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The paper deals with two inverse problems for Sturm--Liouville operator$Ly=-y" +q(x)y$ on the finite interval $[0,\pi]$. The first one is the problemof recovering of a potential by two spectra. We associate with this problem themap $F:\, W^\theta_2\to l_B^\theta,\ F(\sigma) =\{s_k\}_1^\infty$, where$W^\theta_2 = W^\theta_2[0,\pi]$ are Sobolev spaces with $\theta\geqslant 0$,$\sigma=\int q$ is a primitive of the potential $q$ and $l_B^\theta$ arespecial Hilbert spaces which we construct to place in the regularized spectraldata $\bold s = \{s_k\}_1^\infty$. The properties of the map $F$ are studied indetails. The main result is the theorem on uniform stability. It gives uniformestimates from above and below of the norm of the difference $\|\sigma-\sigma_1\|_\theta$ by the norm of the difference of the regularized spectraldata $\|\bold s -\bold s_1\|_\theta$ where the last norm is taken in$l_B^\theta$. A similar result is obtained for the second inverse problem whenthe potential is recovered by the spectral function of the operator $L$generated by Dirichlet boundary conditions. The results are new for classicalcase $q\in L_2$ which corresponds to the value $\theta =1$.

Inverse problems for Sturm--Liouville operators with potentials from Sobolev spaces. Uniform stability