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We study spaces of Hölder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Hölder type space of harmonic functions with prescribed modulus of continuity $\omega =\omega \left(h\right)$ and the second is the variable exponent harmonic Hölder space with the continuity modulus ${\left|h\right|}^{\lambda (·)}$. We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

For the convolution operators $A_a^{\alpha }$ with symbols ${a\left(\right|\xi \left|\right)\left|\xi \right|}^{-\alpha }expi\left|\xi \right|$, 0 ≤ Re α < n, $a\left(\right|\xi \left|\right)\in L_{\infty }$, we construct integral representations and give the exact description of the set of pairs (1/p,1/q) for which the operators are bounded from $L_p$ to $L_q$.

Mathematics Subject Classification: 47B38, 31B10, 42B20, 42B15. We obtain the Lp → Lq - estimates for the fractional acoustic potentials in R^n, which are known to be negative powers of the Helmholtz operator, and some related operators. Some applications of these estimates are also given. * This paper has been supported by Russian Fond of Fundamental Investigations under Grant No. 40–01–008632 a.