We define the class of integral holomorphic functions over Banach spaces; these are functions admitting an integral representation akin to the Cauchy integral formula, and are related to integral polynomials. After studying various properties of these functions, Banach and Fréchet spaces of integral holomorphic functions are defined, and several aspects investigated: duality, Taylor series approximation, biduality and reflexivity.

We use polymeasures to characterize when a multilinear form defined on a product of $C(K,X)$ spaces is integral.

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ${\ell }^{q̅}\left(L_v^{p̅}\right)$ into ${\ell }^q\left(L_u^p\right)$. For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted $L^p$-spaces. Amalgams of the form ${\ell }^q\left(L_w^p\right)$, 1 < p,q < ∞ , q ≠ p, $w\in A_p$, are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

We give new characterizations of Banach spaces not containing ${\ell }_1$ in terms of integral and $p$-dominated polynomials, extending to the polynomial setting a result of Cardassi and more recent results of Rosenthal.

Let ${ℬ}_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty$. Let ${ℬ}_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty$. Define ${𝒜}_c^n$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in ${ℬ}_c$. Similarly with ${𝒜}_r^n$ from ${ℬ}_r$. A type of integral is defined on distributions in ${𝒜}_c^n$ and ${𝒜}_r^n$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb{N}$, the spaces...

Let $\theta \in (0,1)$, $\lambda \in [0,1)$ and $p,p_0,p_1\in (1,\infty ]$ be such that $(1-\theta )/p_0+\theta /p_1=1/p$, and let $\varphi ,{\varphi }_0,{\varphi }_1$ be some admissible functions such that $\varphi ,{\varphi }_0^{p/p_0}$ and ${\varphi }_1^{p/p_1}$ are equivalent. We first prove that, via the $\pm$ interpolation method, the interpolation $\langle L_{{\varphi }_0}^{p_0),\lambda }\left(𝒳\right),L_{{\varphi }_1}^{p_1),\lambda }\left(𝒳\right),\theta \rangle$ of two generalized grand Morrey spaces on a quasi-metric measure space $𝒳$ is the generalized grand Morrey space $L_{\varphi }^{p),\lambda }\left(𝒳\right)$. Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.

Banach operator ideal properties of the inclusion maps between Banach sequence spaces are used to study interpolation of orbit spaces. Relationships between those spaces and the method-of-means spaces generated by couples of weighted Banach sequence spaces with the weights determined by concave functions and their Janson sequences are shown. As an application we obtain the description of interpolation orbits in couples of weighted $L_p$-spaces when they are not described by the K-method. We also develop...

In recent years the study of interpolation of Banach spaces has seen some unexpected interactions with other fields. (...) In this paper I shall discuss some more interactions of interpolation theory with the rest of mathematics, beginning with some joint work with Coifman [CS]. Our basic idea was to look for the methods of interpolation that had interesting PDE's arising as examples.

The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $Ce{s}_p\left(I\right)$ is an interpolation space between $Ce{s}_{p₀}\left(I\right)$ and $Ce{s}_{p₁}\left(I\right)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $Ce{s}_p[0,1]$ is not an interpolation space between Ces₁[0,1] and $Ce{s}_{\infty }[0,1]$.