( ) ( ) [2],[3],[4],[5],[6],[7]

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1 ( ) ( ) Lebesgue Takagi, - []., [2],[3],[4],[5],[6],[7].,. [] M. Hata and M. Yamaguti, The Takagi function and its generalization, Japan J. Appl. Math., (984), [2] T. Sekiguchi and Y. Shiota, A generalization of Hata Yamaguti s results on the Takagi function, Japan J. Indust. Appl. Math., 8(99), [3] T. Okada, T. Sekiguchi and Y. Shiota, Applications of binomial measures to power sums of digital sums, J. Number Theory, 52(995), pp [4] T. Okada, T. Sekiguchi and Y. Shiota, A generalization of Hata Yamaguti s results on the Takagi function II: Multinomial case, Japan J. Indust. Appl. Math., 3(996), pp [5] Z. Kobayashi, Digital Sum Problems for the Gray Code Representation of Natural Numbers, Interdisciplinary Information Sciences, 8(2002), [6] K. Muramoto, T. Okada, T. Sekiguchi and Y. Shiota, Power and exponential sums of digital sums with information per digits, Math. Rep. Toyama Univ., 26(2003), [7] Y. Kamiya, T. Okada, T. Sekiguchi and Y. Shiota, Digital Sum Problems for Generlized Code Systems, manuscript.

2 Real Analysis Symposium 204 October 3st - November 2nd, 204 Toyama University On the generalized σ-lipschitz spaces and the generalized fractional integrals Katsuo Matsuoka College of Economics, Nihon University November 2, 204 For r > 0, let Q r = {y R n : y < r} or Q r = {y = (y, y 2,, y n ) R n : max i n y i < r}, and for x R n, let Q(x, r) = x + Q r = {x + y : y Q r }. For a measurable set G R n, we denote the Lebesgue measure of G by G and the characteristic function of G by χ G. Further, for a function f L loc (Rn ) and a measurable set G R n with G > 0, let f G = f(y) dy = f(y) dy and let G G G N 0 = N {0}. First, we recall the definitions of the non-homogeneous central Morrey space B p,λ (R n ) and the λ-central mean oscillation (λ-cmo) space CMO p,λ (R n ). Definition. For p < and n/p λ <, { ( ) /p B p,λ (R n ) = f L p loc (Rn ) : f B p,λ = sup f(y) p dy < }. r r λ Q r On the other hand, we introduce the new function space, i.e., the generalized σ-lipschitz space Lip (d) β,σ (Rn ) (see Nakai and Sawano (202), M. (to appear); cf. Komori-Furuya, M., Nakai and Sawano (203)). Definition 2. Let U = R n or U = Q r with r > 0. For d N 0 and 0 β, the continuous function f will be said to belong to the generalized Lipschitz space on U, i.e., Lip (d) β (U) if and only if where k h f (d) Lip = sup β (U) x,x+h U,h 0 h β d+ h f(x) <, is a difference operator, which is defined inductively by 0 hf = f, hf = h f = f( + h) f( ), k hf = k h f( + h) k h f( ), k = 2, 3,. Definition 3. For d N 0, 0 β and 0 σ <, the continuous function f will be said to belong to the generalized σ-lipschitz (σ-lip) space, i.e., Lip (d) β,σ (Rn ) if and only if f (d) Lip = sup β,σ r r f <. σ Lip (d) β (Qr) In particular, Lip β,σ (R n ) = Lip (0) β,σ (Rn ) and BMO (d) σ (R n ) = Lip (d) 0,σ(R n ), BMO σ (R n ) = BMO (0) σ (R n ).

