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In this work we construct the heat kernel of the 1/2-order Laplacian perturbed by the first-order gradient term in H\"older space and the zero-order potential term in generalized Kato's class, and obtain sharp two-sided estimates as well as the gradient estimate of the heat kernel.
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Barrios B., Colorado E., Pablo A.de, Sánchez U.: On some critical problems for the fractional Laplacian operator. J. Diffenertial Equations., 252(2012), 6133-6162
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Bogdan K., Jakubowski T. : Estimates of Green Function for the Fractional Laplacian Perturbed by Gradient. Potential Anal., 36(2012), 455-481
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Jakubowski T. and Szczypkowski K.: Time-dependent gradient perturbations of fractional Laplacian. J. Evol. Equ., 10(2010), 319-339
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Jakubowski, T.: Fractional Laplacian with singular drift. Studia Math. 207, no. 3, 257-273(2011)
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Maekawa, Y. and Miura, H.: Upper bounds for fundamental solutions to non-local diffusion equations with divergence free drift, J. Funct. Anal, 264, 2245-2268(2013)
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