This paper presents an elementary proof of the following theorem: Given {$r$_{$j$}}_{$j$}^{$m$}=1 with m=d+1, fix $$fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } $$ and let Q=[−R, R]^{d}. Then any f∈ L^{2}(Q) is completely determined by its averages over cubes of side r_{j}that are completely contained in Q and have edges parallel to the coordinate axes if and only if r_{j}/r_{k}is irrational for j≠k. When$d$=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.

A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^{μ}in$R$^{$n$},$n$≥1, is obtained, for μ∈(0,1/2$n$)/(1,1/2$n$−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev.

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Inequalities of the form $$\sum\nolimits_{k = 0}^\infty {|\hat f(m_k )|/(k + 1) \leqslant C||f||_1 } $$ for all$f$∈$H$^{1}, where {$m$_{$k$}} are special subsequences of natural numbers, are investigated in the vector-valued setting. It is proved that Hardy's inequality and the generalized Hardy inequality are equivalent for vector valued Hardy spaces defined in terms ff atoms and that they actually characterize$B$-convexity. It is also shown that for 1<$q$<∞ and 0<α<∞ the space$X=H$(1,$q$,γa) consisting of analytic functions on the unit disc such that $$\int_0^1 {(1 - r)^{q\alpha - 1} M_1^q (f,r) dr< \infty } $$ satisfies the previous inequality for vector valued functions in$H$^{1}($X$), defined as the space of$X$-valued Bochner integrable functions on the torus whose negative Fourier coefficients vanish, for the case {$m$_{$k$}}={2^{k}} but not for {$m$_{$k$}}={$k$^{$a$}} for any α ∈ N.