Hardy Type Inequalities for Abstract Differential Operators by Werner O. Amrein
By Werner O. Amrein
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Extra resources for Hardy Type Inequalities for Abstract Differential Operators
Example text
36) will be used in several contexts . It clearly "\> s h o w s t h e r o l e o f t h e a u x i l i a r y f u n c t i o n ip : it s e r v e s t o m o d e r a t e t h e h growth of tp' which cannot be dominated by the other terms , unless 2 (2-a)P + Q gives a strong positive contribution . e. cp = 0 , and the arguments are slightly simpler : for example , there is no need for the norm || . || and S = S T, , one can take p = 1 . The term 2Im(Pv,tMcp v) can be estimated in two ways , depen3 2 ding on whether (p belongs to C or only to C .
This suggests that , if our operator is of order m , then 9 = m-1/2 should be the critical value in the corresponding Hardy type inequality . As an example in support of this conjecture , we shall show how , from a mixed Hardy-Carleman type inequality with slowly growing weight , one can deduce Hardy type inequalities with rapidly growing weights . This is related to the discussion of part D of the Introduction . 1 and 5«2 of our preprint [51 ) . Now let v > 0 , take x = nv/2 , multiply the preceding inequality by 2n/n!
Proof : (i) Let v € (0,1] and v € P(r(v)) . 44) . 6. 46) || nv||2 < c || nL(