On the boundedness of generalized Cesàro operators on Sobolev spaces

Abstract

For $\beta >0$ and $p\ge 1$, the generalized Cesàro operator $${\mathcal{C}}_{\beta }f(t):=\frac{\beta }{t^{\beta }}{\int }_0^t(t-s{)}^{\beta -1}f(s)ds$$ and its companion operator ${\mathcal{C}}_{\beta }^{\ast }$ defined on Sobolev spaces ${\mathcal{T}}_p^{(\alpha )}(t^{\alpha })$ and ${\mathcal{T}}_p^{(\alpha )}(|t{|}^{\alpha })$ (where $\alpha \ge 0$ is the fractional order of derivation and are embedded in $L^p(\RR^+)$ and $L^p(\RR)$ respectively) are studied. We prove that if $p>1$, then ${\mathcal{C}}_{\beta }$ and ${\mathcal{C}}_{\beta }^{\ast }$ are bounded operators and commute on ${\mathcal{T}}_p^{(\alpha )}(t^{\alpha })$ and ${\mathcal{T}}_p^{(\alpha )}(|t{|}^{\alpha })$. We show explicitly the spectra $\sigma ({\mathcal{C}}_{\beta })$ and $\sigma ({\mathcal{C}}_{\beta }^{\ast })$ and its operator norms (which depend on $p$). For $1<p\le 2$, we prove that $\widehat{{\mathcal{C}}_{\beta }(f)}={\mathcal{C}}_{\beta }^{\ast }(\hat{f})$ and $\widehat{{\mathcal{C}}_{\beta }^{\ast }(f)}={\mathcal{C}}_{\beta }(\hat{f})$ where $\hat{f}$ is the Fourier transform of a function $f\in L^p(\RR)$.

24 pages

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