Effective Dimensions in Grothendieck--Hölder Inequalities
What becomes of Grothendieck's inequality when its Hilbert-space vectors are replaced by vectors in $\ell_p^r$ and $\ell_{p'}^r$? For an $m\times n$ matrix, we show that the universal dimensional loss is governed not by $r$ alone, but by the effective dimension \[ d=\min\{r,m,n\}. \] More precisely, the optimal constant is bounded above and below, up to dimension-free factors, by \[ d^{|1/p-1/2|}. \] The upper bound rests on a pointwise Hilbertian stabilization: for every matrix $A$, \[ G_2^{(r)}(A)=G_2^{(d)}(A). \] Combining this stabilization with Lewis's Euclidean distortion estimate gives the effective-dimensional Grothendieck--Hölder bound. Beyond the universal power law, we retain the finer rectangular geometry. At $p=\infty$ we prove the exact compression formula \[ Γ_\infty^\K(r;m,n) = ρ\!\left( \ell_1^m(\K), \ell_1^{\min\{r,n\}}(\K) \right), \] and derive from it endpoint-sensitive lower bounds throughout the full Hölder scale. Over the complex field, interpolation between the Hilbertian problem and the exact rectangular endpoints yields corresponding upper bounds. Thus the effective dimension determines the universal exponent, whereas the tensor-norm endpoint retains additional aspect-ratio information. We determine the boundedness threshold when $p$ approaches $2$: it is governed by \[ |1/p-1/2|\log d. \] For complex two-row matrices, the endpoint tensor ratios are explicit, and the general rectangular estimates give quantitative two-sided bounds for every $1\le p\le\infty$.