Given a fixed$p$≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $ , provided that the dimension$d$is sufficiently large. The method also yields new$L$^{$q$}space-time regularity results for solutions of the wave equation in high dimensions.

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In this paper, we show how under the continuum hypothesis one can obtain an integral representation for elements of the topological dual of the space of functions of bounded variation in terms of Lebesgue and Kolmogorov-Burkill integrals.

An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $\alpha \in (0, N)$: \[ \|I_\alpha u\|_{L^{N/(N-\alpha)}(\mathbb{R}^N)} \leq C \|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space $\mathcal{H}^1(\mathbb{R}^N)$ and provides a new family of $L^1$-Sobolev inequalities for the Riesz fractional gradient.