In this thesis we consider the polarized deep inelastic scattering in the region of low values of Bjorken variable, $x$. We formulate the evolution equations for the unintegrated parton distributions which include a complete resummation of the double logarithmic contributions, $ln^2(1/x)$. Afterwards, these equations are completed with the standard LO and NLO DGLAP evolution terms, in order to obtain the proper behaviour of the parton distributions at moderate and large values of $x$. The equations obtained are applied to the following observables and processes: (i) to the nucleon structure function, $g_1$, in the polarized deep inelastic scattering, (ii) to the structure function of the polarized photon, $g_1^{gamma}$, in the scattering of a lepton on a polarized photon, and (iii) to the differential structure function, $x_J d^2g_1/dx_J dk_J^2$, in the polarized deep inelastic scattering accompanied by a forward jet. Case (iii) is proposed to be a test process for the presence and the magnitude of the $ln^2(1/x)$ contributions. For each process the consequences of including the logarithmic corrections are studied in a detail. After integrating out the structure function, $g_1$, the moments of the nucleon structure function are obtained. The contribution of the region of low $x$ to these moments is estimated, and then discussed in the context of the spin sum rules. Finally, some predictions for the observables, the asymmetry and the cross sections, in the processes (i)-(iii) are given. They are important to planned experiments with the polarized HERA and linear colliders, which will probe the region of low values of Bjorken $x$.
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