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On a family of weighted spaces. (English) Zbl 0862.43002
Let \(G\) be a locally compact abelian group with dual group \(\widehat G\) and \(w\) a real-valued, measurable, locally bounded function on \(G\) such that \(1\leq w(x)\), \(w(x +y) \leq w(x) w(y)\) for \(x,y\in G\). Let \(L^p_w(G)\) \((1\leq p< \infty)\) be the set of functions \(f\) such that \(wf \in L^p(G)\) with norm \(|f|_{p,w} = |wf |_p\). The paper is devoted to study the weighted spaces \(A^{p,q}_{w,\omega} (G)\) of functions \(f\in L^p_\omega (G)\) with generalized Fourier transform \(\widehat f\in L^q_\omega (\widehat G)\) \((1\leq q< \infty)\) equipped with the sum norm \(|f|^{p,q}_{w, \omega} (G)= |f|^p_w + |\widehat f|^q_\omega\). Classical and generalized Fourier transforms are used to prove the properties of spaces \(A^{p,q}_{w,\omega} (G)\). Some of the results obtained generalize well-known assertions for the case of non-weighted space \(A_p(G) = A_{1,1}^{1,p'} (G)\).
Reviewer: A.A.Kilbas (Minsk)

MSC:
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
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References:
[1] DOMAR Y.: Harmonic analysis based on certain commutative Banach algebras. Acta Math. 96 (1956), 1-56. · Zbl 0071.11302 · doi:10.1007/BF02392357
[2] BRAUN W.-FEICHTINGER H. G.: Banach spaces of distributions having two modul structures. J. Funct. Anal. 51 (1983), 174-212. · Zbl 0515.46045 · doi:10.1016/0022-1236(83)90025-3
[3] FEICHTINGER H. G.: Banach convolution algebras of Wiener type. Functions, Series, Operators. Colloq. Math. Soc. János Bolyai 35, North-Holland, Amsterdam, 1980, pp. 509-523.
[4] FEICHTINGER H. G.: On a new Segal algebra. Mh. Math. 92 (1981), 269-289. · Zbl 0461.43003 · doi:10.1007/BF01320058 · eudml:178048
[5] FEICHTINGER H. G.: Un espace de Banach distributions temperees sur les grupes localement compacts abelians. C. R. Acad. Sci. Paris Sér. I 290 (1980), 791-794. · Zbl 0433.43002
[6] FEICHINGER H. G.: On a class of convolution algebras of functions. Ann. Inst. Fourier (Grenoble) 27 (1977), 135-162.
[7] FEICHTINGER H. G.: Compactness in · Zbl 0515.46044 · doi:10.1016/0022-247X(84)90173-2
[8] FEICHTINGER H. G.-GÜRKANLI A. T.: On a family of weighted convolution algebras. Internat. J. Math. Math. Sci. 13 (1990), 517-526. · Zbl 0752.43001 · doi:10.1155/S0161171290000758 · eudml:46728
[9] GULICK S. L.-LIU T. S.-Van ROOIJ A. C M.: Group algebra modules 2. Canad. J. Math. 19 (1967), 151-173. · Zbl 0148.12004 · doi:10.4153/CJM-1967-008-4
[10] GÜRKANLI A. T.: Some results in the weighted \(A_p(\mathbb R^n)\) spaces. Demonstratio Math. 19 (1986), 825-830 (517-526).
[11] KESAVA MURTHY G. N.-UNNI K. R.: Multipliers on Weighted Space. Lecture Notes in Math. 399, Springer, New York-Berlin, 1973. · Zbl 0302.42005
[12] KÖTHE G.: Topological Vector Spaces. Springer Verlag, Berlin-Heidelberg-New York, 1969. · Zbl 0179.17001
[13] LAI H. C.: On some properties of \(A^p(G)\) algebras. Proc. Japan Acad. 45 (1969), 577-581. · Zbl 0186.46203
[14] LARSEN R.-LIU T. S.-WANG J. K.: On functions with Fourier transforms in Lp. Michigan Math. J. 11 (1964), 369-378. · Zbl 0123.09803
[15] LARSEN R.: Introduction to the Theory of Multipliers. Springer Verlag, Berlin-Heidelberg-New York, 1971. · Zbl 0213.13301
[16] LOOMIS L. H.: An Introduction to Abstract Harmonic Analysis. D. Van Nostrant Company, INC, Toronto-New York-London, 1953. · Zbl 0052.11701
[17] MARTIN J. C.-YAP L. Y. H.: The algebra of functions with Fourier transform in \(L_p\). Proc. Amer. Math. Soc. 24 (1970), 217-219. · Zbl 0187.38701 · doi:10.2307/2036732
[18] REITER H.: Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, Oxford, 1968. · Zbl 0165.15601
[19] RUDIN W.: Fourier Analysis on Groups. Interscience Publishers, New York, 1962. · Zbl 0107.09603
[20] STEIN E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, N. J.-1970. · Zbl 0207.13501
[21] WANG H. C: Homogeneous Banach Algebras. Marcel Dekker INC, New York-Basel, 1977. · Zbl 0346.43003
[22] WARNER C. R.: Closed ideals in the group algebra \(L^1 (G) \cap L^2(G)\). Trans. Arner. Math. Soc. 121 (1966), 408-423. · Zbl 0139.30603 · doi:10.2307/1994487
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