On Sj\"{o}lin-Soria-Antonov type extrapolation for locally compact groups

Sj\"{o}lin-Soria-Antonov type extrapolation theorem for locally compact $\sigma$-compact non-discrete groups is proved. As an application of this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class $L\log^{+}L\log^{+}\log^{+}\log^{+}L$.