Related papers: A note on the limit of Orlicz norms
The classical Banach space $L_1(L_p)$ consists of measurable scalar functions $f$ on the unit square for which $$\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.$$ We show that $L_1(L_p)$ $(1 < p < \infty)$ is primary,…
In this paper, we present a characterization of support functionals and smooth points in $L_{0}^{\Phi}$, the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of $L_{0}^{\Phi}$ is also obtained.…
We consider weighted norm inequalities for the Riesz potentials $I_\alpha$, also referred to as fractional integral operators. First we prove mixed $A_p$-$A_\infty$ type estimates in the spirit of [13, 15, 17]. Then we prove strong and weak…
The boundedness (continuity) of composition operators from some function space to another one is significant, though there are few results about this problem. Thus, in this study, we provide necessary and sufficient conditions on the…
We give a complete description of the horofunction boundary of finite-dimensional $\ell_p$ spaces for $1\leq p\leq \infty$. We also study the variation norm on $\mathbb{R}^{\mathcal{N}}$, $\mathcal{N}=\{1,...,N\}$, and the corresponding…
Let $L^p(\mathbf{T})$ be the Lesbegue space of complex-valued functions defined in the unit circle $\mathbf{T}=\{z: |z|=1\}\subseteq \mathbb{C}$. In this paper, we address the problem of finding the best constant in the inequality of the…
In this paper we establish uniform oscillation estimates on $L^p(X)$ with $p\in(1,\infty)$ for the polynomial ergodic averages. This result contributes to a certain problem about uniform oscillation bounds for ergodic averages formulated by…
For $1< p <2$ we obtain sharp inequalities for the supremum of products of homogeneous polynomials on $L_p(\mu)$, whenever the number of factors is no greater than the dimension of these Banach spaces (a condition readily satisfied in the…
We study minimizers of non-autonomous functionals \begin{align*} \inf_u \int_\Omega \varphi(x,|\nabla u|) \, dx \end{align*} when $\varphi$ has generalized Orlicz growth. We consider the case where the upper growth rate of $\varphi$ is…
We address the question and related controversy of the formulation of the $q$-entropy, and its relative entropy counterpart, for models described by continuous (non-discrete) sets of variables. We notice that an $L_p$ normalized functional…
In the present note, we are interested in bounded 2-functionals and 2-dual spaces of $L^p[0,1]$. The 2-dual spaces of the sequence space $l^p$ is considered in the literature. But interestingly an explicit computation of $Lp$ spaces has not…
Let \Omega\subset\mathbb{R}^n be a bounded domain with C^\infty boundary. We show that a harmonic function in \Omega that is Lipschitz along a family of curves transversal to b\Omega is Lipschitz in \Omega. The space of Lipschitz functions…
We obtain lower bounds for the $l_1$-norm of the Fourier transform of functions on $\mathbb{Z}_p^d$.
We prove sharp $L^p-L^q$ estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve…
A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to…
We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\frac{\pi}{\sin\frac{\pi}{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big(…
Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. In this paper, we investigate their Boyd indices. Bounds on the Boyd indices in terms of the Matuszewska-Orlicz indices of the defining functions are…
Under appropriate integrability conditions the risk measure of the sample measures for a law invariant risk measure converge almost surely to the risk measure of the sampled random variable. The results follow from general convergence…
In this paper we study asymptotic behavior of solutions of obstacle problems for $p-$Laplacians as $p\to \infty.$ For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case,…
On any metric space, I provide an intrinsic characterization of those complex-valued functions which are uniform limits of Lipschitz functions. There are applications to function theory on complete Riemannian manifolds and, in particular,…