The purpose of this note is to establish further properties of the variable Lorentz spaces $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$ introduced by L. Ephremidze, V. Kokilashvili and S. Samko, which will allow us to apply the theory of Hardy spaces associated with ball quasi-Banach function spaces, and so define the variable Hardy-Lorentz spaces associated with $\mathfrak{L}^{p(\cdot), q(\cdot)}(\mathbb{R}^n)$. Then, the finite and infinite atomic decompositions for these spaces will be deduced immediately. We also obtain the boundedness of singular integrals and fractional type operators in variable Hardy-Lorentz spaces, and provide a Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $r$-convexification of variable Lorentz spaces.
In this article we consider polynomials orthogonal with respect to the inner product $$\langle P,Q \rangle_S = \int_{\mathbb{R}}P(x)Q(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x + \lambda\int_{\mathbb{R}}P^\prime(x)Q^\prime(x) \mathrm{e}^{-V(x)} \, \mathrm{d}x$$ where $V$ is a polynomial of even degree (at least four) and pos...
The well-known characterization of Hardy spaces $\mathrm{H}_{p}(\mathbb D)$, $0<p<\infty$, in terms of the Littlewood-Paley g-function $$ S f (\zeta) = \left(\int_0^1 |f'(r \zeta)|^2 (1 - r) \mathrm dr\right)^{1/2} \in \mathrm{L}_{p} $$ is generalized to Hardy-type spaces $X_A$ corresponding to quasi-Banach lattices $X...
E. Abakumov, Evgueni Doubtsov, D. Rutsky· 0 citations
Let $(\mathcal{X},d,\mu)$ be a doubling metric measure space, $X$ a ball quasi-Banach function space on $\mathcal{X}$, and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernel satisfies a Gaussian upper bound. In this article, we study the local Hardy space $h_{X,L}(\mathcal{X})$ associated...
We study the boundedness of the Fourier transform on variable Lebesgue spaces. We obtain necessary conditions and, independently, sufficient conditions on the exponents $p(\cdot)$ and $q(\cdot)$ under which the inequality $$\|\hat{f}\|_{q(\cdot)}\leq C\|f\|_{p(\cdot)}$$ holds for some constant $C>0$ and all $f\in L^{p(...
Rodrigo Cardeccia, M. I. Caruso, Martin Mazzitelli et al.· 0 citations
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 2 citations
We identify properties of a Banach space $\mathcal{B}$ of analytic functions on the open unit disk $\mathbb{D}$ in the complex plane ensuring that a multiplication operator $M_\psi: \mathcal{B} \to\mathcal{B}$ is Fredholm if and only if its symbol $\psi$ is bounded away from $0$ near $\partial \mathbb{D}$. The properti...
P. Bourdon, M. Fatehi· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
