The space $\widehat{\ell }_k$ of absolutely almost convergent series was introduced and studied by Das et al [4]; which plays an important role in summability theory, approximation theory, Fourier analysis, etc. In the present paper we generalize the space making use of some factors and weighted mean transformations, investigate its topological structures and relations between classical sequence spaces. Also we characterize certain matrix operators on it.
Recently, in [Bor4], Bor proved a main theorem dealing with $|\bar{N}, p_{n}|_{k}$ summability factors of infinite series. In the present paper, we have generalized that theorem for $|A, p_{n}|_{k}$ summability method by taking normal matrices in place of weighted mean matrices.
We prove weighted restriction type estimates for Grushin operators. These estimates are then used to prove sharp spectral multiplier theorems as well as Bochner-Riesz summability results with sharp exponent.
A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $(p,σ)$-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banac…
We show that, in every weighted Dirichlet space on the unit disk with superharmonic weight, the Taylor series of a function in the space is $(C,α)$-summable to the function in the norm of the space, provided that $α>1/2$. We further show that the constant $1/2$ is sharp, in marked contrast with the classical case of the disk algebra.
We introduce the notion of index of summability for pairs of Banach spaces; for Banach spaces E; F, this index plays the role of a kind of measure of how the m-homogeneous polynomials from E to F are far from being absolutely summing. In some cases the optimal index of summability is computed.
The aim of this paper is to generalize a main theorem concerning weighted mean summability to absolute matrix summability which plays a vital role in summability theory and applications to the other sciences by using quasi-$f$-power sequences.
In this note we show that the Taylor series of a function in a weighted Dirichlet space is (generalized) Nörlund summable, provided that the sequence determining the Nörlund operator is non-decreasing and has finite upper growth rate. In particular the Taylor series is Nörlund summable for all $α>1/2$, and the rate of convergence is of the order $O(n^{-1/2})$. The inequality $α>1/2$ is sharp. On the other hand if the Taylor series is Nörlund summable and the partial sums of the determining sequence enjoy a certain growth condition then the determining sequence has finite lower growth rate. An …
In this paper, we have generalized a main theorem dealing with weighted mean summability method for absolute matrix summability method which plays a vital role in summability theory and applications to the other sciences by using quasi-monotone sequences. The main result in this paper extends the results in
We prove new summability properties for multilinear operators on $\ell_p$ spaces. An important tool for this task is a better understanding of the interplay between almost summing and absolutely summing multilinear operators.
Let $1 \leq p <\infty$. A sequence $\lef x_n \rig$ in a Banach space $X$ is defined to be $p$-operator summable if for each $\lef f_n \rig \in l^{w^*}_p(X^*)$, we have $\lef \lef f_n(x_k)\rig_k \rig_n \in l^s_p(l_p)$. Every norm $p$-summable sequence in a Banach space is operator $p$-summable, while in its turn every operator $p$-summable sequence is weakly $p$-summable. An operator $T \in B(X, Y)$ is said to be $p$-limited if for every $\lef x_n \rig \in l_p^w(X)$, $\lef Tx_n \rig$ is operator $p$-summable. The set of all $p$-limited operators form a normed operator ideal. It is shown that ev…
It is well known that not every summability property for non linear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. We analyze the class of linear operators that are defined by a summability inequality involving a homogeneous map. Our construction includes the cases of absolutely $p$-summing linear operators, $(p,σ)$-absolutely continuous linear operat…
In this paper, we study the $L^p$-Bochner-Riesz mean summability problem related to the spectrum of some particular Sturm-Liouville operators in the weighted $L^p([a,b],ω).$ Our purpose is to establish suitable conditions under which the Bochner-Riesz expansion of a function $f \in L^p([a,b],ω)$,$1<p<\infty$, in two generalisations of Slepian's basis, converges to $f$ in $L^p([a,b],ω)$.
Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\inω}$ in $A$ and unconditionally convergent series $\sum_{n\inω}x_n$ in $X$ the series $\sum_{n\inω}a_nx_n$ is unconditionally convergent. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^*\in Y^*$ the operator $D_{y^*}:X\to A^*$, $D_{y^*}(x)(a)=y^*(ax)$, is absolutely summing. Combining …
Let $X(t,ω),$ $t \in \textit{R}$ be a symmetric stable process with index $α\in (1,2]$ and $a_n$ be the Fourier-Jacobi coefficients of $f \in L^p,$ where $p \geq α.$ For $γ, δ> 0,$ $t \in [-1,1],$ define $A_n(ω)=\int_{-1}^1 P_n^{(γ,δ)}(t)ρ^{(γ,δ)}dX(t,ω)$ where $P_n^{(γ,δ)}(t)$ are orthogonal Jacobi polynomials. The $A_n(ω)$ exists in the sense of mean. In this paper, it is shown that the random Fourier-Jacobi series $\sum_{n=0}^\infty a_n A_n(ω)P_n^{(γ,δ)}(y)$ converges to the stochastic integral $\int_{-1}^1f(y,t)ρ^{(γ,δ)}dX(t,ω)$ in the sense of mean and the sum function is weakly continuou…
We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak$^*$-compact set. We prove that a weak$^*$-compact set has summable Szlenk index if and only if its weak$^*$-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from [Draga and Kochanek 2016] regarding the behavior of summability of Szlenk index under $c_0$ direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from [Draga and Kochanek 2017]. As a f…
We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending previous work by Langenbruch. As a consequence we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.
In this paper, we give necessary and sufficient conditions for the boundedness of rough Hausdorff operators on Herz, Morrey and Morrey-Herz spaces with absolutely homogeneous weights. Especially, the estimates for operator norms in each case are worked out. Moreover, we also establish the boundedness of the commutators of rough Hausdorff operators on the such spaces with their symbols belonging to Lipschitz space.
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
We give analytic description for the completion of $C_0^\infty ( \mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, ω):= \{ u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely continuous on} \ \mathbf{R}_+ \ {\rm and}\ \| u^{'}\|_{L^p(\mathbf{R}_+, ω)}<\infty \}$, for given continuous weight $ω$, in terms of the local $B_p$ conditions due to Kufner and Opic, where $1<p<\infty$. Moreover, we propose applications of our results to: analysis of Hardy inequalities, boundary value problems, complex interpolation theory, and to derivation of new Morrey type inequalities.
In the paper we consider $T_{1},..., T_{d}$ absolute contractions of von Neumann algebra $\M$ with normal, semi-finite, faithful trace, and prove that for every bounded Besicovitch weight $\{a(\kb)\}_{\kb\in\bn^d}$ and every $x\in L_{p}(\M)$, ($p>1$) the averages A_{\Nb}(x)=\frac{1}{|\Nb|}\sum\limits_{\kb=1}^{\Nb}a(\kb)\Tb^{\kb}(x). converge bilaterally almost uniformly in $L_{p}(\M)$.
We characterize a weighted norm inequality which corresponds to the embedding of a class of absolutely continuous functions into the fractional order Sobolev space. The auxiliary result of the paper is of independent interest. It comprises of several types of necessary and sufficient conditions for the boundedness of the Hardy--Steklov operator (an integral operator with two variable boundaries of integration) in weighted Lebesgue spaces.
