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.

We study the integral representation properties of limits of sequences of integral functionals like $\int f(x,Du)\mathrm{d}x$ under nonstandard growth conditions of $(p,q)$-type: namely, we assume that$${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$Under weak assumptions on the continuous function $p\left(x\right)$, we prove $\Gamma$-convergence to integral functionals of the same type. We also analyse the case of integrands $f(x,u,Du)$ depending explicitly on $u$; finally we weaken the assumption allowing $p\left(x\right)$ to be discontinuous on nice sets.

We study the integral representation properties of limits of sequences of integral functionals like  $\int f(x,Du)\mathrm{d}x$  under nonstandard growth conditions of (p,q)-type: namely, we assume that $${\left|z\right|}^{p\left(x\right)}\le f(x,z)\le {L(1+|z|}^{p\left(x\right)}).$$ Under weak assumptions on the continuous function p(x), we prove Γ-convergence to integral functionals of the same type. We also analyse the case of integrands f(x,u,Du) depending explicitly on u; finally we weaken the assumption allowing p(x) to be discontinuous on nice sets.

In this paper, we give an integral representation for the boundary values of derivatives of functions of the de Branges–Rovnyak spaces $\mathscr{H}\left(b\right)$, where $b$ is in the unit ball of $H^{\infty }\left({ℂ}_{+}\right)$. In particular, we generalize a result of Ahern–Clark obtained for functions of the model spaces $K_b$, where $b$ is an inner function. Using hypergeometric series, we obtain a nontrivial formula of combinatorics for sums of binomial coefficients. Then we apply this formula to show the norm convergence of reproducing kernel $k_{\omega ,n}^b$ of evaluation...

The paper gives new integral representations of the $g$-Drazin inverse of an element $a$ of a $C^{*}$-algebra that require no restriction on the spectrum of $a$. The representations involve powers of $a$ and of its adjoint.

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...