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Vietnam Journal of Mathematics 38:4 (2010) 425-433

 

Behavior of the Sequence of Norms of Primitives of a Function in Lorentz Space

Ha Huy Bang and Bui Viet Huong

 

Abstract.  Let $f\in \Psi(\mathbb R)$ and $I^nf\in N_\Psi(\mathbb R)$, for all $n = 1, 2, ...$. Then

\[\mathop {\lim }\limits_{n \to \infty } ||I^n f||_{N_\Psi  ()}^{1/n}  = \sigma ^{ - 1},\]

where $\sigma := \inf\{|\xi|: \xi \in \supp f\}, ||.||_N_\Psi(\mathbb R)$ is the norm in the Lorentz space $N_\Psi (\mathbb R)$, and for $g\in S'(\mathbb R)$, the tempered generalized function $Ig$ is a primitive of $g$ if $D(Ig) = g$, that is

\[<Ig,\varphi^'>\-<g,\varphi>, \forall \varphi \in S(\mathbb R)\]

and $S(\mathbb R)$ is the Schwartz space of rapidly decreasing functions.

In other words, in this paper we characterize behavior of the sequence of $N_\Psi(\mathbb R) - norms of primitives of a function by its spectrum (the support of its Fourier transform).

2000 Mathematics Subject Classification: 46E30.

Keywords: Lorentz spaces, primitives of generalized functions.

 

 

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