### A characterization of localized Bessel potential spaces and applications to Jacobi and Henkel multipliers

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from ${L}^{p}\left(\right)$ to ${L}^{q}\left(\right)$. We also give a condition on p which is necessary if this operator maps ${L}^{p}\left(\right)$ into L²().

We strengthen the Carleson-Hunt theorem by proving ${L}^{p}$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $r>\mathrm{\U0001d696\U0001d68a\U0001d6a1}\{{p}^{\text{'}},2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of ${L}_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

Four theorems of Ahmad [1] on absolute Nörlund summability factors of power series and Fourier series are proved under weaker conditions.

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with $\Sigma \subset {\mathbb{R}}^{n-1}$, a bounded domain of class ${C}^{1,1}$, are obtained in the space ${L}^{q}(\mathbb{R};L\xb2\left(\Sigma \right))$, q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

For 1 ≤ q < ∞, let ${}_{q}\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_{q}\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction q >...