Dimension free estimates for the bilinear Riesz transform
Let \(R_k, k=1, 2, \ldots, n\), denote the bilinear Riesz transforms given by \[ R_k(f,g)(x)= b_n \lim_{\varepsilon \to 0}\int_{|y|>\varepsilon} f(x-y)g(x+y)\frac{y_k}{|y|^{n+1}}dy, \] where \(b_n\) is chosen so that \(R_k\) has operator norm 1 as a bilinear map from \(L^2(\mathbb R^n) \times L^2(\mathbb R^n) \mapsto L^1(\mathbb R^n)\). Using the method of rotations and some transference results, the authors prove that, for \(1< p_1, p_2 < \infty\) and \(1/p_1+1/p_2=1/p_3\), (i) if \(p_3 \geq 1\), then \(\|R_k(f,g)\|_{p_3} \leq C \|f\|_{p_1} \|g\|_{p_2}\); (ii) if \(p_3 > 1\), then \(\|(\sum_{k=1}^n |R_k(f,g)|^2)^{1/2}\|_{p_3} \leq C \|f\|_{p_1} \|g\|_{p_2}\), where the constant \(C\) is independent of \(n\).
scientific article; zbMATH DE number 5615095
scientific article