## 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

### Keywords:

fractional integration; circle method

Zbl 0945.42009
Full Text:

### References:

 [1] M. Christ,Endpoint bounds for singular fractional integrals operators, unpublished ms. [2] A. Cordoba,Translation invariant operators, inFourier Analysis, Proc. Seminar El Escorial, June 17–23, 1979, Assoc. Matem. Espanola, # 1, pp. 117–176. [3] J. Gerver,The differentiability of the Riemann function at certain rational multiples of {$$\pi$$}, Amer. J. Math.92 (1970), 35–55. · Zbl 0203.05904 [4] A. Greenleaf, A. Seeger and S. Wainger,On X-ray transforms for rigid line complexes and integral curves in $$\mathbb{R}$$4, to appear in Proc. Amer. Math. Soc. · Zbl 0934.44003 [5] G. H. Hardy and J. E. Littlewood,Some problems in Diophantine approximations II, Acta Math.37(1914), 193–239. · JFM 45.0305.03 [6] G. H. Hardy and E. M. Wright,The Theory of Numbers, 3rd edition, Oxford, 1954. · Zbl 0058.03301 [7] C. Hooley,On Hypothesis K* in Waring’s problem, inSieve Methods, Exponential Sums, and their Applications in Number Theory, London Math. Soc. Lecture Notes #237, Cambridge University Press, 1997. [8] H. L. Montgomery,Ten Lectures in the Interface between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc, Providence, RI, 1991. [9] F. Ricci and E. M. Stein,Harmonic analysis on nilpotent groups and singular integrals III. Fractional integration along-sub-manifolds, J. Funct. Anal.86 (1989), 360–389. · Zbl 0684.22006 [10] E. M. Stein and S. Wainger,Discrete analogues of singular Radon transforms, Bull. Amer. Math. Soc.23 (1990), 537–544. · Zbl 0718.42015 [11] E. M. Stein and S. Wainger,Discrete analogues in harmonic analysis I: l2estimates for singular Radon transforms, to appear in Amer. J. Math. [12] E. M. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton, 1971. · Zbl 0232.42007 [13] E. C. Titchmarsh,The Theory of the Riemann Zeta-Function, Oxford University Press, 1951. · Zbl 0042.07901 [14] I. M. Vinogradov,The Method of Trigonometric Sums in the Theory of Numbers, Interscience, London, 1954. · Zbl 0055.27504 [15] A. Walfisz,Gitterpunkte in mehrdimensional Kuglen, Polish Scientific Publishers, Warsaw, 1957. [16] Z. Zalcwasser,Sur les polynomes associés aux fonctions modulaires {$$\theta$$}, Studia Math.7 (1938), 16–35. · JFM 64.0228.02 [17] A. Zygmund,Trigonometric Series, Cambridge, 1959. · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.