Revistes Catalanes amb Accés Obert (RACO)

Local Riesz transforms characterization of local Hardy spaces

Marco M. Peloso, Silvia Secco


For $0 < p \leq 1$, let $h^ p(\mathrm R^n)$ 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 $(r_kf)\textasciicircum (\xi) = -i(1 - \hat{\theta}(\xi))\xi_ k/|\xi| \hat {f}(\xi)$ be the local Riesz transform and define $(r_0 f)\textasciicircum (\xi) = (1 - \hat{\theta}(\xi)) \hat{f}(\xi)$. Let $\Psi$ be a fixed Schwartz function with $\int{\Psi} dx = 1$, let $M > 0$ be an integer and suppose $(n - 1)/(n + M - 1) < p \leq 1$. We show that a tempered distribution $f$ which is restricted at infinity belongs to $h^ p(\mathrm R^n)$ if and only if $\theta \ast f \in h^ p(\mathrm R^n)$ and there exists a constant $A > 0$ such that for all $\varepsilon$ with $0 <\varepsilon\leq 1 $ we have $\sum _{M\leq|\alpha|\leq M+1} ||r^\alpha(f) \ast \Psi_{\varepsilon||L^p(R^n)}\leq A$. Here, $\Psi_\varepsilon (x) = \varepsilon^{-n} \Psi (x/\varepsilon), \alpha = (\alpha_0, . . . , \alpha_n) \in N^{n+1}, r^\alpha$, as usual, denotes the composition $r_{0}^{\alpha 0}$ o. . .o $r_{n}^{\alpha n}$ . This result extends to the local Hardy spaces the analogous characterization of the classical Hardy spaces $h^p(\mathrm R^n)$ (see e.g. [9, Chapter III.5.16]).

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