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In the present paper, we give some characterization of the $L^{2}$ maximal estimate for the operator $P_{a,\gamma}^{t}f\big(\Gamma(x,t)\big)$ along curve with complex time, which is defined by $$P_{a,\gamma}^{t}f\big(\Gamma(x,t)\big) =\int_{\mathbb{R}} e^{i\Gamma(x,t)\xi}e^{it|\xi|^{a}} e^{-t^{\gamma}|\xi|^{a}} \hat{f}(\xi)d\xi,$$ where $t,\gamma>0$ and $a\geq2,$ curve $\Gamma$ is a function such that $\Gamma:\mathbb{R}\times[0,1]\rightarrow\mathbb{R},$ and satisfies H\"{o}lder's condition of order $\sigma$ and bilipschitz conditions. The authors extend the results of the Schr\"{o}dinger type with complex time of Bailey \cite{Bailey} and Cho, Lee and Vargas \cite{CLV} to Schr\"{o}dinger operators along the curves.
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참고문헌 (12)
참고문헌 신청
A. D. Bailey / 2013 / Boundedness of maximal operators of Schrodinger type with complex time / Rev. Mat. Iberoam 29(2):531~546
L. Carleson / 1980 / Some analytic problems related to statistical mechanics / Lecture Notes in Math 779:5~45
C. -H. Cho / 2012 / Problems on pointwise convergence of solutions to the Schrodinger equation / J. Fourier Anal. Appl 18(5):972~994
B. E. J. Dahlberg / 1981 / A note on the almost everywhere behavior of solutions to the Schrodinger equation / Lecture Notes in Math 908:205~209
Y. Ding / 2017 / Weighted maximal estimates along curve associated with dispersive equations / Anal. Appl. (Singap.) 15(2):225~240
L. Carleson / 1980 / Some analytic problems related to statistical mechanics / Lecture Notes in Math 779:5~45
C. -H. Cho / 2012 / Problems on pointwise convergence of solutions to the Schrodinger equation / J. Fourier Anal. Appl 18(5):972~994
B. E. J. Dahlberg / 1981 / A note on the almost everywhere behavior of solutions to the Schrodinger equation / Lecture Notes in Math 908:205~209
Y. Ding / 2017 / Weighted maximal estimates along curve associated with dispersive equations / Anal. Appl. (Singap.) 15(2):225~240
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