Abstract
For the third author the research was supported by grant No. 15-01-02732 of the Russian Fund of Basic Research
Highlights
Let p > and denote by p the conjugate parameter defined by p + = (p= ∞ when p = )
One weakness with many of these results is that the authors do not refer to the fact that already in it was given necessary and sufficient conditions for ( ) to hold and with sharp constant and general kernel of degree
The following conditions are equivalent: (i) The constant κp(a, b) < ∞. (ii) The inequality k(x, y)f (x)g(y) dx dy ≤ Cfpgp holds for some finite constant C for all f ∈ Lp and g ∈ Lp . (iii) The inequality k(x, y)f (x) dy dx ≤ Cp f p(x) dx holds for the same finite constant C as in ( ) and all f ∈ Lp. (iv) The sharp constant in both ( ) and ( ) is C = κp(a, b)
Summary
One weakness with many of these results is that the authors do not refer to the fact that already in (see [ ] and [ ]) it was given necessary and sufficient conditions for ( ) to hold and with sharp constant and general kernel of degree – . Theorem Let p ≥ , the kernel k(x, y) satisfy ( ) and κp be the constant defined by ( ). The following statements are equivalent: (i) The inequality k(x, y)f (x)g(y) dx dy ≤ Cfpgp holds for some finite constant C and all f ∈ Lp and g ∈ Lp .
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