Resumen de Local Riesz transforms characterization of local Hardy spaces

Marco M. Peloso , Silvia Secco

• For 0 < p ? 1, let h^p (R^n) denote the local Hardy space. Let ? be a smooth, compactly supported function, which is identically one in a neighborhood of the origin. For k = 1, . . . , n, let (r_k f )�(?) = -i(1 - ?^(?))?_k /|?|f^(?) be the local Riesz transform and define (r_0 f )�(?) = (1 -?^(?))f^(?). Let ? be a fixed Schwartz function with \int ? dx = 1, let M > 0 be an integer and suppose (n - 1)/(n + M - 1) < p ? 1.

  We show that a tempered distribution f which is restricted at infinitybelongs to h^p (R^n ) if and only if ? * f belongs to h^p (R^n ) and there exists a constant A > 0 such that for all ? with 0 < ? ? 1 we have \sum_{M ?|?|?M +1} || r^? (f ) *??||_{L^p (R^n)} ? A. Here, ?? (x) = ?-n ?(x/?), ? = (?0 , . . . , ?n ) belongs to N^{n+1} , r^? , as usual, denotes the composition r_0^{a_0} ? · · · ? r_n^{a_n} . This result extends to the local Hardy spaces the analogous characterization of the classical Hardy spaces H^p (R^n ) (see e.g. [9, Chapter III.5.16]).