Riesz transform and fractional integral operators generated by non-degenerate elliptic differential operators

Document Type: Original Article

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Abstract

‎The Morrey boundedness‎ ‎is proved‎ ‎for the Riesz transform and the inverse operator‎ ‎of the non-degenerate elliptic differential operator of divergence form‎ ‎generated by a vector-function in $(L^\infty)^{n^2}$‎, ‎and for the inverse operator of the Schr\"{o}dinger operators whose non-negative potentials satisfy a certain integrability condition‎. ‎In this note‎, ‎our result is not obtained directly from the estimates of integral formula‎, ‎which reflects the fact that the solution of the Kato conjecture‎ ‎did not use any integral expression of the operators‎. ‎One of the important tools in the proof‎ ‎is the decomposition of the functions in Morrey spaces‎ ‎based on the elliptic differential operators in question‎. ‎In some special cases‎ ‎where the integral kernel comes into play‎, ‎the boundedness property of the Littlewood--Paley operator‎ ‎was already obtained by Gong‎. ‎So‎, ‎the main novelties of this paper‎ ‎are the decomposition results associated with elliptic differential operators‎ ‎and the result in the case where the explicit formula of the integral kernel of the heat semigroup‎ ‎is unavailable.

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