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Discrete analogues in harmonic analysis. II: Fractional integration. (English) Zbl 0972.42010

The present work represents a continuation of that initiated in [Am. J. Math. 121, No. 6, 1291-1336 (199; Zbl 0945.42009)] in which the authors prove \(L^{p}\) estimates for discrete analogues of singular Radon transforms. As noted by the authors, the main idea in that case was to replace a nondegeneracy condition pertaining to the submanifold of integration by regarding the domain of the corresponding discrete operator as the image of a polynomial map. A corresponding approach for discrete analogues of fractional integrals is unavailable because no adequate geometric theory exists for fractional integrals defined on submanifolds. Even for simple geometric circumstances, analysis of discrete analogues of fractional integrals is tied to unresolved problems in number theory. In view of these constraints the authors limit their analysis to two special operators, \[ I_{\lambda }(f)(n)=\sum_{m=1}^{\infty }\frac{f(n-m^{2})}{ m^{\lambda }} \quad \text{and} \quad J_{\lambda}(f)(n_{1},n_{2})=\sum_{m=1}^{\infty }\frac{ f(n_{1}-m,n_{2}-m^{2})}{m^{\lambda}}. \] The \((l^{p},l^{q})\) continuity of \( I_{\lambda }\) follows from weak-type estimates and Marcinkiewicz interpolation when \(1<p<q<\infty \) and \(1/q\leq 1/p-1+\lambda \). Other values of \((p,q)\) require much more subtle estimates. The main result about \( I_{\lambda }\) asserts that when \(1/2<\lambda <1\), if: (i) \( 1/q<1/p-(1-\lambda)/2\), (ii) \(1/q=1/p-(1-\lambda)/2\) and \(p\leq 2\leq q\) or (iii) \(p<1/(1-\lambda)\), \(q>1/\lambda \) then \(I_{\lambda }\) is bounded from \(l^{p}(Z)\) to \(l^{q}(Z)\). The Marcinkiewicz region lies strictly inside the region to which the theorem applies.
The main technique of the proof is to show that the multiplier \(m_{\lambda }(\theta)=\sum_{n=1}^{\infty }\frac{e^{2\pi in^{2}\theta }}{n^{\lambda }}\) belongs to weak-\(L^{r}\) when \(1/2<\lambda <1\) and \(r=2/(1-\lambda)\). In turn, this reduces to fairly intricate estimates for \(\sum_{j=0}^{\infty }\int_{2^{-j-1}}^{2^{j}}y^{-1+\lambda /2}\sum_{n=-\infty }^{\infty }e^{-\pi n^{2}(y-2i\theta)}dy\) which rely on the circle method. This boils down to dividing the \(\theta \)-circle into arcs depending on whether \(\theta \) is well-approximated by a rational with a denominator on the order of \(2^{j/2}\) . Similar estimates are used for \(J_{\lambda }\) which yield the following result when \(1/2<\lambda <1\), if: (i) \(1/q<1/p-(1-\lambda)/3\), (ii) \( 1/q=1/p-(1-\lambda)/2\) and \(p\leq 2\leq q\) or (iii) \(p<1/(1-\lambda)\), \( q>1/\lambda \) then \(J_{\lambda }\) is bounded from \(l^{p}(Z^{2})\) to \( l^{q}(Z^{2})\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
26A33 Fractional derivatives and integrals
44A12 Radon transform

Citations:

Zbl 0945.42009
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References:

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