Abstract
Let {δt}t>0 be a non-isotropic dilation group on R n. Let τ: R n → [0,∞) be a continuous function that vanishes only at the origin and satisfies τ(δ t x) = tτ(x), t > 0, x ∈ R n. In this paper we obtain two-sided inequalities for spherical means of the form \(\int_{S^{n-1}}\tau(r_1\omega_1,\cdots,r_n\omega_n)^{-\alpha}d\sigma (\omega),\) where α is a positive constant, and r1,…, rn are positive parameters.
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References
F. Abi-Khuzam and B. Shayya, A sharp norm inequality for non-isotropic distance functions on Rn, Math. Inequal. Appl. 5 (2002), 361–368.
E. H. Lieb AND M. Loss, Analysis, 2nd edition, American Mathematical Society, 2001.