Weighted Hardy operators in local generalized Orlicz-Morrey spaces

In this paper, we find sufficient conditions on general Young functions $(\Phi, \Psi)$ and the functions $(\varphi_1,\varphi_2)$ ensuring that the weighted Hardy operators $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ are of strong type from a local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ into another local generalized Orlicz-Morrey space $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$. We also obtain the boundedness of the commutators of $A_\omega^\alpha$ and ${\mathcal A}_\omega^\alpha$ from $M^{0,\,loc}_{\Phi,\,\varphi_1}(\mathbb R^n)$ to $M^{0,\,loc}_{\Psi,\,\varphi_2}(\mathbb R^n)$.

Lower bounds for Orlicz eigenvalues

<p style='text-indent:20px;'>In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.</p><p style='text-indent:20px;'>We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.</p>

An Analytic Characterization of p , q -White Noise Functionals

In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

Weighted Orlicz algebras on hypergroups

Let K be a hypergroup, w be a weight function and let (?,?) be a complementary pair of Young functions. We consider the weighted Orlicz space L??(K) and investigate some of its algebraic properties under convolution. We also study the existence of an approximate identity for the Banach algebra L?w(K). Further, we describe the maximal ideal space of the convolution algebra L?w(K) for a commutative hypergroup K.

The maximal operator in weighted variable spacesLp(⋅)

We study the boundedness of the maximal operator in the weighted spacesLp(⋅)(ρ)over a bounded open setΩin the Euclidean spaceℝnor a Carleson curveΓin a complex plane. The weight function may belong to a certain version of a general Muckenhoupt-type condition, which is narrower than the expected Muckenhoupt condition for variable exponent, but coincides with the usual Muckenhoupt classApin the case of constantp. In the case of Carleson curves there is also considered another class of weights of radial type of the formρ(t)=∏k=1mwk(|t-tk|),tk∈Γ, wherewkhas the property thatr1p(tk)wk(r)∈Φ10, whereΦ10is a certain Zygmund-Bari-Stechkin-type class. It is assumed that the exponentp(t)satisfies the Dini–Lipschitz condition. For such radial type weights the final statement on the boundedness is given in terms of the index numbers of the functionswk(similar in a sense to the Boyd indices for the Young functions defining Orlich spaces).

The Maximal Operator in Variable Spaces 𝐿 𝑝(·)(Ω, ρ) with Oscillating Weights

We study the boundedness of the maximal operator in the spaces 𝐿 𝑝(·)(Ω, ρ) over a bounded open set Ω in 𝑅𝑛 with the weight , where 𝑤𝑘 has the property that belongs to a certain Zygmund-type class. Weight functions 𝑤𝑘 may oscillate between two power functions with different exponents. It is assumed that the exponent 𝑝(𝑥) satisfies the Dini–Lipschitz condition. The final statement on the boundedness is given in terms of index numbers of functions 𝑤𝑘 (similar in a certain sense to the Boyd indices for the Young functions defining Orlicz spaces).