# Endpoint bounds for convolution operators with singular measures

Colloquium Mathematicae (1998)

• Volume: 76, Issue: 1, page 35-47
• ISSN: 0010-1354

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## Abstract

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Let $S\subset {ℝ}^{n+1}$ be the graph of the function $\varphi :{\left[-1,1\right]}^{n}\to ℝ$ defined by $\varphi \left({x}_{1},\cdots ,{x}_{n}\right)={\sum }_{j=1}^{n}{|{x}_{j}|}^{{\beta }_{j}},$ with 1<${\beta }_{1}\le \cdots \le {\beta }_{n},$ and let $\mu$ the measure on ${ℝ}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is ${L}^{p}$-${L}^{q}$ bounded.

## How to cite

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Ferreyra, E., Godoy, T., and Urciuolo, M.. "Endpoint bounds for convolution operators with singular measures." Colloquium Mathematicae 76.1 (1998): 35-47. <http://eudml.org/doc/210551>.

@article{Ferreyra1998,
abstract = {Let $S\subset \mathbb \{R\}^\{n+1\}$ be the graph of the function $\varphi :[ -1,1]^n\rightarrow \mathbb \{R\}$ defined by $\varphi ( x_1,\dots ,x_n) =\sum _\{j=1\}^n| x_j|^\{\beta _j\},$ with 1<$\beta _1\le \dots \le \beta _n,$ and let $\mu$ the measure on $\mathbb \{R\}^\{n+1\}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is $L^p$-$L^q$ bounded.},
author = {Ferreyra, E., Godoy, T., Urciuolo, M.},
journal = {Colloquium Mathematicae},
keywords = {convolution; singular measure},
language = {eng},
number = {1},
pages = {35-47},
title = {Endpoint bounds for convolution operators with singular measures},
url = {http://eudml.org/doc/210551},
volume = {76},
year = {1998},
}

TY - JOUR
AU - Ferreyra, E.
AU - Godoy, T.
AU - Urciuolo, M.
TI - Endpoint bounds for convolution operators with singular measures
JO - Colloquium Mathematicae
PY - 1998
VL - 76
IS - 1
SP - 35
EP - 47
AB - Let $S\subset \mathbb {R}^{n+1}$ be the graph of the function $\varphi :[ -1,1]^n\rightarrow \mathbb {R}$ defined by $\varphi ( x_1,\dots ,x_n) =\sum _{j=1}^n| x_j|^{\beta _j},$ with 1<$\beta _1\le \dots \le \beta _n,$ and let $\mu$ the measure on $\mathbb {R}^{n+1}$ induced by the Euclidean area measure on S. In this paper we characterize the set of pairs (p,q) such that the convolution operator with $\mu$ is $L^p$-$L^q$ bounded.
LA - eng
KW - convolution; singular measure
UR - http://eudml.org/doc/210551
ER -

## References

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1. [B-S] Bennett C. and Sharpley R., Interpolation of Operators, Pure and Appl. Math. 129, Academic Press, 1988. Zbl0647.46057
2. [C] Christ M., Endpoint bounds for singular fractional integral operators, UCLA preprint, 1988.
3. [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.
4. [O] Oberlin D., Convolution estimates for some measures on curves, Proc. Amer. Math. Soc. 99 (1987), 56-60. Zbl0613.43002
5. [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. Zbl0684.22006
6. [S] Stein E.M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. Zbl0207.13501
7. [St] Stein E.M. , Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, 1993.
8. [S-W] Stein E. and Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.

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