Previous |  Up |  Next


integral operator; Hardy space
Let $A_{1},\dots ,A_{m}$ be $n\times n$ real matrices such that for each $1\leq i\leq m,$ $A_{i}$ is invertible and $A_{i}-A_{j}$ is invertible for $i\neq j$. In this paper we study integral operators of the form $$ Tf( x) =\int k_{1}( x-A_{1}y) k_{2}( x-A_{2}y) \dots k_{m}( x-A_{m}y) f( y) {\rm d} y, $$ $k_{i}( y) =\sum _{j\in \mathbb Z}2^{jn/{q_{i}}}\varphi _{i,j}( 2^{j}y) $, $1\leq q_{i}<\infty ,$ $1/{q_{1}}+1/{q_{2}}+\dots +1/{q_{m}}=1-r,$ $0\leq r<1,$ and $\varphi _{i,j}$ satisfying suitable regularity conditions. We obtain the boundedness of $T\colon H^{p}( \mathbb {R} ^{n}) \rightarrow L^{q}( \mathbb {R}^{n}) $ for $ 0<p<1/{r}$ and $1/{q}=1/{p}-r.$ We also show that we can not expect the $H^{p}$-$H^{q}$ boundedness of this kind of operators.
[1] Ding, Y., Lu, S.: Boundedness of homogeneous fractional integrals on $L^{p}$ for $N/\alpha \leq p\leq \infty$. Nagoya Math. J. 167 (2002), 17-33. DOI 10.1017/S0027763000025411 | MR 1924717 | Zbl 1031.42015
[2] Gelfand, I. M., Shilov, G. E.: Generalized Functions, Properties and Operations. Vol. 1, Academic Press Inc. (1964). MR 0166596
[3] Godoy, T., Urciuolo, M.: On certain integral operators of fractional type. Acta Math. Hung. 82 (1999), 99-105. DOI 10.1023/A:1026437621978 | MR 1658586 | Zbl 0937.47032
[4] Ricci, F., Sjogren, P.: Two-parameter maximal functions in the Heisenberg group. Math. Z. 199 (1988), 565-575. DOI 10.1007/BF01161645 | MR 0968322
[5] Rocha, P., Urciuolo, M.: On the $H^{p}$-$L^{p}$ boundedness of some integral operators. Georgian Math. J. 18 (2011), 801-808. DOI 10.1515/GMJ.2011.0043 | MR 2897664 | Zbl 1230.42030
[6] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton N. J. (1970). MR 0290095 | Zbl 0207.13501
[7] Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton N. J. (1993). MR 1232192 | Zbl 0821.42001
[8] Stein, E. M., Weiss, G.: On the theory of harmonic functions of several variables I: The theory of $H^{p}$ spaces. Acta Math. 103 (1960), 25-62. DOI 10.1007/BF02546524 | MR 0121579
[9] Taibleson, M. H., Weiss, G.: The molecular characterization of certain Hardy spaces. Astérisque 77 (1980), 67-151. MR 0604370 | Zbl 0472.46041
Partner of
EuDML logo