2000 character limit reached
Maximal operators given by Fourier multipliers with dilation of fractional dimensions
Published 27 May 2024 in math.CA | (2405.16855v1)
Abstract: In this paper, we investigate $Lp$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For the Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $Lp$ bounds of the maximal operators for each $p$. Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.
- Bourgain, J. Averages in the plane over convex curves and maximal operators. J. Analyse Math. 47 (1986), 69–85.
- Calderón, A.-P. Intermediate spaces and interpolation, the complex method. Studia Math. 24 (1964), 113–190.
- Calderón, C. P. Lacunary spherical means. Illinois J. Math. 23, 3 (1979), 476–484.
- Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type. Math. Z. 249, 1 (2005), 223–240.
- Maximal operators associated to Fourier multipliers with an arbitrary set of parameters. Proc. Roy. Soc. Edinburgh Sect. A 128, 4 (1998), 683–696.
- On maximal functions for Mikhlin-Hörmander multipliers. Adv. Math. 204, 2 (2006), 363–378.
- Kiprijanov, I. A. The fractional differentiation operator and powers of elliptic operators. Soviet Math. Dokl. 1 (1960), 222–224.
- Maximal operators associated with Fourier multipliers and applications. J. Funct. Anal. 284, 8 (2023), Paper No. 109857, 37.
- A note on the dimensions of Assouad and Aikawa. J. Math. Soc. Japan 65, 2 (2013), 343–356.
- Lototsky, S. V. Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations. Methods Appl. Anal. 7, 1 (2000), 195–204.
- Miyachi, A. On some Fourier multipliers for Hp(𝐑n)superscript𝐻𝑝superscript𝐑𝑛H^{p}({\bf R}^{n})italic_H start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( bold_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 1 (1980), 157–179.
- Spherical maximal functions and fractal dimensions of dilation sets. Amer. J. Math. 145, 4 (2023), 1077–1110.
- Calderón-Zygmund theory for operator-valued kernels. Adv. in Math. 62, 1 (1986), 7–48.
- Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications, Edited and with a foreword by S. M. Nikol’skiĭ, Translated from the 1987 Russian original, Revised by the authors.
- Pointwise convergence of spherical means. Math. Proc. Cambridge Philos. Soc. 118, 1 (1995), 115–124.
- Stein, E. M. Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A. 73, 7 (1976), 2174–2175.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.