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In this article, we prove exact estimates for the $W$-invariant Dunkl kerneland heat kernel, for the root system of type $A$ with arbitrary positivemultiplicities. We apply the estimates of the $W$-invariant Dunkl heat kernelto compute sharp estimates for the Newton kernel and for the $s$-stablesemigroups generated by a fractional power of the $W$-invariant DunklLaplacian.

Sharp estimates for $W$-invariant Dunkl and heat kernels in the $A_n$ case