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Proof of Lions-Perthame-Tadmor conjecture and Jabin's conjecture on regularity for scalar conservation laws in higher dimension

[Note: Currently the proof is incomplete as we are using the lemma 3.2 which is not true in general]. We offer a complete resolution of a conjecture by Lions-Perthame-Tadmor mentioned in their celebrated work (1994, [34]). We prove the optimal regularizing effect to W^{s,p}_{loc} with the best exponent s, of the entropy solution for scalar conservation laws in higher dimension. In addition, we prove t (div f^\p(u))\in\mathcal{M}^1_{loc} for arbitrary flux f which was conjectured by Jabin (2010, [29]). Here we follow a new approach to understand the regularity issues in multi-dimensional scalar conservation laws.