Resumen de Local Riesz transforms characterization of local Hardy spaces

Marco M. Peloso , Silvia Secco

• For 0 p≤1, leth p(R n) denote the local Hardy space. Let \hat \theta be a smooth, compactly supported function, which is identically one in a neighborhood of the origin. For k=1, ..., n, let \left( {r_k f} \right)^ \wedge \left( \xi \right) = - i\left( {1 - \hat \theta \left( \xi \right)} \right){{\xi _k } \mathord{\left/ {\vphantom {{\xi _k } {\left| \xi \right|\hat f}}} \right. \kern-\nulldelimiterspace} {\left| \xi \right|\hat f}}\left( \xi \right) be the local Riesz transform and define \left( {r_0 f} \right)^ \wedge \left( \xi \right) = \left( {1 - \hat \theta \left( \xi \right)} \right)\hat f\left( \xi \right). Let Ψ be a fixed Schwartz function with ∫ Ψdx=1, letM>0 be an integer and suppose (n−1)/(n+M−1) p<-1.

  We show that a tempered distributionf which is restricted at infinity belongs toh p(R n) if and only if θ*ƒ∈h p(R n) and there exists a constant A > 0 such that for all ε with 0 ε ≤ 1 we have \sum\nolimits_{M \leqslant \left| \alpha \right| \leqslant M + 1} {\left\| {r^\alpha \left( f \right)*\Psi _\varepsilon } \right\|_{L^p \left( {R^n } \right)} \leqslant A} . Here, Ψε(x)=ε−n Ψ(x/ε), α=(α0, …, αn) ∈N n+1,r α, as usual, denotes the compositionr α 0 0…or α n n . This result extends to the local Hardy spaces the analogous characterization of the classical Hardy spacesH p(R n) (see e.g. [9, Chapter III.5.16]).