Artículo
Abstract:
A measure μ on Rn is called locally and uniformly h-dimensional if μ(Br(x)) ≤ h(r) for all x ∈ Rn and for all 0 < r < 1, where h is a real valued function. If f ∈ L2(μ) and Fμf denotes its Fourier transform with respect to μ, it is not true (in general) that Fμf ∈ L2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L2(μ) the L2-norm of its Fourier transform restricted to a ball of radius r has the same order of growth as rnh(r-1) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L2(μ)-norm of f (Theorem 3.2). By imposing certain restrictions on the measure μ, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which h(x) = xα.
Registro:
| Documento: | Artículo |
| Título: | A fractal Plancherel theorem |
| Autor: | Molter, U.M.; Zuberman, L. |
| Filiación: | Depto. de Matemática FCEyN, Univ. de Buenos Aires, Cdad. Univ. Pab. I, Capital Federal, 1428, Argentina |
| Palabras clave: | Dimension; Fourier transform; Hausdorff measures; Plancherel |
| Año: | 2009 |
| Volumen: | 34 |
| Número: | 1 |
| Página de inicio: | 69 |
| Página de fin: | 86 |
| Título revista: | Real Analysis Exchange |
| Título revista abreviado: | Real Anal. Exch. |
| ISSN: | 01471937 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01471937_v34_n1_p69_Molter |
Referencias:
- Agmon, S., Hörmander, L., Asymptotic properties of solutions of differ-ential equations with simple characteristics (1976) J. Anal. Math, 30, pp. 1-38
- Benedetto, J.J., Lakey, J.D., The definition of the Fourier transform for weighted inequalities (1994) J. Funct. Anal, 120 (2), pp. 403-439
- Cabrelli, C., Mendivil, F., Molter, U., Shonkwiler, R., On the h-Hausdorff measure of Cantor sets (2004) Pacific J. Math, 217 (1), pp. 29-43
- Falconer, K.J., (1985) The Geometry of Fractal Sets, , Cambridge University Press, Cambridge
- Falconer, K.J., (1997) Techniques in Fractal Geometry, , John Wiley & Sons, New York
- Lau, K.-S., Fractal measures and mean p-variations (1992) J. Funct. Anal, 108 (2), pp. 427-457
- Lau, K.S., Wang, Y., (1997) Characterizations of Lp-solutions for the two scale dilation equations, , Preprint
- Mattila, P., (1995) Geometry of Sets and Measures in Euclidean Spaces, , Cambridge University Press, Cambridge
- Rogers, C.A., (1998) Hausdorff Measures, , Cambridge University Press, Cambridge, UK, 2nd Edition
- Strichartz, R., Fourier asymptotics of fractal measures (1990) J. Funct. Anal, 89, pp. 154-187
- Strichartz, R., Self-similar measures and their Fourier transforms I (1990) Indiana Univ. Math. J, 39 (3), pp. 797-817
- Strichartz, R., Self-similar measures and their Fourier transforms II (1993) Trans. Amer. Math. Soc
- Strichartz, R., Self-similarity in harmonic analysis (1994) J. Fourier Anal. Appl, 1 (1), pp. 1-37
Citas:
---------- APA ----------
Molter, U.M. & Zuberman, L. (2009) . A fractal Plancherel theorem. Real Analysis Exchange, 34(1), 69-86.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01471937_v34_n1_p69_Molter [ ]
---------- CHICAGO ----------
Molter, U.M., Zuberman, L. "A fractal Plancherel theorem" . Real Analysis Exchange 34, no. 1 (2009) : 69-86.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01471937_v34_n1_p69_Molter [ ]
---------- MLA ----------
Molter, U.M., Zuberman, L. "A fractal Plancherel theorem" . Real Analysis Exchange, vol. 34, no. 1, 2009, pp. 69-86.Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01471937_v34_n1_p69_Molter [ ]
---------- VANCOUVER ----------
Molter, U.M., Zuberman, L. A fractal Plancherel theorem. Real Anal. Exch. 2009;34(1):69-86.Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01471937_v34_n1_p69_Molter [ ]
