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## Studia Mathematica

1997 | 125 | 1 | 35-56
Tytuł artykułu

### Pointwise multipliers on weighted BMO spaces

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $ϕ: X×ℝ_{+} → ℝ_{+}$, we denote by $bmo_{ϕ,p}(X)$ the set of all functions $f ∈ L^{p}_{loc}(X)$ such that $sup_{a ∈ X, r>0} 1/ϕ(a,r) (1/μ(B(a,r)) ʃ_{B(a,r)} |f(x) -f_{B(a,r)}|^p dμ)^{1/p} < ∞$, where B(a,r) is the ball centered at a and of radius r, and $f_{B(a,r)}$ is the integral mean of f on B(a,r). Let $bmo_{ϕ}(X) = bmo_{ϕ,1}(X)$ and $bmo(X) = bmo_{1,1}(X)$. In this paper, we characterize $PWM(bmo_{ϕ1,p_1}(X), bmo_{ϕ2,p_2}(X))$. The following are examples of our results. $PWM(bmo_{(log(1/r))^{-α}}(𝕋^n),bmo_{(log(1/r))^{-β}}(𝕋^n)) = bmo_{(log(1/r))^{α-β-1}}(𝕋^n)$, 0≤β < α < 1, $PWM (bmo_{(log(1/r))^{-1}}(𝕋^n),bmo(𝕋^n)) = bmo_{(log log(1/r))^{-1}}(𝕋^n),$ $PWM (bmo(ℝ^n),bmo_{log(|a|+r+1/r),p}(ℝ^n)) = bmo(ℝ^n)$, 1 < p < ∞, etc.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
35-56
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-04-05
poprawiono
1997-02-10
Twórcy
autor
• Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka 582, Japan
Bibliografia
• [1] H. Aimar, Singular integrals and approximate identities on spaces of homogeneous type, Trans. Amer. Math. Soc. 292 (1985), 135-153.
• [2] S. Bloom, Pointwise multipliers of weighted BMO spaces, Proc. Amer. Math. Soc. 105 (1989), 950-960.
• [3] R. R. Coifman et G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
• [4] R. R. Coifman et G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
• [5] Y. Gotoh, On multipliers for $BMO_ϕ$ on general domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 339-354.
• [6] S. Janson, On functions with conditions on the mean oscillation, Ark. Mat. 14 (1976), 189-196.
• [7] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426.
• [8] P. G. Lemarié, Algèbres d'opérateurs et semi-groupes de Poisson sur un espace de nature homogène, Publ. Math. Orsay 84-3 (1984).
• [9] R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257-270.
• [10] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
• [11] B. Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1974), 101-106.
• [12] E. Nakai, On the restriction of functions of bounded mean oscillation to the lower dimensional space, Arch. Math. (Basel) 43 (1984), 519-529.
• [13] E. Nakai, Pointwise multipliers for functions of weighted bounded mean oscillation, Studia Math. 105 (1993), 105-119.
• [14] E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan 37 (1985), 207-218.
• [15] E. Nakai and K. Yabuta, Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type, Math. Japon. 46 (1997), to appear.
• [16] S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 19 (1965), 593-608.
• [17] D. A. Stegenga, Bounded Toeplitz operators on $H^1$ and applications of the duality between $H^1$ and the functions of bounded mean oscillation, Amer. Math. 98 (1976), 573-589.
• [18] K. Yabuta, Pointwise multipliers of weighted BMO spaces, Proc. Amer. Math. Soc. 117 (1993), 737-744.
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Bibliografia
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