We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^p$ and $H^q$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces $H_u^p$. We also provide a reduction of $H^{\infty }$ problems to $H_u^p$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.

We study continuity envelopes of function spaces $B_{p,q}^s(ℝⁿ,w)$ and $F_{p,q}^s(ℝⁿ,w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

We consider Littlewood-Paley functions associated with a non-isotropic dilation group on ${ℝ}^n$. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^p$ spaces, $1<p<\infty$, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

Let $L$ be a non-negative self-adjoint operator acting on $L^2\left({ℝ}^n\right)$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_r$ weight on ${ℝ}^n\times {ℝ}^n$, $1<r<\infty$. In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L,w}^p({ℝ}^n\times {ℝ}^n)$, $0<p\le 1$ associated to $L$. Based on the atomic decomposition, we show the dual relationship between $H_{L,w}^1({ℝ}^n\times {ℝ}^n)$ and ${\mathrm{BMO}}_{L,w}({ℝ}^n\times {ℝ}^n)$.

We prove several optimal Moser–Trudinger and logarithmic Hardy–Littlewood–Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere $S^2$, on a bounded domain $\Omega \subset {ℝ}^2$ and on all of ${ℝ}^2$. In some cases we also address the question of existence of minimizers.

We study the boundedness of the one-sided operator $g{⁺}_{\lambda ,\phi }$ between the weighted spaces $L^p\left(M¯w\right)$ and $L^p\left(w\right)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g{⁺}_{\lambda ,\phi }$. For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g{⁺}_{\lambda ,\phi }$ from $L^p\left({\left(M¯\right)}^{[p/2]+1}w\right)$ to $L^p\left(w\right)$, where ${\left(M¯\right)}^k$ denotes the operator M¯ iterated k times.

In this paper, characterizations of the embeddings between weighted Copson function spaces ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $$\begin{array}{cc}\hfill d({\int }_0^{\infty }& {\left({\int }_0^{t}f{\left(\tau \right)}^{p_2}{v}_2\left(\tau \right)\mathrm{d}\tau \right)}^{q_2/p_2}{u}_2\left(t\right)\mathrm{d}t{)}^{1/q_2}\hfill \\ & \le c{\left({\int }_0^{\infty }{\left({\int }_t^{\infty }f{\left(\tau \right)}^{p_1}{v}_1\left(\tau \right)\mathrm{d}\tau \right)}^{q_1/p_1}{u}_1\left(t\right)\mathrm{d}t\right)}^{1/q_1},\hfill \end{array}d$$ where $p_1,p_2,q_1,q_2\in (0,\infty )$, $p_2\le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques...

The main aim of this paper is to prove that the maximal operator ${\sigma }^{}$ is bounded from the Hardy space $H_{1/2}$ to weak-$L_{1/2}$ and is not bounded from $H_{1/2}$ to $L_{1/2}$.

Let $A_{\alpha }f\left(x\right)={\left|B\right(0,\left|x\right|\left)\right|}^{-\alpha /n}{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{\alpha }$ is bounded from $L^p$ to $L^{p_{\alpha }}$ with $p_{\alpha }=np/(\alpha p-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{\alpha ,Y}$, which is strictly larger than X, and a ’target’ space $T_Y$, which is strictly smaller than Y, under the assumption that $A_{\alpha }$ is bounded from X into Y and the Hardy-Littlewood...

The aim of this paper is to introduce some new fixed point results of Hardy-Rogers-type for $\alpha$-$\eta$-$GF$-contraction in a complete metric space. We extend the concept of $F$-contraction into an $\alpha$-$\eta$-$GF$-contraction of Hardy-Rogers-type. An example has been constructed to demonstrate the novelty of our results.

