Endpoint bounds for convolution operators with singular measures

ArticleOriginal scientific text

Title

Authors ¹, ¹, ¹

Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, Argentina

Let

S \subset R^{n + 1}

be the graph of the function

φ : {[- 1, 1]}^{n} \to R

defined by

φ (x_{1}, \dot{s}, x_{n}) = \sum_{j = 1}^{n} {| x_{j} |}^{β_{j}},

with 1$!\beta_1\leq \dots \leq \beta_n,

and \leq t

\mu

t h e m e a s u r e o n

\R^{n+1}

\in d u c e d b y t h e E u c l a n a r e a m e a s u r e o n S . I n t h i s p a p e r w e c h a r a c t e r i z e t h e s e t o f p a i r s (p, q) s u c h t \hat{t} h e c o n v o l u t i o n o p e r a \to r w i t h

\mu

i s

L^p

-

L^q!$! bounded.

[B-S] Bennett C. and Sharpley R., Interpolation of Operators, Pure and Appl. Math. 129, Academic Press, 1988.
[C] Christ M., Endpoint bounds for singular fractional integral operators, UCLA preprint, 1988.
[F-G-U] Ferreyra E., Godoy T. and Urciuolo M., $L^{p}$ - $L^{q}$ estimates for convolution operators with $n$ -dimensional singular measures, J. Fourier Anal. Appl., to appear.
[O] Oberlin D., Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), 56-60.
[R-S] Ricci F. and Stein E.M., Harmonic analysis on nilpotent groups and singular integrals. III, Fractional integration along manifolds, J. Funct. Anal. 86 (1989), 360-389.
[S] Stein E.M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970.
[St] Stein E.M. , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.
[S-W] Stein E. and Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.