Integral operators with homogeneous kernels in grand Lebesgue spaces
The author considers a class multidimensional integral operators acting in grand Lebesgue spaces \(L_a^{p)}(\mathbb{R}^n)\) and \(L_a^{p)}(\mathbb{R}_+)\) in the form: \[ \mathbf K f(x)= \int\limits_{\mathbb{R}^n} K(x,y) f(y) \text{d}y, \,x \in \mathbb{R}^n, \,n \geq2, \] and \[ \mathbf K f(x)= \int\limits_0^\infty K(x,y) f(y) \text{d}y \text{ for } n=1 \] where the kernel \(K(x, y)\) is homogeneous of degree \(-n\) : \(K(\lambda x,\lambda y) = \lambda^{-n} K(x,y)\), \(\lambda>0\) and \(K (x, y)\) is invariant with respect to rotations: \(K (\omega(x), \omega(y)) \equiv K (x, y)\), \(\omega\) is an arbitrary rotation in \(\mathbb{R}^n\). The function \(a(x) \in L^1(\Gamma_\delta^N)\) for any \(0 < \delta < N < \infty\), where \( L^1(\Gamma_\delta^N) = \{ x \in \mathbb{R}^n : \delta < |x| < N \}\) is called a grandizer. In the work, sufficient and necessary conditions on the grandizer are obtained to guarantee that the one-dimensional and multidimensional integral operators with homogeneous kernels are bounded in grand Lebesgue spaces \(L_a^{p)}(\mathbb{R}^n)\) and \(L_a^{p)}(\mathbb{R}_+)\) . In addition, two-sided estimates of the grand norms of such operators are established. In the special case of a radial kernel two-sided estimates for the norms of multidimensional operators in terms of spherical means are proved and it is proved that this result is ``stronger'' than the inequalities for the norms of operators with a nonradial kernel. As an application of the obtained results sufficient conditions for boundedness in grand spaces for one-dimensional operators of fractional Riemann-Liouville integration and for multidimensional Hilbert-type operators are obtained.
integral operator with homogeneous kernel
scientific article; zbMATH DE number 6845175
scientific article