Let $u$ be a weight on $(0,\infty )$. Assume that $u$ is continuous on $(0,\infty )$. Let the operator $S_u$ be given at measurable non-negative function $\varphi$ on $(0,\infty )$ by $$S_u\varphi \left(t\right)=\underset{0<\tau \le t}{sup}u\left(\tau \right)\varphi \left(\tau \right).$$ We characterize weights $v,w$ on $(0,\infty )$ for which there exists a positive constant $C$ such that the inequality $${\left({\int }_0^{\infty }{\left[S_u\varphi \left(t\right)\right]}^{q}w\left(t\right)dt\right)}^{\frac{1}{q}}\lesssim {\left({\int }_0^{\infty }{\left[\varphi \left(t\right)\right]}^{p}v\left(t\right)dt\right)}^{\frac{1}{p}}$$ holds for every $0<p,q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k,p}\left(G\right)$; $k\in {\mathbb{N}}^{*}$, $1\le p\le \infty$ and $G$ is the open unit disk or the annulus of the complex space $ℂ$.

We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality for $1\le p<\infty$. More precisely, given any measurable set $E_0$, the estimate $$w\left(\{x\in {ℝ}^n:M^{+,d}\left({𝒳}_{E_0}\right)\left(x\right)>t\}\right)\le \frac{C{\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}^p}{t^p}v\left(E_0\right)$$ holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}\left(ℛ\right)$, that is, $$\frac{\left|E\right|}{\left|Q\right|}\le {\left[(w,v)\right]}_{A_p^{+,d}\left(ℛ\right)}{\left(\frac{v\left(E\right)}{w\left(Q\right)}\right)}^{1/p}$$ for every dyadic cube $Q$ and every measurable set $E\subset Q^{+}$. The proof follows some ideas appearing in S. Ombrosi (2005). We also obtain a similar quantitative characterization for the...

Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z,p}\left(f\right)={(1-|zₙ\left|²\right)}^{1/p}f\left(zₙ\right)$. Necessary and sufficient conditions on zₙ are given such that $T_{z,p}$ maps the Hardy space $H^p$ boundedly into the sequence space ${\ell }^q$. A corresponding result for Bergman spaces is also stated.

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $Aₙf\left(x\right)=2^{-n}{\int }_x^{x+2ⁿ}f$. The square function is defined as $Sf\left(x\right)=({\sum }_{n=-\infty }^{\infty }|Aₙf\left(x\right)-A_{n-1}f\left(x\right){\left|²\right)}^{1/2}$. The local version of this operator, namely the operator $S₁f\left(x\right)=({\sum }_{n=-\infty }^0|Aₙf\left(x\right)-A_{n-1}f\left(x\right){\left|²\right)}^{1/2}$, is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^p$ into itself (p > 1) and $L^{\infty }$ into BMO. We prove that the operator S not only maps $L^{\infty }$ into BMO but it also maps BMO into BMO. We also prove that the $L^p$ boundedness...

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${\left|\right|Mf\left|\right|}_{p),w}{\le c\left|\right|f\left|\right|}_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_p$, and therefore iff ${\left|\right|Mf\left|\right|}_{p,w}{\le c\left|\right|f\left|\right|}_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small...

Let $X={\sum }_{i=1}^k{a}_i{U}_i$, $Y={\sum }_{i=1}^k{b}_i{U}_i$, where the $U_i$ are independent random vectors, each uniformly distributed on the unit sphere in ℝⁿ, and $a_i,b_i$ are real constants. We prove that if $b{²}_i$ is majorized by $a{²}_i$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^p$ Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts...

We consider the operator $-d^2/d{r}^2-V$ in $L_2\left({ℝ}_{+}\right)$ with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound $\mathrm{tr}{(-d^2/d{r}^2-V)}_{-}^{\gamma }\le C_{\gamma ,\alpha }{\int }_{{ℝ}_{+}}{(V\left(r\right)-1/\left(4r^2\right))}_{+}^{\gamma +(1+\alpha )/2}{r}^{\alpha }dr$ for any $\alpha \in [0,1)$ and $\gamma \ge (1-\alpha )/2$. This includes a Lieb-Thirring inequality in the critical endpoint case.