Off-diagonal bounds for spectral projector kernels

We study the growth of the spectral function of the Laplace-Beltrami operator on a Riemannian manifold $M$ of dimension $n$. For $n\ge 3$ we show that for any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ $$E_{\mathrm{\Lambda }}(x,y)=O_{M,x,y,\epsilon }({\mathrm{\Lambda }}^{n\mathrm{/}2}(\log \mathrm{\Lambda }{)}^{3\mathrm{/}2}(\log \log \mathrm{\Lambda }{)}^{1\mathrm{/}2+\epsilon })\mathrm{.}$$ For $n=2$ we prove a stronger statement. For any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ $$E_{\mathrm{\Lambda }}(x,y)=O_{M,x,y,\epsilon }({\mathrm{\Lambda }}^{5\mathrm{/}6}(\log \mathrm{\Lambda }{)}^{1\mathrm{/}6}(\log \log \mathrm{\Lambda }{)}^{1\mathrm{/}6+\epsilon })\mathrm{.}$$