Dimension free estimates for the bilinear Riesz transform
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Autores: Óscar Blasco de la Cruz
, T. Alastair Gillespie
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 60, Fasc. 3, 2009, págs. 249-259
- Idioma: español
- DOI: 10.1007/bf03191370
- Enlaces
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Resumen
It is shown that for 1 < p1, p2 < 1, 1/p3 = 1/p1 + 1/p2, p3 ≥ 1 there existsC 1 (independent ofn) such that \left\| {R_k (f,g)} \right\|_{L^{p_3 } (\mathbb{R}^n )} \leqslant C_1 \left\| f \right\|_{L^{p_1 } (\mathbb{R}^n )} \left\| g \right\|_{L^{p_2 } (\mathbb{R}^n )} where R_k (f, g)(x) = b_n \mathop {\lim }\limits_{\varepsilon \to 0} \int_{\left| y \right| > \varepsilon } { f} (x - y)g(x + y)\frac{{y_k }}{{\left| y \right|^{n + 1} }}dy, andb_n is chosen so thatR_k has norm 1 as a bilinear map fromL^2(ℝ^n) ×L^2(ℝ^n) →L^1(ℝ^n). In the casep_3 > 1 it is even shown that \left\| {\left( {\sum\limits_{k = 1}^n {\left| {R_k (f, g)} \right|^2 } } \right)^{1/2} } \right\|_{L^{p_3 } (\mathbb{R}^n )} \leqslant C_2 \left\| f \right\|_{L^{p_1 } (\mathbb{R}^n )} \left\| g \right\|_{L^{p_2 } (\mathbb{R}^n )} for some constantC_2 independent of the dimension.
