Boundedness of high order Riesz-Bessel transformations generated by generalized shift operator on B\(a\) spaces. (English) Zbl 1059.47053
It is shown that the Riesz–Bessel transformations \(R_{B_j}\), \(j=1,2,\dots, n-1, \) are bounded on the space \(B_a\), where
\[
B_a= \{ f: f \in \cap_{m=1}^{\infty} B_m, R_f>0 \},
\]
\[ \| f\| _{B_a}=\inf_{\alpha>0}\{ 1/\alpha: I(f,\alpha)\leq 1\}, \]
\[ B_m= L_{p_m,\nu} (\mathbb{R}_n^+)=\left\{ f: \| f\| _{ L_{p_m, \nu}}\equiv \bigg(\int_{\mathbb{R}^+_n} | f(x)| ^{p_m} x_n^{2 \nu} dx \bigg)^{1/p_m} < \infty \right\} \] and \(R_f\) is the radius of the convergence of the series \[ I(f,\alpha)=\sum_{m=1}^{\infty}a_m \| f\| _{B_m}^m \alpha^m, \] if and only if there exist constants \(\alpha\) and \(\beta\) such that the sequence \(\{p_m\}\) satisfies \(1<\alpha<p_m<\beta\) for all \(a_m\neq 0\).
\[ \| f\| _{B_a}=\inf_{\alpha>0}\{ 1/\alpha: I(f,\alpha)\leq 1\}, \]
\[ B_m= L_{p_m,\nu} (\mathbb{R}_n^+)=\left\{ f: \| f\| _{ L_{p_m, \nu}}\equiv \bigg(\int_{\mathbb{R}^+_n} | f(x)| ^{p_m} x_n^{2 \nu} dx \bigg)^{1/p_m} < \infty \right\} \] and \(R_f\) is the radius of the convergence of the series \[ I(f,\alpha)=\sum_{m=1}^{\infty}a_m \| f\| _{B_m}^m \alpha^m, \] if and only if there exist constants \(\alpha\) and \(\beta\) such that the sequence \(\{p_m\}\) satisfies \(1<\alpha<p_m<\beta\) for all \(a_m\neq 0\).
Reviewer: Alexander Meskhi (Pisa)
MSC:
47G10 | Integral operators |
45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
References:
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