A Generalization of Calderón Transfer Principle

Sakin Demir

Dumlupınar Mah. 38. Sok. No. 23 Daire 7, Demirtaş Nah. Osmangazi, 16245, Bursa, TURKEY.


Let T be an operator defined on the space of locally integrable functions on the real line with the following properties: the values of T are continuous functions on the real line, T is sublinear and commutes with the translations, and T is semilocal. Assume that X is a measure space which is totally  -finite and t U is a one-parameter group of measure-preserving transformations of X . Let us also assume that for every measurable function f on X the function ( ) t f U x is measurable in the product of X with the real line. Given a function f on X let F(t, x)  f (Ut x) . If f is the sum of two functions which are bounded and integrable, respectively, then F(t, x) is locally integrable function of t for almost all x and thus G(t, x) T(F(t, x)) is a well-defined continuous function of t for almost all x . Define # T f G(0, x) . Let ( ) n S and ( ) n K be the sequences of operators with the above properties and define sup | | n n Sf  S f and sup | | n n Kf  K f . Let # # sup | | n n S f  S f and # # sup | | n n K f  K f . We prove that if there exists a constant C  0 such that p p Sf  C Kf for all ( ) p f L (ℝ), 1 p   , then # # p p S f  C K f for all ( ) p f L X , 1 p   Mathematics Subject Classification: 47A35, 28D05, 47A64.

Keywords :Translation Invariant Operator, Calderón Transfer Principle, Operator Inequality.

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