### (${L}_{p}$,${L}_{q}$) mapping properties of convolution transforms

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An inequality, which generalizes and unifies some recently proved Carlson type inequalities, is proved. The inequality contains a certain number of “blocks” and it is shown that these blocks are, in a sense, optimal and cannot be removed or essentially changed. The proof is based on a special equivalent representation of a concave function (see [6, pp. 320-325]). Our Carlson type inequality is used to characterize Peetre’s interpolation functor $\u27e8{\u27e9}_{\phi}$ (see [26]) and its Gagliardo closure on couples of...

In this note we present an affirmative answer to the problem posed by M. Baronti and C. Franchetti (oral communication) concerning a characterization of Lp-spaces among Orlicz sequence spaces. In fact, we show a more general characterization of Orlicz spaces isometric to Lp-spaces.

Hagler and the first named author introduced a class of hereditarily ${l}_{1}$ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily ${l}_{p}$ Banach spaces for $1\le p<\infty $. Here we use these spaces to introduce a new class of hereditarily ${l}_{p}\left({c}_{0}\right)$ Banach spaces analogous of the space of Popov. In particular, for $p=1$ the spaces are further examples of hereditarily ${l}_{1}$ Banach spaces failing the Schur property.

For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ${\omega}_{m}\left(t\right)/\omega \u2099\left(t\right)\to \infty $ as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...

Some boundedness and VMO results are proved for a function f integrable on a cube ${Q}_{0}$, starting from an integral bound.