$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by $\newcommand{\eA}{\mathscr{A}}$ $\eA$ the space of symmetric positive semidefinite operators $A:\bsV\to \bsV$. To each $A\in \eA$ we can associate in a canonical fashion a centered Gaussian measure $\gamma_A$ on $\bsV$ which is concentrated on $(\ker A)^\perp$. For example, if $A$ is nondegenerate, then

$$ \gamma_A(dv)= \frac{1}{\sqrt{\det 2\pi A}} e^{-\frac{1}{2} (A^{-1}v,v)}dv, $$

while if $A=0$, then $\gamma_0$ is the Dirac measure concentrated at the origin.

Fix a locally Lipschitz function $f:\bsV\to\bR$ which is positively homogeneous of degree $\alpha \geq 2$. For any $A\in \eA$ we denote by $\bsE_A(f)$ the expectation of $f$ with respect to the probability measure $\gamma_A$ on $\bsV$. Consider the function

$$ \eA\ni A\mapsto \bsE_A(f)\in \bR. $$

This function is continuous and positively homogeneous of degree $\frac{\alpha}{2}$, i.e.,

$$ \bsE_{tA}(f)=t^{\frac{\alpha}{2}} \bsE_A(f),\;\;\forall t>0,\;\;A\in\eA. $$

I am interested in its modulus of uniform continuity on the ball

$$\eA_1:=\bigl\lbrace A\in\eA;\;\;\Vert A\Vert\leq 1\bigr\rbrace, $$

I was able to prove that on this ball the above function is Holder continuous, with Holder exponent $\frac{1}{2N+3}$. This suffices for the applications I have in mind, but I strongly suspect that it is far from optimal. I believe that the Holder exponent $\frac{1}{2}$ is *uniformly* optimal in the following sense: there exist $C, r>0$ so that for any $A, B\in\eA_1$ satisfying

$$\Vert A- B\Vert \leq r, $$

we have

$$\bigl\vert \bsE_A(f)-\bsE_B(f)\bigr\vert\leq C\Vert A-B\Vert^{\frac{1}{2}}. \tag{1} $$

**Remark.** To see that the exponent $\frac{1}{2}$ is the best one can hope for consider the case $\bsV=\bR^2$, $f(x,y)=|xy|$ and $\newcommand{\ve}{{\varepsilon}}$ and the Gaussian measures

$$ \gamma_{A_\ve}=\frac{1}{2\pi\ve} e^{-\frac{1}{2\ve^2}x^2-\frac{1}{2}y^2} |dxdy| $$

Then $\Vert A_\ve-A_0\Vert =\ve^2$,

$$\bsE_{A_0}(f)=0,\;\; \bsE_{A_\ve}(f)=\left(\int_{\bR}|x|e^{-\frac{1}{2}x^2} |dx|\right)^2 \ve. $$

My question is the following: *have you encountered continuity results of this sort, and if so, can you indicate some references that deal with this?* Thanks!

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