Multivariate Hausdorff operators on the spaces \(L^p(\mathbb R^n)\)
If \(\mu\) is a \(\sigma\)-finite measure on \(\mathbb{R}^n\), \(c:\mathbb{R}^n\to \mathbb{C}\) is a Borel measurable function and if \({\mathcal A}= [a_{jk}]\) is a \(\mu\)-a.e. non-singular matrix of Borel measurable functions \(a_{jk}: \mathbb{R}^n\to\mathbb{C}\), \(j> 1\), \(k> 1\), then the multivariate Hausdorff operator \(H= H(\mu,c,{\mathcal A})\) is defined by \[ H(f)(x)= \int_{\mathbb{R}^n} c(t) f({\mathcal A}(t)x) d\mu(t). \] If \(1\leq p\leq \infty\), \((1/p)+(1/p^*)= 1\), then the results of this paper involve estimates and adjoint operator product formula in the form (i) \(\|H(f)\|_p= k_p\|f\|_p\), where \(k_p= \int_{\mathbb{R}^n}|c(t)|\;|\text{det }{\mathcal A}^1(t)|^{1/p} d\mu(t)\); (ii) \(\int_{\mathbb{R}^n}g(x)H(f)(x)dx=\int_{\mathbb{R}^n} f(x) H^*(g)(x) dx\); (iii) \(H^*= H(\mu, c|\text{det }{\mathcal A}^1|,{\mathcal A}^1)\); (iv) \(\|H^*\|_{p^*\to p^*}=\|H\|_{p\to p}\). The Fourier transform multiplier representation of \(H\) is also stated.
Fourier transform multiplier representation
scientific article; zbMATH DE number 1836764