Endpoint bounds for convolution operators with singular measures

E. Ferreyra; T. Godoy; M. Urciuolo

Colloquium Mathematicae (1998)

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

Abstract

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Let S ⊂ ℝ n + 1 be the graph of the function ϕ : [ - 1 , 1 ] n → ℝ defined by ϕ ( x 1 , ⋯ , x n ) = ∑ j = 1 n | x j | β j , with 1< β 1 ≤ ⋯ ≤ β n , and let μ 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 μ 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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