Some Recent Relations

For a Banach space B on Sd−1 which satisfies ‖Cnf‖B ≤ C‖f‖B for some one has (see [10, Theorems 6.4, 6.5 and Section 9A])

K˜α(f, t2α)B ≤ Ct2α ∑

and

K˜α(f, t2α)B ≤ Ct2α ∫ 1 t

K˜β(f, u2β)B u2α+1

du, α < β. (9.9)

We observe that special cases of the above were known much earlier. For relations between K˜r(f, t2r)p with different p we have (see

[17, Section 10])

K˜α(f, t2r)q ≤ C (∫ t

du

u

)1/q1 (9.10)

where 1 ≤ p < q ≤ ∞ and q1 = { q, q <∞ 1, q =∞.