Sobolev-type theorem for commutators of Hardy operators in grand Herz spaces

Babar Sultan; Mehvish Sultan

doi:10.3842/umzh.v76i7.7546

Babar Sultan Department of Mathematics, Quaid-I-Azam University, Islamabad, Pakistan
Mehvish Sultan Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan

DOI: https://doi.org/10.3842/umzh.v76i7.7546

Keywords: exponent, Lebesgue spaces, BMO spaces, weighted estimates, Hardy operators, grand Herz spaces

Abstract

UDC 517.5

The higher-order commutators of fractional Hardy-type operators of variable order $\zeta(z)$ are shown to be bounded from the grand variable Herz spaces ${\dot{K} ^{a(\cdot), u),\theta}_{ p(\cdot)}(\mathbb{R}^n)}$ into the weighted space ${\dot{K} ^{a(\cdot), u),\theta}_{\rho, q(\cdot)}(\mathbb{R}^n)},$ where $\rho=(1+|z_1|)^{-\lambda}$ and $\displaystyle {1 \over q(z)}={1 \over p(z)}-{\zeta (z) \over n}$ if $p(z)$ is not necessarily constant at infinity.

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