, No 12, pp. 3589-1597. [6] I. Daubechies, 1992. Ten lectures on wavelets. NSF-CBMS Regional Conference Series in Apll. Math., 61, SIAM Publication, Philadelphia. [7] David L. Donoho, 1993. Nonlinear wavelet methods for recovery of signal, densities, and spectra from indirect and noisy data. Proc. Sympos. Appl. Math. 47, AMS, Providence, RI. [8] Lars H .. ormander, 1964. Linear partial differential operators. Springer-Verlag, Berline- Gothigen-Heidelberg. [9] S.V.Kozyrev, 2002. Wavelet thoery as p-adic spectral analysis. Izv. Ross. Akad. Nauk Ser. Mat 66, No 2, 1490158. [10] Y. Meyer, 1990. Ondelettes et Operators. I, II, III, Hermann, Pari . 6