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Denote by $L_D$ the Sturm-Liouville operator $Ly=-y" +q(x)y$ on the finiteinterval $[0,\pi]$ with Dirichlet boundary conditions $y(0)=y(\pi)=0$. Let$\{\lambda_k\}_1^\infty$ and $\{\alpha_k\}_1^\infty$ be the sequences of theeigenvalues and norming constants of this operator. For all $\theta \geqslant0$ we study the map $F: W_2^{\theta} \to l_D^\theta$ defined by $F(\sigma)=\{s_k\}_1^\infty$. Here $\sigma= \int q $ is the primitive of $q$, $\bold s =\{s_k\}_1^\infty$ be regularized spectral data defined by $s_{2k}=\sqrt{\lambda_k}-k,\ s_{2k-1}=\alpha_k-\pi/2$ and $l_D^\theta$ are specialHilbert spaces which are constructed in the paper as finite dimensionalextensions of the usual weighted $l_2$ spaces. We give a completecharacterization of the image of this nonlinear operator, show that it islocally invertible analytic map, find explicit form of its Frechet derivative.The main result of the paper are the uniform estimates of the form$\|\sigma-\sigma_1\|_\theta \asymp \|\bold s -\bold s_1\|_\theta$, providedthat the spectral data $\bold s$ and $\bold s_1$ run through special convexsets in the spaces $l_D^\theta$.

Properties of the map associated with recovering of the Sturm-Liouville  operator by its spectral function. Uniform stability in the scale of Sobolev  spaces