3 Next we recall the definition of modified fractional integral Ĩα. Definition 4. For 0 < α < n, ( Ĩ α f(x) = f(y) R x y χ ) Q (y) dy. n n α y n α Recently, in M. and Nakai (20), from the B σ -Morrey-Campanato estimate for Ĩ α we obtained the following as the corollary. Theorem (M. and Nakai (20); cf. Komori-Furuya and M. (200)). Let 0 < α < n, n/α < p < and n/p λ < α. If β = α n/p and σ = λ + n/p, then Ĩ α : B p,λ (R n ) Lip β,σ (R n ). Now we define the generalized fractional integral Ĩα,d. Definition 5. For 0 < α < n and d N 0, we define the generalized fractional integral (of order α), i.e., Ĩα,d, as follows : For f L loc (Rn ), Ĩ α,d f(x) = f(y) R K α(x y) x l l! (Dl K α )( y) ( χ Q (y)) dy, n where {l: l d} K α (x) = x n α and for x = (x, x 2,, x n ) R n and l = (l, l 2,, l n ) N n 0, l = l + l l n, x l = x l x l 2 2 x ln n and D l is the partial derivative of order l, i.e., D l = ( / x ) l ( / x 2 ) l2 ( / x n ) ln. Then as one of the results for a generalized fractional integral Ĩα,d we can get the following estimate on B p,λ (R n ), which extends Theorem. Theorem 2 ([M]). Let 0 < α < n, n/α < p <, d N 0 and n/p + α + d λ + α < d +. If β = α n/p and σ = λ + n/p, then References Ĩ α,d : B p,λ (R n ) Lip (d) β,σ (Rn ). [M] K. Matsuoka, Generalized fractional integrals on central Morrey spaces and generalized σ - Lipschitz spaces, in Current Trends in Analysis and its Applications: Proceedings of the 9th ISAAC Congress, Kraków 203, Springer Proceedings in Mathematics and Statistics, Birkhäuser Basel, to appear.

4 A note on Herz type inequalities Gaku Sadasue(Osaka Kyoiku University) Let (Ω, F, P ) be a probability space, and {F n } n 0 a nondecreasing sequence of sub-σ-algebras of F such that F = σ( n F n). The expectation operalor and the conditional expectation operator relative to F n are denoted by E and E n, respectively. A sequence of integrable random variables f = (f n ) n 0 is called a martingale relative to {F n } n 0 if, for every n, f n is F n measurable and satisfies E n [f m ] = f n (n m). If f L p, p [, ), then (f n ) n 0 with f n = E n f is an L p -bounded martingale and converges to f in L p ([6]). For this reason a function f L and the corresponding martingale (f n ) n 0 with f n = E n f will be denoted by the same symbol f. We now introduce two martingale Hardy spaces. Let M be the set of all martingale f = (f n ) n 0 relative to {F n } n 0 such that f 0 = 0. Then the maximal function of a martingale f are defined by f n = sup 0 m n f m, f = sup f n. n 0 Denote by Λ the collection of all sequences (λ n ) n 0 of nondecreasing, nonnegative and adapted functions, and set λ = lim n λ n. For f M and 0 < p <, let Λ[P p ](f) = {(λ n ) n 0 Λ : f n λ n, λ L p }. We define two martingale spaces by Hp = {f M : f H p = f p < }, P p = {f M : f Pp = inf λ p < }. (λ n ) n 0 Λ[P p ](f) We next introduce two martingale BMO spaces. For f L, let f BMO = sup E n f E n f, n Then, we define two martingale BMO spaces: BMO = {f L : f BMO < }, BMO f BMO = sup E n f E n f. n = {f L : f BMO < }.

5 In [3], Herz discussed the duality between H and BMO and proved the following inequality for martingales: E[fφ] 2 f P φ BMO (f L, φ BMO). This inequality is generalized by many authors. In this talk, we give an extension of Herz type inequality with a different proof. References. [] C. Fefferman, E. M. Stein, H p spaces of several variables. Acta Math. 29 (972), no. 3-4, [2] A. M. Garsia, Martingale inequalities: Seminar notes on recent progress. Mathematics Lecture Notes Series. W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 973. viii+84 pp. [3] C. Herz, Bounded mean oscillation and regulated martingales. Trans. Amer. Math. Soc. 93 (974), [4] S. Ishak and J. Mogyoródi, On the P Φ -spaces and the generalization of Herz s and Fefferman s inequalities. II. Studia Sci. Math. Hungar. 8 (983), no. 2-4, [5] R. L. Long, Martingale spaces and inequalities, Peking University Press, Beijing, 993. [6] J. Neveu, Discrete-parameter martingales, North-Holland, Amsterdam, 975. [7] F. Weisz, Probability theory and applications, 47 75, Math. Appl., 80, Kluwer Acad. Publ., Dordrecht, 992. [8] F. Weisz, Martingale Hardy spaces and their applications in Fourier analysis, Lecture Notes in Mathematics, 568, Springer-Verlag, Berlin,

6 204 B u w(e)- The purpose of this talk is to introduce Bw(E)-funciton u spaces which unify many function spaces, Lebesgue, Morrey-Campanato, Lipschitz, B p, CMO, local Morreytype spaces, etc. We investigate the interpolation property of Bw(E)-funciton u spaces and apply it to the boundedness of linear and sublinear operators, for example, the Hardy-Littlewood maximal operator, singular and fractional integral operators, and so on, which contains previous results and extends them to Bw(E)-funciton u spaces.